Parametric invariance

Abstract

We examine the development of the concept of parametric invariance in classical mechanics, quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of pressure caused by vibration, and the general approach we follow here is based on his. Specific calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant, which is a volume in phase space, is extended to the case of a variable number of particles. We show that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as a parametric invariance.

Full text

arXiv:2203.07262v1 [cond-mat.stat-mech] 14 Mar 2022
Parametric invariance
M´ario J. de Oliveira
Universidade de S˜ao Paulo, Instituto de F´ısica, Rua do Mat˜ao, 1371, 05508-090 S˜ao Paulo, SP, Brazil
We examine the development of the concept of parametric invariance in classical mechanics,
quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to
entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization
rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of
pressure caused by vibration, and the general approach we follow here is based on his. Specific
calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant,
which is a volume in phase space, is extended to the case of a variable number of particles. We show
that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as
a parametric invariance.
I. INTRODUCTION
When a mechanical system is under the influence of a
disturbance caused by a time variation of one of its pa-
rameters, we expect its properties to change. However,
it was found that there are some properties that remain
invariant if the parameter changes very slowly. It is cus-
tomary to trace the origin of this type of invariance to
the Solvay Congress held in Brussels in 1911 [1]. During
the discussion that followed the Einstein lecture, Lorentz
remembered a conversation he had with Einstein some-
time earlier. In this conversation, he asked him how the
energy of a simple pendulum varies when its lengths is
shortened by holding the string between two fingers and
sliding down. Einstein replied that, if the length of the
pendulum is changed in an infinitely slow manner, the en-
ergy varies in proportion to the frequency of oscillations.
In other words, the ratio between energy and frequency
remains invariant.
This concept of invariance appeared more consistently
in the writings of Ehrenfest [2–6] on the formulation of
the theory of quanta, later called old quantum theory
[2, 7, 8]. He used these invariants to formulate his proce-
dure for obtaining quantized states. To this end, he in-
troduced in 1913 [9, 10] a hypothesis according to which
the allowed motions of a system are transformed into al-
lowed motions if the system is affected by a reversible
adiabatic change. In a paper published in the following
year, Einstein used the Ehrenfest hypothesis, and called
it the adiabatic hypothesis [11]. The invariant quanti-
ties resulting from a reversible adiabatic change Ehren-
fest called adiabatic invariants in a paper of 1916 [12–14].
In that same paper Ehrenfest explained what he meant
by a reversible adiabatic change: it is an influence on the
system in which the parameters change in an infinitely
slow way.
In spite of the explanation given by Ehrenfest, the in-
fluence of the infinitely slow change of a parameter be-
came associated to the term adiabatic. Jeans considered
the term not particularly a happy one [15]. Accordingly,
we find it more appropriate to name it by its definition
and not by its consequences, and call it a slow paramet-
ric action, which in addition avoids the reference to any
thermodynamic meaning. The invariant that results from
this action we call parametric invariant. We reserve the
term adiabatic invariant for a thermodynamic quantity
that is constant along a slow adiabatic process, an exam-
ple of which is the well known Poisson relation between
pressure and volume for an ideal gas.
Here we review the concept of parametric invariance
through the critical analysis of its evolution and how it is
treated in classical mechanics and in quantum mechanics
as well as its relation with thermodynamics particularly
with entropy. The parametric invariance is included as a
subject of classical mechanics [16–22], usually connected
with the technique of action-angle variables. It is treated
in quantum mechanics [23–26], kinetic theory and sta-
tistical mechanics [27–29], dynamics of charged particles
[30–33], and has been applied to specific problems by
several authors [34–39], particularly in modern computer
calculations to determine entropy and free energy by the
method of adiabatic switching [40, 41]
II. EHRENFEST PRINCIPLE
Ehrenfest enunciated his hypothesis in 1916 in the fol-
lowing terms [12–14]: If a system be affected in a re-
versible adiabatic way, allowed motions are transformed
into allowed motions. In the paper of 1914, Einstein
stated it in the following terms [11]: With reversible
adiabatic changes of a parameter, every quantum-the-
oretically possible state changes over into another possi-
ble state. In both statement, reversible adiabatic change
is to be understood as a slow variation of a parameter.
In his paper of 1916 [12, 13], Ehrenfest considered a pe-
riodic system and showed that the time integral of twice
the kinetic energy Kover a period,
I=Z2Kdt, (1)
is a parametric invariant. Defining the time average ¯
K
of the kinetic energy by
¯
K=νZKdt, (2)

2
where νis the frequency, the inverse of the period, the
invariant is equivalent to 2 ¯
K/ν . For a harmonic oscillator
the energy Eis twice the kinetic energy and the invariant
becomes E/ν.
Ehrenfest had already presented the invariant (1) in
his publication of 1913 but now he provided a demon-
stration through the use of the Lagrange analytical the-
ory. Considering that the kinetic energy Kof a system
with many degrees of freedom is a quadratic form in the
variables ˙qk, the time derivative of the coordinates qk,
the Euler theorem on homogeneous functions allows us
to write
K=1
2X
k
pk˙qk,(3)
where pk=∂K/∂ ˙qkis the momentum conjugate to qk.
Using this expression Ehrenfest writes the integral (1) in
the form
I=X
kZpkdqk.(4)
The geometrical interpretation of this expression was
given by Ehrenfest as follows. In the phase space, the rep-
resentative point of the system describes a closed curve
which projects closed curves on each one of the planes
(qk, pk). Each term
Ik=Zpkdqk(5)
of the sum in (4) represents the area of each one of the
projected closed curves.
In 1915, Wilson [42] and by Sommerfeld [43], inde-
pendently postulated the quantization rule by the use of
phase integrals,
Zpidqi=nih, (6)
where niis an integer number and his the Planck con-
stant. These phase integrals were shown by Schwarzshild
[44] and by Epstein [45, 46] to emerge when it is possible
to separate variables by using the Hamilton-Jacobi the-
ory to systems in which the variables can be separable.
The Wilson-Sommerfeld rule is then applied to each pair
of these separable canonically conjugate variables, called
action and angle variables by Schwarzschild [44].
At the end of his paper of 1916, Eherenfest asked him-
self whether the phase integral (6) appearing in the pa-
pers of Schwarzschild and Epstein could also be an in-
variant. The demonstration that indeed each one of these
phase integrals is an invariant was shown by Burgers in
1916 [47, 48]. According to Burgers, if the momentum pk
in the integral (5) depends only on qkthen Ikis an in-
variant. Notice that Ehrenfest had showed that the sum
of integrals of the type (5) is an invariant, nothing being
said about each one of them.
In 1913 Bohr proposed his atomic model based on the
assumptions that the electron describes stationary or-
bits around the nucleus [49]. Bohr assumed that the fre-
quency νof the radiation emitted is half the frequency of
revolution of the electron and that the amount of energy
emitted is hν times an integer n. From these assump-
tions he obtained the binding energy Eof the electron
as
E=2π2me4
h2n2,(7)
where eis the charge of the electron and mits mass.
Another fundamental assumption made by Bohr was as
follows. When the electron passes from one stationary or-
bit to another, the loss of energy in the form of radiation
is equal to hν.
As a way of justifying the stationarity of the or-
bits, Bohr employed the Ehrenfest hypothesis, which he
named the principle of mechanical transformability, and
appeared in 1918 in his paper on the quantum theory of
line-spectra [50]. Bohr explains that this name indicate
in a more direct way the content of the principle. This
reference on the Bohr paper of 1918 turned the Ehrenfest
hypothesis widely known but at the same time it became
closely linked to Bohr’s work [6]. The same can be said
of the Burger’s paper on the invariance of the phase in-
tegrals [6].
Although the Ehrenfest principle explained the per-
manence of a system in a given state, it did not explain
why the states are discretized. Neither did the Wilson-
Sommerfeld rule as it was introduced as a postulate. The
explanation came with the emergence of quantum me-
chanics around 1925 which replaced classical mechanics
in the explanation of the motion at the microscopic level.
Within quantum mechanics, the Wilson-Sommerfeld rule
was found to be valid at higher quantum numbers. As
to the Ehrenfest principle, the works of Born [51], Fermi
and Persico [52], and Born and Fock [53] turned it into a
theorem of quantum mechanics [5, 6].
III. WAVE MECHANICS
The quantization of the electronic orbits of the hydro-
gen atom used by Bohr and the quantization rule used by
Sommerfeld explained accurately the spectrum of the hy-
drogen including the fine structure of the hydrogen lines.
In spite of its successful explanation of the spectrum of
atoms and several problems in atomic physics, the quan-
tum physics up to 1925 was a collection of quantum rules
without a unifying principle [2].
In 1925, two quantum theories were proposed which
were latter shown to be equivalent. Heisenberg proposed
a matrix theory [54] and Schr¨odinger [55, 56] proposed a
wave theory. The point of departure of the Schr¨odinger
theory was the relation between the wave theory of light
and geometric optics [2]. Hamilton had shown that there
is an analogy between the principle of least action of

3
mechanics and the Fermat principle of geometric optics.
The principle of least action is
δZ2Kdt = 0,(8)
where Kis the kinetic energy and the minimization of the
action is subject to trajectories where the energy T+V
is conserved, and can be written in the form
δZp2m(E−V)ds = 0.(9)
The Fermat principle of geometric optics is
δZds
v= 0,(10)
where vis the velocity of light. Thus the Fermat princi-
ple can be regarded as the principle of least action where
v−1plays the role of the integrand of (9) [21]. As there is
a wave theory of light, which reduces to geometric optics
for small wavelength, the Schr¨odinger theory is under-
stood as a wave theory that reduces to the mechanics.
The wave representation of a quantum theory by
Schr¨odinger was suggested by de Broglie who associated
a wave to the motion of a particle which he called wave
phase [57]. According to de Broglie, the wavelength λof
the wave associated to a particle of momentum pis given
p=h/λ, where his the Planck constant. The use of
wave naturally leads to quantization. For instance, the
possible states of a standing wave are the normal modes
of vibration, and the possible values of the wavelengths
of a standing wave forms a discretized set of values. More
generally, the quantization comes from the fact that the
solution of the wave equation naturally result in the so-
lution of an eigenvalue problem, as stated by Schr¨odinger
in the title of his paper on wave mechanics.
In the first part of his paper on wave mechanics,
Schr¨odinger introduced the time independent equation
for an electron under the action of the inverse square
force,
∇2ψ+2m
K2(E+e2
r)ψ= 0,(11)
where K, according to Schr¨odinger, must have the value
K=h/2πso that the discrete spectrum corresponds to
the Balmer series. In the second paper, he considered
the one-dimensional oscillator whose equation he wrote
in the abbreviated form
d2ψ
dx2+ ( a
√b−x2)ψ= 0,(12)
and determined the proper values of a/√b, which are
1,3,5,... by the use of the known solution of equation
(12) in terms of Hermite orthogonal functions. From
this result the allowed energies of the oscillator are E=
hν(n+ 1/2). In the forth part, Schr¨odinger postulates
that the wave equation is a first order in time and writes
∇2ψ−8π2
h2V ψ ∓4πi
h
∂ψ
∂t = 0.(13)
In the year following the publication of the wave theory
by Schr¨odinger there appears an approximation method
proposed independently by Wentzel [58], by Brillouin [59]
and by Kramers [60]. This approximation corresponds
to a perturbation expansion in powers of the Planck con-
stant. The zero order approximation gives the classical
result. The first order results in the Wilson-Sommerfeld
rule of the old quantum mechanics. The approximation
is obtained by writing the wave function in the form [23]
ψ=AeiS/~,(14)
where Sdoes not depend on ~and Ais independent of
time and is an expansion in powers of ~. Replacing it in
the time independent Schr¨odinger equation,
−~2
2m
∂2ψ
∂x2+V ψ =Eψ, (15)
the equation containing only terms of order zero in ~is
1
2m(∂S
∂x )2+V=E, (16)
and the equation coming from terms of first order in ~is
∂A
∂x
∂S
∂x +A
2
∂2S
∂x2= 0.(17)
The equation (17) can be integrated with the result
A2proportional to the reciprocal of (∂S/∂x). It is now
left to solve the equation (16). We consider two cases
according to the sign of E−V. If E≥V, we define
κ=p2m(E−V),(18)
and the solutions of the equations (16) and (17) are
S=±Zκ(x)dx, A =1
pκ(x).(19)
If E < V , we define
γ= 2m(V−E),(20)
and the solutions are
S=±iZγ(x)dx, A =1
pγ(x).(21)
Let us suppose that the first condition occurs when
a≤x≤b, where V(a) = Eand V(b) = E, are the
classical turning points. The connection of the solutions
at the turning points leads to the condition [23]
Zb
a
κdx =~π(n+1
2).(22)

4
B
CD
θ
(a) A(b)
Γ
Γ
Γ
2
1
FIG. 1: (a) The pendulum with variable length. In the set
up employed by Rayleigh the point A is fixed and the ring B
move vertically causing the variation of the length BC of the
pendulum. In the set up used by Bossut and Lecornu, the ring
B remains immobile and the point A moves vertically. (b) The
forces acting on the ring when it is free to move upward, where
Γ is the tension of the string, Γ1= Γ cos θ, and Γ2= Γ sin θ.
The net force is upward and equals T(1 −cos θ).
Considering that the classical momentum pis κor −κ,
we may write this condition as
Ipdx =h(n+1
2),(23)
which is the Wilson-Sommerfeld rule except for the 1/2
term.
As the phase integral in the left hand side of equation
(23) is invariant, it follows that the quantum number n
is an invariant. That is, if a parameter of the system
is slowly varying in time, it remains in the same state
with the same quantum number. Nevertheless, a demon-
stration of invariance of the quantum state was provided
for the new quantum mechanics, without referring to the
phase integral. In 1926, one year after the introduction
of the quantum wave by Schr¨odinger, a demonstration of
the invariance in quantum mechanics was given by Born
[51] and by Fermi and Persico [52]. Two years later, a
more general demonstration was provided by Born and
Fock [53]
IV. RAYLEIGH APPROACH
A. Pendulum of variable length
In a paper of 1902 [61], Rayleigh analyzed a simple
pendulum with its string being varied very slowly. Mo-
tivated by the theoretical demonstration of the radiation
pressure by Maxwell and its experimental confirmation
by Lebedev, Rayleigh inquired whether any other kinds
of vibration, such as sound vibrations, would also cause
pressure. To answer this question, he posed the problem
of finding the force acted by a vibrating pendulum on its
pivot when its length changes slowly and continuously
with time.
The length of the pendulum is changed by the use of a
ring through which passes the string, as shown in figure
1. The ring is constrained to move vertically and as it
moves the length BC of the pendulum changes although
the total length ABC remains constant. The problem
is to determine the force that tends to move the ring
upwards as the pendulum swings.
The ring is acted by two vertical forces, one of them
is upward and equal to the tension Γ of the string and
the other is downward and equal to Γ cos θwhere θis the
angle BCD. The net upward force on the ring is thus F=
Γ(1 −cos θ). Now the potential energy of the pendulum
is V=P ℓ(1 −cos θ) where Pis the weight of the bob,
and ℓis the length BC of the pendulum. Considering
that for small oscillations Γ is approximately equal to P,
one finds F=V/ℓ. As the mean value of the potential
energy is one half of the total energy Eof the pendulum,
Rayleigh concludes that the upward mean force Fon the
ring is
F=E
2ℓ.(24)
As the work done on the ring is equal to decrease in the
energy of the pendulum, then dE =−F dℓ, and dE =
−Edℓ/2ℓwhich by integration gives E=a/√ℓ, where
ais a constant. Although, Rayleigh did not mention it
explicitly, it follows from this result that the quantity
I=E√ℓ(25)
is an invariant quantity when the length of the pendulum
changes slowly with time. As the frequency of oscillation
ωof a simple pendulum executing small oscillations is
inversely proportional to √ℓit follows that the quantity
I=E
ω(26)
is an invariant as well.
Rayleigh also treated in the same paper the problem
of the force exerted by a vibrating stretched string on
the points where it is attached. One end of the stretched
string is fixed and the other is allowed to move by the
use of a ring as shown in figure 2. The position of the
ring determines the length of the vibrating string, which
we denote by ℓ. Rayleigh argues that the mean force F
acting on the ring is related to the total energy Eof the
vibrating string by
F=E
ℓ,(27)
and is thus equal to the energy per unit length. Again,
the work done on the ring equals the decrease in energy,
dE =−F dℓ =−Edℓ/ℓ and, after integration, E=a/ℓ
where ais a constant, and now Eis inversely proportional
to ℓ. Thus,
I=Eℓ (28)
is invariant.

5
BA C
Q
P
FIG. 2: The stretched string APBC is fixed at the points A
and C. The ring B restricts the vibration to occurs only be-
tween A and B. The length of the vibrating string changes by
the motion of the ring along the direction ABC.The string ten-
sion remains unaltered when the position of the ring changes.
B. Invariance
The main result of the approach by Rayleigh can be
stated as follows. Let us consider a periodic system and
the force facted by the system on the environment at a
point which moves as a result of the change of a param-
eter λ. As the system is periodic the force foscillates
in time but its time average Fover one cycle is nonzero.
If the parameter changes slowly Fvaries slowly and so
does the energy Eof the system. These two quantities
are related by
dE
dλ =−F, (29)
if the parameter changes very slowly with time. To show
this result we proceed as follows.
We consider a system with several degrees of freedom
described by the Lagrangian L=K−V, where Kis
the kinetic energy and Vis the potential energy. The
Lagrangian L(q, ˙q, λ) depends on the coordinates qiand
velocities ˙qi, which we are denoting collectively by qand
˙q, respectively, and on a parameter λwhich depends on
time. The Lagrange equations of motion are
d
dt
∂L
∂˙qi
=∂L
∂qi
.(30)
If we define the momentum
pi=∂L
∂˙q,(31)
then the equations of motion are written as
dpi
dt =∂L
∂qi
,(32)
and the differential of Las
dL =X
i
pid˙qi+X
i
˙pidq +∂L
∂t dt. (33)
If we perform the Legendre transformation
H=X
i
pi˙qi−L, (34)
then
dH =X
i
˙qidpi−X
i
˙pidq −∂L
∂t dt, (35)
from which follows the Hamilton equations of motion
˙qi=∂H
∂pi
,˙pi=−∂H
∂qi
,(36)
where H(q, p, λ) is the Hamiltonian, a function of the
coordinates and momenta, and depends on time through
the parameter λ. It also follows that
∂H
∂t =−∂L
∂t ,(37)
where the first derivative is determined at constant qand
pwhereas the second at constant qand ˙q.
The Hamiltonian Hequals the total energy K+Vif
Kis a quadratic form in the velocities. Its variation with
time is
dH
dt =∂H
∂t ,(38)
and as it depends on time through the parameter λ, it is
not conserved. Defining the function fby
f=−∂H
∂λ ,(39)
the equation (38) can be written as
dH
dt =−cf, (40)
where c=dλ/dt is the rate of variation of the parameter.
Now we proceed as follows. We divide the time axis
into intervals equal to the cycle period T. The integration
of the equation (40) over one cycle beginning at time t0
and ending at time t1gives
∆H
T=−1
TZt1
t0
cf(q, p, λ)dt, (41)
where
∆H=H(q1, p1, λ1)−H(q0, q0, λ0),(42)
and the subindexes refer to the beginning and ending of
the interval.
We use an approximation in which the trajectory in
phase space is replaced by a trajectory (x(t), y(t)) which
is the solution of the equations of motion by considering
that the parameter λis kept unchanged and equal to its
value at the beginning at the interval. The variation of
Hbecomes
∆H=H(x1, y1, λ1)−H(x0, y0, λ0).(43)
The variation ∆λ=λ1−λ0of the parameter in the
interval is equal to ∆λ=T c and cis small because the

6
parameter varies slowly with time. Therefore ∆H/T =
c∆H/∆λcan be approximated by
cd
dλ H(x, y, λ),(44)
calculated at the beginning at the interval.
Using again the same approximation, we replace qby
x(t), and pby y(t) in the right hand side of (41). Defining
F=1
TZt1
t0
f(x, y, λ)dt, (45)
the equation (41) becomes
dE
dλ =−F, (46)
where cwas assumed cto be constant during the cycle.
In this equation, Eis equal to H(x, y , λ) and depends
only on λbecause it is conserved along one cycle, and
the quantity Fis understood as the time average of f
along one cycle and also depends only on λ. Since Fis a
function of λ, this relation is understood as a differential
equation in λ. Its solution gives the explicit dependence
of Eon λ, and on time since λis a given function of time.
Let us use the notation
f=1
TZt0+T
t0
fdt, (47)
for the time average of fover one cycle of period T.
The function fdepends on the parameter λ, which is
considered to be fixed. The main result can then be
written in the abbreviated form
dE
dλ =∂H
∂λ ,(48)
and E=H. Notice that, as His conserved if λis fixed,
then Ecoincides with E=Hduring one period and this
time average is immaterial.
C. Examples
Let us apply this approach to the Rayleigh pendulum
The Lagrangian function is given by
L=1
2mℓ2˙
θ2−1
2mgℓθ2.(49)
To determined f=−∂H/λ, we observe that using the
equality (37), it can also be determined by f=∂L/∂λ.
Deriving Lwith respect to ℓ, we find
f=mℓ ˙
θ2−1
2mgθ2.(50)
The energy Eis given by
E=1
2mℓ2˙
θ2+1
2mgℓθ2.(51)
The equation of motion, keeping the parameter ℓun-
changed is
ℓ¨
θ=−gθ, (52)
whose solution is θ=ccos ωt, where ω=pg/ℓ. Re-
placing the solution in the expression for Hand f, and
taking the time average over one cycle, we obtain
E=1
2mgc2ℓ, F =1
4mgc2,(53)
which gives F=E/2ℓ, the result (24). The integration
of dE/dℓ =−Fgives the result E√ℓan invariant, or
E/ω an invariant, results already found.
For a particle of mass mbounded to a spring of coef-
ficient k, the Lagrangian function is
L=1
2m˙x2−1
2kx2,(54)
and the energy is
E=1
2m˙x2+1
2kx2.(55)
The equation of motion is m¨x=−kx whose solution is
x=acos ωt, where ω=pk/m. Replacing these results
in the expression for the energy, E=ka2/2.
Let us suppose that the spring coefficient is the varying
parameter. Then f=−x2/2 and F=−a2/4. Therefore,
F=−E/2kand the integration of dE/dk =−Fgives
E/√kas an invariant, or E/ω.
Suppose now that the mas is the varying parameter.
Then, f= ˙x2/2 and F=ka2/4m. Therefore F=E/2m
and the integration of dE/dm =−Fgives E√mas an
invariant, or E/ω.
Another example consists of a free particle of mass m
that moves with speed vbetween two walls that are a
distance ℓapart. The mean force Fon the wall is the
change of its momentum 2mv divided by the time 2ℓ/v
between two collisions, that is, F=mv2/ℓ Considering
that the kinetic energy is E=mv2/2 one finds F= 2E/ℓ
and the integration of dE /dm =−Fgives Eℓ2as an
invariant.
Let us consider now the vibrating stretched string. De-
noting by uthe transverse displacement PQ of the string
at the point Qand by xthe distance from this point to
the fixed end A, as shown in figure 2, the equation of
motion for uis the wave equation
∂2u
∂t2=c2∂2u
∂x2,(56)
where cis the velocity of the wave and equal to pΓ/µ,
where Γ is the tension of the string and µis the mass
per unit length. The wave equation is understood as the
equation of motion associated to the Lagrangian L=
T−Vwhere Kis the kinetic energy,
K=µ
2Zℓ
0∂u
∂t 2
dx, (57)

7
and Vis the potential energy,
V=Γ
2Zℓ
0∂u
∂x 2
dx. (58)
The energy is H=T+V.
With the purpose of revealing the explicit dependence
on ℓ, we change the variable of integration from xto
y=x/ℓ, with the result
K=µℓ
2Z1
0∂u
∂t 2
dy, (59)
and Vis the potential energy,
V=Γ
2ℓZ1
0∂u
∂y 2
dy. (60)
Using these expression on L=K−V, we determine
f=∂L/∂ ℓ, which gives the following result f=H/ℓ,
where H=K+V. Therefore F=E/ℓ, which is the
result (27), and, using the relation dE/dℓ =−F, we find
that E=A/ℓ and Eℓ is an invariant as already found.
The solution of the wave equation for vibrations be-
tween the fixed ends at x= 0 and x=ℓis the standing
wave
u=Asin kx cos ωt, (61)
where k=nπ/ℓ and nis an integer, and ω=kc. Re-
placing these results into the expression for Kand Vand
taking the time average over one cycle, we find
E=µA2c2n2π2
4ℓF=µA2c2n2π2
4ℓ2,(62)
showing that indeed E=a/ℓ and F=E/ℓ.
V. EHRENFEST INVARIANT
A. Invariance
To demonstrate the invariance of expression (1) or its
equivalent form (4) we proceed as follows. We start by
writing F, given by (45), in the form
F=−1
TZT
0
∂H
∂λ dt, (63)
where we have used the expression (39) for f, and we
are considering the variables λand tindependent of each
other. Recalling that E=H, we obtain
dE
dλ =1
TX
iZT
0 ∂H
∂pi
∂pi
∂λ +X
i
∂H
∂qi
∂qi
∂λ +∂H
∂λ !dt.
(64)
Replacing it in the equation (46), written in the form
dE
dλ +F= 0,(65)
we find
X
iZT
0∂H
∂pi
∂pi
∂λ +∂H
∂qi
∂qi
∂λ dt = 0.(66)
Using the equations of motion (36), we find
X
iI∂pi
∂λ dqi−∂qi
∂λ dpi= 0,(67)
which can be written in the form
dI
dλ = 0,(68)
where
I=1
2X
iI(pidqi−qidpi).(69)
An integration by parts gives
I=X
iIpidqi.(70)
Since dI/dλ = 0, it follows that Iis indeed an invariant.
We recall that this expression can also be written as
I=ZT
0
2Kdt, (71)
where Kis the kinetic energy and Tis the period of the
cycle.
The demonstration just carried out shows that the in-
variant (70) is a sum terms of the type
Ii=Ipidqi.(72)
It does not say whether or not each term Iiis an invari-
ant. However, if the momentum piin this integral (72)
depends only on qi, which is a statement that the pair of
variables (qi, pi) is separable from the others, then Iiis
an invariant, which is the result obtained by Burgers. To
show this result, it suffices to write (72) as
Ii=Zpi˙qidt. (73)
In this form we see that the integrand is twice the ki-
netic energy of a system with one degree of freedom, as
no other variables are involved. But this is the total ki-
netic energy of a system with one degree of freedom and
is thus the Ehrenfest invariant. It is worth mentioning
however that the separation of variables may only occur if
an appropriate transformation of variables is performed.

8
B. Systems with one and two degrees of freedom
The Ehrenfest invariant for a system with one degree
of freedom reduces to the phase integral
I=Ipdq, (74)
where pis the momentum conjugate to q. The general
form of a Lagrangian describing a conservative system
with one degree of freedom is
L=1
2m˙q2−V, (75)
where mmight depend on q, and Vis a function of q
only, and p=∂L/∂ ˙q. As the energy is conserved we
write E=H(q, p) which describes a closed curve on the
phase space. Solving this equation for pand replacing
the result in the phase integral, we find
I= 2 Zp2m(E−V)dq, (76)
where the integral is performed in the interval between
the two points of return.
For a particle of mass munder the action of a harmonic
force, the energy is given by
p2
2m+1
2mω2x2=E, (77)
where ωis the frequency of oscillation. This equation de-
scribes in the phase space an ellipse of semi-axis equal to
√2mE and p2E/mω2. The phase integral is the area of
this ellipse and equals 2πE/ω, which is thus an invariant.
Another example is given by a free particle that moves
along an axis and collides with walls that are a distance
ℓapart. The integral (74) becomes equal to 2pℓ which is
thus an invariant. Taking into account that the kinetic
energy of the particle is E=p2/2m, it follows that Eℓ2
is an invariant as the wall moves slowly.
Let us consider now a particle of mass mmoving in a
plane under a central force. Using polar coordinates, the
Lagrangian is given by
L=1
2m( ˙r2+r2˙
θ2)−V, (78)
where Vis the potential that depends on rbut not on θ.
The momenta conjugate to the rand θare, respectively,
pr=m˙rand pθ=mr2dotθ.
One equation of motion is
dpr
dt =mr ˙
θ2+f, (79)
where f=−dV/dr is the centripetal force. The other
equation of motion is dpθ/dt = 0 from which follows that
the angular momentum pθ=mr2θis constant. Denoting
by athis constant, then ˙
θ=a/mr2which replaced in the
equation of motion for r, gives
dpr
dt =a2
mr3+f. (80)
This equation tell us that the pair of variables (r, pr) is
separable and that
I1=Iprdr (81)
is an invariant.
To determine I1explicitly, we multiply (80) by pr/m =
˙rand integrate in time to find
p2
r
2m+V+a2
2mr2=E, (82)
where Eis a constant. Solving for prand replacing in
the integral I1, we get
I1= 2 Zp2m(E−V)−(a2/r2)dr. (83)
We remark that the integral
I2=Ipθdθ (84)
is also an invariant, in fact it is a constant, I2= 2πa.
C. Particle under a central force
We wish to determine now the invariants of a system
with three degrees of freedom corresponding to a particle
under the action of a central force as is the case of the
Kepler problem where this force is proportional to inverse
of the square of the distance [16, 17, 19, 62]. In spherical
coordinates, the variables are separable and for each pair
of conjugate variables there corresponds an invariant of
the form (74). Using spherical coordinates r,θ, and φ,
with origin at the center of force, the kinetic energy is
given by
K=m
2˙r2+m
2r2˙
θ2+m
2r2˙
φ2sin2θ, (85)
and the Lagrangian is L=T−Vwhere V(r) is the
potential energy that depends on ronly.
The conjugate momenta are
p0=m˙r, (86)
p1=mr2˙
θ, (87)
p2=mr2sin2θ˙
φ, (88)
and the equations of motion are
dp0
dt =mr ˙
θ2+mr sin2θ˙
φ2+f, (89)

9
dp1
dt =mr2˙
φ2sin θcos θ, (90)
dp2
dt = 0,(91)
where f=−dV/dr is the central force that depends on r
only. From the last equation it follows that p2is constant.
Setting this constant equal to a, then ˙
φ=a/mr2sin2θ,
which replaced in the equation of motion for p1gives
dp1
dt =a2cos θ
mr2sin3θ.(92)
Multiplying this equation by p1=mr2˙
θand integrating
in time, we find
p2
1+a2
sin2θ=b2,(93)
where bis a constant. Therefore p1in the integral
I1=Ip1dθ (94)
depends only on θand is an invariant.
Let us write equation (89) in the form
dp0
dt =p2
1
mr3+p2
2
mr3sin2θ+f. (95)
Replacing the result (93) in this equation and bearing in
mind that p2=a, we find
dp0
dt =b2
mr3+f. (96)
Multiplying by p0/m = ˙rand integrating in time, we find
p2
0
2m+V+b2
2mr2=E, (97)
where Eis a constant. Since p0depends only on r, it
follows that
I0=Ip0dr (98)
is an invariant. Solving equation (97) for p0and replacing
in this integral we reach the result
I0= 2 Zp2m(E−V)−(b2/r2)dr. (99)
We remark that the motion of a particle under a central
force is restricted to the plane defined by the velocity
and the center of force. Therefore the problem could
be reduced two a system with two degrees of freedom
like we have done previously. However, there might be
parameters that could remove the motion from this plane.
In this case it is necessary to consider the problem in
three dimensions as we have just done.
D. Particle on a magnetic field
Let us consider the motion of a particle of mass mand
charge ein a uniform magnetic field Bwhich is parallel to
the zaxis. The Lagrangian Lin cylindrical coordinates
is given by [63]
L=1
2m( ˙r2+r2˙
θ2+ ˙z2) + 1
2eBr2˙
θ. (100)
The momenta conjugate to r,θand zare, respectively,
p0=m˙r, (101)
p1=mr2˙
θ+1
2eBr2,(102)
p2=m˙z, (103)
and the equations of motion are
dp0
dt =mr ˙
θ2+eBr ˙
θ, (104)
dp1/dt = 0, and dp2/dt = 0. From these two last equa-
tions, it follows that ˙z=b, a constant, and that
mr2˙
θ+1
2eBr2=a, (105)
where ais another constant, or
˙
θ=a
mr2−eB
2m.(106)
Replacing this last result in the equation of motion for
p0we find
dp0
dt =a2
mr3−e2B2r
4m.(107)
Multiplying this equation by p0/m = ˙rand integrating
in time, we find
p2
0
2m+a2
2mr2+e2B2r2
8m=E, (108)
where Eis a constant. As p0depends only on r, it follows
that the phase integral
I0=Ip0dr (109)
is an invariant.
VI. DYNAMIC APPROACH
The approaches to the mechanical problem of paramet-
ric action treated up to now involve an approximation in
which the parameter is held constant while the system

10
completes a full cycle. This is the case of the Rayleigh
approach just presented as well as that of Ehrenfest. We
may say that the parameter varies in time in steps, the
time of each step being equal to the period of a cycle
in which the parameter is held constant. In other word,
the parameter as a function of time looks like a staircase.
Nevertheless, these approaches give correct results in the
limit of infinitely slow variation of the parameter.
In the following, the problem is treat without consid-
ering the parameter fixed in a cycle but still considering
that the variation of the parameter is slow. In other
word, the parameter varies continuously in time rather
than increasing by steps as was the case of the Rayleigh
and of the Ehrenfest approaches.
We wish to determine the properties of a system in
the regime of slow parametric action which is defined as
follows. Let a parameter λvaries linearly in time, that
is, λ=λ0(1 + εt) where εis small. This regime is defined
for times smaller that 1/ε and a quantity is an invariant
if it varies little in this interval [18, 20].
A. Pendulum of variable length
The problem of the pendulum with variable length was
treated by Lecornu in 1895 [64] and previously by Bossut
in 1778 [65], although they did not draw the relevant con-
clusion of Rayleigh concerning the relation between en-
ergy and the length of the pendulum. Bossut imagined
the oscillations of an unguided bucket during its ascent
in a mine well. Bearing mind the figure 1, the problem is
formulated by considering that the ring is fixed and that
the string is moved upward. Using this set up, Bossut
and Lecornu otained the equation of motion. Accord-
ing to Lecornu, Bossut reduced the differential equation
of the second order into a Riccati equation but he did
not give a solution. Lecornu reduced the the differential
equation of motion to an equation that could be solved
through the use of Bessel functions. The problem was
examined later on in 1923 by Krutkov and Fock [66] who
demonstrated the invariance of E/ω by using the asymp-
totic form of the Bessel function. In the following we
present the treatment of the problem following the treat-
ment of Lecornu and Krutkow and Fock. More recently,
the problem has been treated by S´anchez-Soto and Zoido
[67].
Let xand ybe the projections of CB in figure 1 along
the horizontal and vertical directions, respectively. They
are related to length ℓof the pendulum and the angle θ
by
x=ℓsin θ, (110)
y=ℓcos θ, (111)
and ℓis a given function of time t. We suppose that
the length of the pendulum to vary linearly with time,
ℓ=ℓ0+ht. The kinetic energy of the pendulum is
K=1
2m( ˙x2+ ˙y2),(112)
where mis the mass of the bob. From the expression of
xand ywe find
˙x=ℓ˙
θcos θ+hsin θ, (113)
˙y=−ℓ˙
θsin θ+hcos θ, (114)
which replaced in the expression for Kgives
K=1
2mℓ2˙
θ2+1
2mh2.(115)
Notice that the second term is the kinetic energy due to
the steady vertical motion of the bob with velocity h.
The potential energy is V=mg(ℓ0−ℓcos θ), where g
is the acceleration of gravity. As we will consider only
small oscillations, θis small and we may write
V=1
2mgℓ θ2+mg(ℓ0−ℓ).(116)
Notice that the second term is the potential energy re-
lated to the vertical motion of the bob.
The equation of motion is derived from the Lagrangian
equation
d
dL
∂L
∂˙
θ=∂L
∂θ ,(117)
where L=K−Vis the Lagrangian, given by
L=1
2mℓ2˙
θ2−1
2mgℓ θ2+1
2mh2−mg(ℓ0−ℓ).(118)
From this expression we reach the equation of motion
ℓ¨
θ+ 2h˙
θ=−gθ, (119)
valid for small oscillations. Changing variable from tto
ℓ=ℓ0+ht, this equation becomes
ℓd2θ
dℓ2+ 2 dθ
dℓ +g
h2θ= 0,(120)
which is the equation derived by Lecornu [64].
It is convenient to define the variable s=ℓ/g or s=
at +b, where a=h/g and b=ℓ0/g, from which we may
write the equation of motion as
sd2θ
ds2+ 2 dθ
ds +θ
a2= 0.(121)
Performing the change of variables defined by z= 2√s/a
and φ=zθ we reach the equation
z2d2φ
dz2+zdφ
dz + (z2−1)φ= 0.(122)

11
In this form, we see that the solutions are the Bessel
functions of first order J1(z) and Y1(z) [68], that is,
φ=A1J1(z) + A2Y1(z),(123)
where A1and A2are constant.
θ=1
z[A1J1(z) + A2Y1(z)],(124)
which gives θas a function of tif we recall that z= 2√s/a
and that s=at +b.
As we wish to get the solution for a very slow variation
of the length of the pendulum, which means that ais very
small, it suffices to consider the solution for large values
of z. For as z= 2√s/a and considering a finite value of
s=at +b,zwill increase as 1/a. Therefore, we use the
asymptotic expression of the Bessel functions [68], as did
Trutkov and Fock [66], namely
J1(z) = 2
πz 1/2
sin z−π
4,(125)
Y1(z) = 2
πz 1/2
cos z−π
4.(126)
The solution can thus be written as
θ=cs−3/4cos 2
a√s−π
4.(127)
The energy Eof the pendulum is the first part of the
kinetic energy given by (115) plus the first part of the
potential energy given by (116),
E=m
2ℓ2˙
θ2+ℓgθ2,(128)
which can be written as
E=m
2g2"a2s2dθ
ds 2
+sθ2#.(129)
Replacing the solution (127) in this equation, we reach
the following expression for the energy
E=mg2c2
2√s=mgc2
2rg
ℓ,(130)
where we have neglected terms of order equal or larger
that a. That is, the energy of the pendulum is propor-
tional to the inverse of √ℓ, the Rayleigh relation. Bearing
in mind that the frequency is ω=pg/ℓ, we may wright
E=mgc2
2ω, (131)
and E/ω is an adiabatic invariant.
In the treatment that we have just given to the pen-
dulum with variable length, we have used the set up em-
ployed by Bossut and Lecornu, which corresponds to keep
the ring of figure 1 fixed, while the point A moves verti-
cally. In the original set up of Rayleigh, the point A is
kept fixed while the ring B moves vertically. In this case
the origin of the axis yshould be placed at the point A,
which is fixed, rather than at the point B, which moves.
The relation between yand the angle θbecomes
y=ℓ0−ℓ+ℓcos θ. (132)
The axis xremains the same and given by (111).
It is straightforward to show that, for small oscillations
and for ℓ=ℓ0+ht, the equation of motion for θfor the
Rayleigh set up is identical to the Bossut set up, given by
equation (119). The kinetic and potential energies for the
Rayleigh set up are given by the first parts of equation
(115) and (116), respectively, since for the Rayleigh set
up there is no vertical net translation, the total energy
being given by (129), with the results (130) and (131).
B. Harmonic oscillator of variable frequency
Let us consider a harmonic oscillator along the uaxis
with variable mass mand variable spring coefficient k.
The equation of motion is
d
dt(m˙u) = −ku. (133)
If the mass varies linearly in time, m=m0+µt, and k
is constant the equation of motion reduces to
m¨u+µ˙u=−ku. (134)
This equation is identical to the equation (119) and can
thus be solved in like manner.
We consider now that mis constant and that the spring
coefficient kvaries varies with time [69–71]. In this case
the equation of motion reduces to
¨u=−su, (135)
where s=k/m From now on we suppose kvaries linearly
with time, that is, s=b+at where ais a small quantity.
Changing variable from tto swe get
a2d2u
ds2=−su. (136)
Making another change of variable from sto z=a−2/3s,
we reach the following equation
d2u
dz2=−zu. (137)
The solutions of this equation are the Airy functions
Ai(−z) and Bi(−z) [68],
u=c1Ai(−z) + c2Bi(−z),(138)
where c1and c2are constants. As we wish to get the
solution for a very slow variation of the spring coefficient,

12
which means that ais small, and bearing in mind that
z=s/a2/3, it suffices to consider the solutions for large
values of z. The asymptotic forms of the Airy functions
are [68]
Ai(z) = 1
√πz−1/4cos(2
3z3/2−π
4),(139)
Bi(z) = −1
√πz−1/4sin(2
3z3/2−π
4).(140)
The solution can thus be written as
u=cs−1/4cos 2
3as3/2+c0,(141)
where cand c0are constants and we recall that sis a
function of time, s=b+at.
The energy Eof the harmonic oscillator is
E=m
2˙u2+k
2u2.(142)
From this expression and using the asymptotic solution,
we find
E=mc2
2√s, (143)
where we have neglected terms of the order equal or
greater than a/s. Recalling that √scan be understood
as the frequency ω=pk/m =√s, it follows that the
energy of a harmonic oscillator of variable frequency is
proportional to the frequency, or that the ratio E/ω is
an invariant.
C. General time dependence
We ask whether an expression of the type
u=r(t) cos θ(t),(144)
could be the solution of the equation of motion for the
harmonic oscillator of variable spring coefficient. If we
replace the expression (144) in the equation (135) we
find that it is indeed a solution as long as the following
equations involving r,θand sare satisfied [69]:
2 ˙rω +r˙ω= 0,(145)
¨r−rω2+sr = 0,(146)
where ˙
θ=ω. The solution of the first equation gives
r=c ω−1/2,(147)
where cis an arbitrary constant and
s=ω2−¨r
r.(148)
Therefore, given ωas a function of time, we determine r
and then s. By the integration of ˙
θ=ω, we determine θ.
Replacing the solution (144) into the expression (142),
and bearing in mind that k/m =s, we find
E=mc2
2ω, (149)
where we have neglected terms of the order equal or larger
than ˙ω/ω. Again we find that E/ω is an invariant.
A simplification arises if we suppose that ωis a finite
function of at+bwhere ais a small quantity. In this case
¨r/r will be of the order a2and can be neglected in the
expression (148), which reduces to
s=ω2.(150)
A possible solutions for the dependence of ωwith time is
ω=at +b, (151)
which gives
θ=1
2at2+bt +c0, r =c(at +b)−1/2,(152)
and, using (150),
s= (at +b)2.(153)
Another solution is
ω= (at +b)1/2,(154)
which gives
θ=2
3a(at +b)3/2+c0, r =c(at +b)−1/4,(155)
and, using (150),
s=at +b. (156)
This solution is identified with that given by equation
(141) if we recall that in (141), sequals at +b. Yet
another solution is
ω= (at +b)−1/2,(157)
which gives
θ=2
a(at +b)1/2+c0, r =c(at +b)1/4,(158)
and, using (150),
s=1
at +b.(159)
This solution is identified with that given by equation
(127) if we recall that in equation (127) sequals at +b.

13
D. Vibrating string of variable length
Let us denote by φthe transverse displacement PQ of
the string at the point Q and by xthe distance from this
point to the fixed end A, as shown if figure 2. The equa-
tion of motion for a uniform string is the wave equation
∂2φ
∂t2=c2∂2φ
∂x2,(160)
where cis the velocity of the wave and equal to pΓ/µ
where Γ is the tension of the string and µis the mass
per unit length. The vibration occurs only for 0 ≤x≤ℓ
where ℓis the distance of the ring B to the fixed end A.
The boundary conditions are φ= 0 for x= 0 and x=ℓ.
We wish to solve this equation as the length changes with
time and determine the energy Eof the vibrating string
which is given by
E=µ
2Zℓ
0∂φ
∂t 2
dx +Γ
2Zℓ
0∂φ
∂x 2
dx. (161)
A closed solution of the wave equation can be obtained
for the following time dependence of the string length
ℓ=ℓ0/(1 + εt).
To solve the wave equation as ℓvaries with time we
proceed as follows. We assume a solution of the type
φ=ei(ax2+θ)sin kx, (162)
where ais a constant and θis a function of t. The co-
efficient kis chosen to be equal to nπ/ℓ where nis an
integer so that φvanishes at x= 0 and x=ℓ, as desired.
As ℓdepends on time, so does k, that is,
k=nπ
ℓ=nπ
ℓ0
(1 + ǫt).(163)
Replacing the solution into the wave equation and bear-
ing in mind that kdepends on time, we find a=k0ε/2c,
where k0=nπ/ℓ0, and ˙
θ=kc, which by integration gives
θ=k0c(t+1
2εt2).(164)
The other solution corresponds to the complex conju-
gate of this expression. Since φis real, we sum the two
solutions to obtain
φ=Acos k0εx2
2c+k0ct +k0cεt2
2sin kx. (165)
Replacing this result in the expression (161) for the en-
ergy E, we obtain the result (62) found before.
VII. QUANTUM MECHANICS
A. Parametric invariance
Born and Fock [53], in their paper of 1928 stated the
invariance in the following terms. If the system was in
a certain state described by a certain quantum number,
the probability to change the state, by a slow variation
of a parameter, is infinitely small, in spite of the fact
that the change in the energy levels be of finite amount.
They considered a discrete and a non-degenerate spec-
trum of energies except for the accidental degeneracy
due to crossing of two energy eigenvalues. Demonstra-
tion of the invariance with less restrictions was given by
Kato in 1950 [72]. Other demonstrations and discussions
of invariance in quantum mechanics are found in several
papers [73–78] and books on quantum mechanics [23–26].
In the following, we show that if the variation of a
parameter is infinitely slow the system remains in the
same quantum state. To this end we use an approach
analogous to the one we employed above for the classical
case. We start by consider a system described by the
Schr¨odinger equation
i~∂ψ
∂t =Hλψ, (166)
where ψis the wave function and Hλis the Hamiltonian
operator, which depends on a parameter λwhich depends
on time.
Let us consider the following quantity
H=hψ|Hλ|ψi,(167)
where ψis a solution of the equation (166).
As the Hamiltonian depends explicitly on time through
the parameter, His not conserved. Its variation in time
is
dH
dt =hψ|∂Hλ
∂t |ψi,(168)
which we write in the form
dH
dt =−chψ|Fλ|ψi,(169)
where Fis the operator
Fλ=−∂Hλ
∂λ ,(170)
and c=dλ/dt.
Now let φλbe a solution of the Schr¨odinger equation
(166) with the condition that the parameter λis kept
unchanged. We define the following quantities
E=hφλ|Hλ|φλi,(171)
and
F=1
TZT
0hφλ|Fλ|φλi.(172)
In accordance with the reasoning given above, for the
classical case, the approximation amounts to replace ψ
by φλ. The resulting equation is
dE
dλ =−F. (173)

14
We now write this equation in the form
ZT
0d
dthφλ|Hλ|φλi − hφλ|Fλ|φλidt = 0.(174)
The equation (174) can be written in the following equiv-
alent form
ZT
0h∂φλ
∂λ |Hλ|φλi+hφλ|Hλ|∂φλ
∂λ idt = 0.(175)
Let us now denote by φλn and Eλn the eigenfunctions
and eigenvalues of of Hλ, where λis considered to be
fixed. We may then expand φλ,
φλ=X
n
aλnφλn ,(176)
We may also expand the derivative of φλwith respect to
λ,
∂φλ
∂λ =X
n
bλnφλn .(177)
Replacing these expansions in equation (175), we find
X
nZT
0
Eλn(aλn b∗
λn +a∗
λnbλn)dt = 0,(178)
where we have assumed that the eigenfunctions are or-
thonormalized. A solution of this equation corresponds
to the case where the coefficients are all zero except one
of them, in which case the expression between paren-
theses vanishes. Therefore if the system is initially in a
certain state, say state φλn it remains in this state as λ
is varied slowly. In other words, the quantum number n
is invariant.
B. Electron on a rotating field
Let us consider the evolution of the spin of an electron
in a rotating magnetic field [25, 77]. The xand ycompo-
nents of the magnetic field are Bcos θand Bsin θwhere
θis the time dependent parameter, θ=ωt. In the repre-
sentation where the component zof the electron spin is
diagonal, the Hamiltonian is given by the square matrix
H=µB(σxcos θ+σysin θ),(179)
where µ=e~/2mis the Bohr magneton and σxand σy
are the Pauli matrix.
We define by χ+and χ−the basis where the Pauli
matrix σzis diagonal, which are the column matrices
with elements 1 and 0, and 0 and 1, respectively. The
eigenvectors and eigenvalues of Hare
φ1=1
√2(e−iθ/2χ++eiθ/2χ−), ε1=µB, (180)
φ2=1
√2(e−iθ/2χ+−eiθ/2χ−), ε2=−µB, (181)
It is useful to know that
dφ1
dt =−iω
2φ2,(182)
dφ2
dt =−iω
2φ1.(183)
The Schr¨odinger equation is
i~dχ
dt =Hχ, (184)
where χis the spinor. Writing χ=xφ1+yφ2, the
Schr¨odinger equation becomes
idx
dt +ω
2y=ωc
2x, (185)
idy
dt +ω
2x=−ωc
2y, (186)
where ωc= 2µB/~=eB/m is the cyclotron frequency.
The solution for the case x= 1 and y= 0 for t= 0 is
x= cos γt −iωc
2γsin γt, (187)
y=iω
2γsin γt, (188)
where
γ=1
2pω2+ω2
c.(189)
We remark that |x|2+|y|2= 1 so that χis normalized.
The probability of the system to be found in the state φ1
and φ2are respectively
|x|2= cos2γt +ω2
c
4γ2sin2γt, (190)
|y|2=ω2
4γ2sin2γt. (191)
In the regime ω/ωc≪1, the probability to change from
state 1 to state 2 is very small, of the order of (ω/ωc)2,
and we recall that ωis the rate of change of the parameter
θ=ωt.
C. Square well with a moving wall
We consider a particle confined in a one-dimensional
box which is equivalent to the motion of a particle under
an infinite square well potential. One wall of the box is

15
fixed and the other moves linearly. Denoting by ℓthe
length of the box, we assume that ℓ=ℓ0(1 + εt). A
closed solution of the Schr¨odinger equation can be found
for this case and in fact a solution was given by Doescher
and Rice [79]. In the following we present the solution
for this problem.
The Schr¨odinger equation to be solved is
i~∂ψ
∂t =−~2
2m
∂2ψ
∂x2,(192)
with the boundary condition that ψvanishes at x= 0
and at x=ℓ.
We assume a solution of the type
ψ=r2
ℓei(αx2+θ)sin kx, (193)
where αand θare functions of t, and k=nπ/ℓ so that
ψvanishes at x= 0 and x=ℓ, as required. Replacing
this expression in the Schr¨odinger equation, we find
α=m
2~
ε
1 + εt,(194)
˙
θ=−~
2mk2,(195)
which integrated gives
θ=−~k2
0t
2m(1 + εt),(196)
where k0=nπ/ℓ0.
Replacing the above results in the equation (193), we
reach the following expression [79]
ψ=r2
ℓexp{i
~
m2εx2−~2k2
0t
2m(1 + εt)}sin kx. (197)
D. Hamiltonian obeying a scaling relation
A closed solution can also be provided when the Hamil-
tonian Hλ(x) that depends on a parameter λobeys the
scaling relation
Hλ(x) = λbˆ
H(ξ)ξ=λax, (198)
where ˆ
H(ξ) only on ξbut not on λ. The eigenfunctions
ˆ
φ(ξ) and the eigenvalues ˆ
Eof ˆ
H(ξ) are related to the
eigenfunctions φλ(x) and eigenvalues Eof the original
Hamiltonian by
Eλ=λbˆ
E, (199)
φλ(x) = λa/2ˆ
φ(ξ).(200)
This last relation follows from the normalization of the
eigenfunctions.
A scaling relation of this type is obeyed by the Hamil-
tonian describing a particle in a box, in which case
a=−1 and b=−2, and λis the length of the box.
It is also obeyed by the Hamiltonian of the harmonic os-
cillator in which case a= 1/2 and b= 1, and λis the
frequency of the oscillation.
We consider the Schr¨odinger equation
i~∂ψ
∂t =Hψ, (201)
where we are omitting the index λin the Hamiltonian
H(x), which depends on time through the parameter λ.
The Hamiltonian operator
H=K+V(202)
is the sum of the kinetic energy operator Kand Vis the
potential energy. In the position representation, which
we use here, Vis a multiplying operator, that is, a func-
tion of x.
Let φ(x) be one of the eigenfunctions of Hand Ethe
corresponding eigenvalue, that is,
Hφ=Eφ, (203)
Again, we are omitting the index λin the eigenfunctions
and eigenvalues but they contain the parameter λand
thus depend on time through this parameter. We wish
to solve the Schr¨odinger equation with the initial condi-
tion such that the wavefunction at t= 0 is one of the
eigenfunctions of the Hamiltonian. Let us consider the
following wave function
ψe=eiθφ, (204)
where θis given by
θ=−1
~Zt
0
Edt′,(205)
which is in accordance with the initial condition. If we
replace it in the Schr¨odinger equation, we see that it is
not a solution because the term i~∂φ/∂t does not cancel
out. We assume then the following form for the solution
ψ=ei(θ+u)φ, (206)
where θis the dynamic phase given by (205) and uis a
function to be found. We look for a real solution for uso
that ψwill differ from φby a phase factor which means
that the system remains in the state described by φ. In
addition the wavefunction ψis normalized because φis
normalized.
Replacing the expression (206) in the Schr¨odinger
equation (201) we get the following equation
i~∂φ
∂t −~∂u
∂t φ=e−iu H(eiuφ)−Eφ. (207)
Taking into account that Vis a multiplying operator,
which is just a function, the right hand side becomes
e−iuK(eiu φ) + Vφ−Eφ, (208)

16
and the first term of this expression is
−~2
2m"i∂2u
∂x2φ−∂u
∂x 2
φ+ 2i∂u
∂x
∂φ
∂x #+Kφ. (209)
Replacing these results into equation (207) we get
i∂φ
∂t −∂u
∂t φ=
=−~
2mi∂2u
∂x2φ−(∂u
∂x )2φ+ 2i∂u
∂x
∂φ
∂x ,(210)
which is an equation for uas φis known.
As we are looking for a real u, its imaginary part should
vanish and the real and imaginary parts of equation (210)
become
−∂u
∂t =~
2m(∂u
∂x )2,(211)
∂φ
∂t =−~
2m∂2u
∂x2φ+ 2 ∂ u
∂x
∂φ
∂x ,(212)
where we have taken into account that φis real, a choice
that is always possible to accomplish because His Hermi-
tian. As we impose that the imaginary part of uvanishes,
this quantity should solve both the equations (211) and
(212).
The first equation (211) can be solved by the separa-
tion of variables. Assuming that u(t, x) = α(t)z(x) and
replacing it in (207) we get
−1
α2
∂α
∂t =~
2m
1
z(∂z
∂x )2.(213)
The left and right hand side should be a constant that
we choose to be equal to the unity, that is,
−∂α
∂t =α2,~
2m(∂z
∂x )2=z. (214)
Integrating,
α=ε
1 + εt, z =m
2~x2,(215)
where εis a constant of integration. Replacing these
results in the second equation (212), it becomes
2(1 + εt)∂φ
∂t =−εφ+ 2x∂ φ
∂x .(216)
Using the scaling laws for φ, we find the equalities
x
φ
∂φ
∂x =ξ
ˆ
φ
∂ˆ
φ
∂ξ ,(217)
1
φ
∂φ
∂t =a
2λ 1 + 2 ξ
ˆ
φ
∂ˆ
φ
∂ξ !dλ
dt .(218)
Replacing these relations in equation (216) we see that it
becomes satisfied as long as
dλ
dt =−λ ε
a(1 + εt).(219)
The integration of this equation gives
λ=λ0(1 + εt)−1/a,(220)
where λ0is a constant of integration, which is the value
of the parameter at t= 0. Therefore, usolves both equa-
tion under the condition (220). In other words, if the
parameter λdepends on time in accordance with (220),
u=αz given by (220) is real and solves both equations
(211) and (212).
We may draw the following conclusions from the above
results. If the parameter λvaries according to relation
(220) and if the scaling (198) is fulfilled, then the wave
function given by equation (206) is an exact solution and
uis real, that is, it is indeed a phase, given by
u=mεx2
2~(1 + εt).(221)
The solution (206) is valid for any value of the parame-
ter ε, and, of course, remains a solution when εis small,
which characterizes a slow variation of the parameter λ
because, according to equation (219) dλ/dt is propor-
tional to ε.
E. Harmonic oscillator of variable frequency
The quantum harmonic oscillator with variable fre-
quency was treated by Husimi in 1953 [80]. The Hamil-
tonian is
H=−~2
2m
∂2
∂x2+1
2mω2x2,(222)
where the frequency ωdepends on time. The eigenfunc-
tions φkn are [25]
φn(x) = mω
π~1/41
√2nn!Hn(ξ)e−ξ2/2,(223)
where Hn(ξ) are the Hermite polynomials and ξ=
xpmω/~. The corresponding eigenvalues are
En=~ω(n+1
2),(224)
where n= 0,1,2... We wish to solve the Schr¨odinger
equation
i~∂ψ
∂t =Hψ, (225)
with the initial condition that the initial state is one
of the eigenstates, say the eigenstate with the quantum
number n.

17
Writing the Hamiltonian in the form
H=~ω(−1
2
∂2
∂ξ2+1
2ξ2),(226)
it becomes manifest that it obeys the scaling relation
(198) with a= 1/2 and b= 1 and ωplaying the role of the
parameter λ. It is clear also that the eigenfunctions and
eigenvalues are in accordance with the scaling relations
(199) and (200).
From the results obtained above, a closed solution for
the time dependent Schr¨odinger equation can be obtained
for the following time dependence of the frequency
ω=ω0
(1 + εt)2.(227)
The solution is
ψ=ei(θ+u)φn,(228)
where φnis one of the eigenfunctions and
θ=−ω0(n+1
2)(t+εt2+1
3ε2t3),(229)
and
u=ε mx2
2~(1 + εt).(230)
F. Raising and lowering operators
We solve again the harmonic oscillator but now we
use a representation in terms of the lowering and raising
operators defined by
a=rmω
2~x+ip
√2m~ω,(231)
a†=rmω
2~x−ip
√2m~ω,(232)
where p=−~∂/∂ x is the momentum operator. They
hold the relations a|ni=√n|niand a†|ni=√n+ 1|ni,
and fulfills the commutation relation [a, a†] = 1.
In terms of these lowering and raising operators, the
Hamiltonian
H=p2
2m+1
2mω2x2(233)
of the harmonic oscillator becomes
H=~ω(a†a+1
2).(234)
The eigenvectors of Hare |ni, that is,
H|ni=En|ni,(235)
and the eigenvalues are En=~ω(n+ 1/2).
We wish to solve the Schr¨odinger equation
i~d
dt|ψi=H|ψi,(236)
considering that the frequency ωis a time dependent pa-
rameter and that at t= 0 the oscillator is in one of its
eigenstates.
As ωis a time dependent parameter the operators a,
a†, and the state vectors |nidepend on time through ω.
It is thus convenient to determine their variation with ω.
From the definitions given by (231) and (232), it follows
that aand a†vary with ωaccording to
∂a
∂ω =1
2ωa†,∂a†
∂ω =1
2ωa. (237)
From these relations we find
dH
dω =∂H
∂ω =mωx2.(238)
From (223) we establish the following variation of the
eigenvectors with ω,
d
dω |ni=1
4ω(aa −a†a†)|ni=i
4~ω(xp +px)|ni.(239)
To find the solution of the Schr¨odinger equation we
consider the following state vector
|ψi=ei(θ+αz)|ni,(240)
where
θ=−1
~Zt
0
Endt′,(241)
and αis a time dependent scalar, and z= (m/2~)x2,x2
being the position operator squared. Replacing |ψiinto
the Schr¨odinger equation, we find
−dα
dt
m
2x2|ni+i~d
dt|ni=e−iαz Heiαz |ni−En|ni.(242)
Now, recalling that zis proportional to x2,
e−iαzH eiαz =1
2me−iαzp2eiαz +1
2mω2x2=
=H+α
2(xp +px) + α2m
2x2,(243)
and the Schr¨odinger equation becomes
−dα
dt
m
2x2|ni+i~dω
dt
d
dω |ni=
=α
2(xp +px)|ni+α2m
2x2|ni.(244)

18
A solution of this equation occurs if
dα
dt =−α2,(245)
from which follows the result
α=ε
1 + εt,(246)
and if
dω
dt =−2αω, (247)
where we have taken into account the relation(239). Solv-
ing this equation one finds the dependence of ωwith time,
ω=ω0
(1 + εt)2.(248)
We conclude that (240) is an exact solution if ωdepends
on time according to (248).
VIII. STATISTICAL MECHANICS
A. Hertz invariant
Let H(q, p, λ) be the Hamiltonian of a system with
ndegrees of freedom where we are denoting by qand
pthe collection of coordinates q1,...,qnand momenta
p1,...,pn, and λis a parameter. In his treatise on sta-
tistical mechanics [81], Gibbs introduced the following
integral
Φ = ZH≤E
dqdp, (249)
which is the hypervolume of the region of the phase space
enclosed by a hypersurface of constant energy E. He also
defines the quantity
Ω = ∂Φ
∂E .(250)
Using the step function ϑ(x), which is equal to zero or
the unity according to whether the argument xis nega-
tive or positive, then Φ can be written as
Φ = Zϑ(E− H(λ))dqdp. (251)
Taking into account that the derivative of the step func-
tion is the Dirac delta function δ(x), then Ω can be writ-
ten as
Ω = Zδ(E− H(λ))dqdp. (252)
Gibbs considers two possible forms for the entropy of
a system. One of them is
S=kln Ω,(253)
which is the form widely used, and the other is
S′=kln Φ.(254)
He states that each of them has its advantage but the
first form is a little more simple than the second and
if simplicity is a criterion, the first form is preferable.
Nevertheless, the two forms differ little from one another
when the number of degrees of freedom is large, a fea-
ture that was recognized by Gibbs [81]. Another appeal
for the use of the first form is that Ω occurs as the nor-
malization of the Gibbs microcanonical distribution [81],
given by
ρ=1
Ωδ(E− H(λ)).(255)
In accordance with Clausius, who introduced the con-
cept of thermodynamic entropy, this quantity is constant
along a reversible adiabatic process, understood as a pro-
cess carried out without the exchange of heat, and slow
enough so that the system can be considered to be in
equilibrium. If Sand S′are to be interpreted as the
thermodynamic entropy, then Φ and Ω should be con-
stant along a reversible adiabatic process. The question
is thus how to define a mechanical procedure that re-
sults in a reversible adiabatic process, without referring
to heat. This question was implicit answered by Paul
Hertz in a paper of 1910 [82]. In this paper he used a
procedure that coincides with what we are calling slow
parametric action to show that Φ is constant. He as-
sumed that the system has an external coordinate that
is changed by external intervention, and that the energy
is a function of q,pand a. The variables qand pvary
according to the equations of motion and the parameter
ais subject to our arbitrariness.
The approach used by Hertz was based on a hint con-
tained in the Gibbs treatise on statistical mechanics [2].
According to Gibbs [81] “the entropy of a body is not
(sensibly) affected by mechanical action, during which
the body is at each instant (sensibly) in a state of thermo-
dynamic equilibrium”, which “may usually be attained
by a sufficiently slow variation of the external coordi-
nates”. The slow variation of the parameter was implicit
in the Hertz use of the Gibbs microcanonical distribu-
tion. If the variation of the parameter is slow, the sys-
tem remains in equilibrium and the Gibbs distribution,
which is understood to be valid for system in equilibrium,
can be used. Therefore, we may say that the use of the
Gibbs distribution for two distinct values of the parame-
ter, resulting from its variation with time, means that the
variation is implicitly slow. The invariance of Φ is thus a
parametric invariance, although Hertz did not used this
terminology but simply stated that Φ is constant.
The demonstration of the invariance of Φ is contained
in some books on statistical mechanics [27–29] and has
been considered by some authors [83–87]. A demonstra-
tion based on the Hertz paper was provided by de Koning
and Antonelli [41]. We demonstrate the invariance of Φ

19
as follows. The variation in energy of a system described
by a time dependent Hamiltonian is
dH
dt =∂H
∂t .(256)
If the Hamiltonian depends on time through a parameter
λwhich varies slowly with time, then in accordance with
(48), the right-hand side of this equation is replaced by
its time average and it becomes
dE
dλ =∂H
∂λ ,(257)
where E=H. The time average is then replaced by the
average in probability,
dE
dλ =h∂H
∂λ i,(258)
h∂H
∂λ i=Z∂H
∂λ ρ dqdp, (259)
and ρis the Gibbs microcanonical distribution, and
E=hHi =ZHρdqdp. (260)
The invariance of Φ means that Φ(E′, λ′) equals
Φ(E, λ) when the parameter changes slowly from λto
λ′causing a change of energy from Eto E′. Considering
small differences, we write
dΦ
dλ =∂Φ
∂λ +∂Φ
∂E
dE
dλ ,(261)
and the invariance means that dΦ/dλ should vanish. Us-
ing the relation (250), the equation (261) can be written
as
dΦ
dλ =∂Φ
∂λ + Ω dE
dλ .(262)
We observe that, using the definition (255) of the mi-
crocanonical probability distribution, the equation (258)
can be written as
ΩdE
dλ =Z∂H
∂λ δ(E− H)dqdp. (263)
We also observe that, using the expression (251), the
derivative of Φ with respect to λis
∂Φ
∂λ =−Z∂H
∂λ δ(E− H)dqdp. (264)
From these two relations, it becomes manifest that the
right hand side of (262) vanishes and so does dΦ/dλ as
desired.
Let us calculate Φ for a system of Nparticles confined
in a container of volume V. The number of degrees of
freedom is 3N. The particles do not interact and the
energy of the system is the kinetic energy
H=X
i
p2
i
2m,(265)
where pidenotes a component of each one of the particles.
The quantity Φ is
Φ = VNZH≤E
d3Np, (266)
the integral being equal to the volume of a sphere of
radius √2mE in a space of dimension 3N, that is,
Φ = VN(2πmE)3N/2
(3N/2)! ,(267)
and
Ω = 3N
2EΦ.(268)
B. Several parameters
Let us generalize the above result for several parame-
ters that we denote by λi. The variation in time of the
Hamiltonian is
dH
dt =X
i
∂H
∂λi
dλi
dt .(269)
Considering that the parameter vary slowly in time this
equation is replaced by
dE =X
i
∂H
∂λi
dλi.(270)
The time averages are in turn replaced by averages on
probability,
dE =X
ih∂H
∂λiidλi.(271)
We write this equation in the form
dE =−X
i
Fidλi,(272)
where
Fi=−h∂H
∂λii=−Z∂H
∂λi
ρdqdp, (273)
and ρis the Gibbs microcanonical distribution
ρ=1
Ωδ(E− H),(274)
where
Ω = Zδ(E− H)dqdp, (275)

20
and depends on λi. Defining the integral
Φ = Zϑ(E− H)dqdp, (276)
that depends on λi, we see that Fican be written as
Fi=1
Ω
∂Φ
∂λi
.(277)
Replacing this results in equation (272), and taking into
account that
Ω = ∂Φ
∂E ,(278)
we find
∂Φ
∂E dE +X
i
∂Φ
∂λi
dλi= 0.(279)
But the left-hand side of this equation is the differential
dΦ. Since it equals zero, it follows that Φ is constant as
we vary Eand the parameters λi.
C. Variable number of particles
The integral Φ given by the equation (249) was shown
to be an invariant when a parameter changes slowly.
However, the variation of the parameter did not change
the number of particles. Here wish to show that the ex-
pression that is invariant when the number nof particles
changes is Φn= Φ/n! with the factor 1/n!.
Let Hnbe the Hamiltonian function corresponding to
a system of nparticles. We use the notation xifor the
positions and momenta of the i-th particle so that the
Hamiltonian Hndepends on the variables from x1to xn
and is invariant under the permutation of the nparticles.
We now consider the removal of a particle from the
system. To simulate this procedure we suppose that at
t= 0 a particle is chosen at random and it its velocity is
slowly reduced and its interaction with the other particles
is slowly decreased. After an interval of time ∆t, its
velocity has vanished and it does not interact with other
particles anymore. We suppose that this procedure is
carried out by means of a parameter λthat takes the
value λ0at t= 0 and the value λ1at t= ∆t. During this
procedure the Hamiltonian Hthat describes the system
is not invariant by permutation of all the particles. If we
chose the particle nto be taken out, His invariant under
the permutation of the remaining n−1 particles.
Supposing that λvaries slowly, then the variation of
the energy Ewith λis given by
dE
dλ =h∂H
∂λ i,(280)
where the average is calculated using the probability dis-
tribution
ρ=1
Ωδ(E− H),(281)
where
Ω = Zδ(E− H)dnx, (282)
and depends on Eand λ. It is convenient to define the
integral
Φ = Zϑ(E− H)dnx, (283)
so that
∂Φ
∂E = Ω.(284)
Deriving (283) with respect to λ, and taking into ac-
count that Hndepends on λ, we find
∂Φ
∂λ =−Z∂H
∂λ δ(E− H)dnx, (285)
from which we obtain after dividing by Ω
dE
dλ =−1
Ω
∂Φ
∂λ .(286)
Replacing (284) into this equation, we reach the fol-
lowing relation
dE
dλ =−∂Φ/∂λ
∂Φ/∂E =∂ E
∂λ Φ
.(287)
This relation tell us that Eand λvaries in such a way
that Φ(E, λ) is invariant.
If Eis the energy when λ=λ0and E′when λ=λ1
then Φ(E, λ0) = Φ(E′, λ1). Let us define Φ∗
nby
Φ∗
n=Zϑ(E− Hn)dnx(288)
then Φ(E, λ0) = Φ∗
n(E) and Φ(E′, λ1) = nΦ∗
n−1(E′), the
factor ncoming from the existence of npossibility of
removing a particle from the system, and the invariance
is written as
Φ∗
n(E) = nΦ∗
n−1(E′) (289)
Defining Φn= Φ∗
n/n! then the invariance becomes
Φn(E) = Φn−1(E′) (290)
From these relation it follows that
Φn(E) = 1
n!Zϑ(E− Hn)dnx(291)
is invariant when one varies Eand n.
For a system of Nnoninteracting particles we use the
result (267) to find
ΦN=VN
N!
(2πmE)3N/2
(3N/2)! .(292)

21
D. Quantum systems
Here we use the occupation number representation to
describe the dynamics of a quantum system [88]. To this
end we introduce the operator a†
iwhich creates a particle
in the single particle state ψiand the operator aiwhich
annihilates a particle in state ψi. The number operator
is a†
iaiand the total number operator is
N=X
i
a†
iai.(293)
We assume the following form for the Hamiltonian
H=X
ij
Kij a†
iaj+X
ijlk
Vijkl a†
ia†
jalak,(294)
where Kij =hφi|K|φjiand Vijkl =hφiφj|V|φkφli, and
Kand Vare the kinetic and potential energy. We re-
mark that Ncommutes with H. If Hdoes not depend
explicitly on time, His conserved and so is N.
Now we wish to treat the case where the number of
particle varies with time, for instance, by introducing or
removing particles from the system. This is accomplished
by assuming that the creation and annihilation operators
depends on a parameter λthat varies with time. We start
with the equation
dE
dλ = Tr ∂H
∂λ ρ, (295)
where ρis the density operator, which we assume to be
given by
ρ=1
Ωδ(E−H),(296)
and
Ω = Trδ(E−H).(297)
In an explicitly form,
dE
dλ =1
ΩTr ∂H
∂λ δ(E−H).(298)
Defining
Φ = Trϑ(E−H),(299)
we find the relations
∂Φ
∂E = Ω,(300)
∂Φ
∂λ =−Tr∂H
∂λ δ(E−H),(301)
from which follows
dE
dλ =−∂Φ/∂λ
∂Φ/∂E =∂ E
∂λ Φ
.(302)
Therefore as one varies Eand λ, Φ remains invariant.
Let us suppose that as λvaries, assuming values λ1,
λ2, and so on, the number of particles in the system are,
respectively, n1,n2, and so on, and the energy are E1,
E2, and so on. If we define
Φ∗
n(E) = Trnϑ(E−H),(303)
where the trace is taken within the subspace of states
with nparticles, then Φ(Ei, λi) = Φni(Ei). Therefore the
invariance Φ(E, λ) when one varies Eand λis equivalent
to the invariance of Φ∗
n(E) when one varies the energy E
and the number of particles.
The interpretation of Φ∗
n(E) is very simple. If we use a
basis of eigenvectors |φ(n)
iiof Hbelonging to the Hilbert
space with nparticles, with the associate eigenvalue E(n)
i,
then
Φ∗
n=X
i
ϑ(E−E(n)
i).(304)
Thus the invariant Φ∗
nis the number of quantum states
with nparticles with eigenvectors less or equal to E.
It is worth taking the classical limit of (303). To this
end, we consider that the system is enclosed in a cubic
vessel of size Land use the eigenfunctions of the momen-
tum
φn(x) = 1
√Leikx,(305)
where k= 2πn. After expressing Φ in terms of the basis
of plane waves, we may take the classical limit. The
result is
Φ∗
N=1
N!Zϑ(E− H)dnxdnp
hn,(306)
where His the classical Hamiltonian, Nis the number
of particles, nis the number of degrees of freedom, and
his the Planck constant. Except for the factor hn, this
coincides with (291).
For a system of Nnoninteracting particles we use the
result (292) to find
Φ∗
N=VN
N!
(2πmE/h2)3N/2
(3N/2)! .(307)
E. Canonical distribution
Let us suppose that instead of the Gibbs microcanon-
ical distribution the system is described by the Gibbs
canonical distribution
ρ=1
Ze−βH,(308)
where βis a parameter, His the Hamiltonian that does
not depend on βbut depend on a parameter λ, and
Z=Ze−βHdqdp (309)

22
We wish to determine the invariant which is analogous
to that of Hertz.
The probability distribution P(E) of the energies is the
marginal probability distribution obtained from ρ(q, p)
and is given by
P(E) = 1
ZΩe−βE ,(310)
where
Ω = Zδ(E− H)dqdp, (311)
which depends on Eand λ, and
Z=Ze−βE ΩdE, (312)
which depends on βand λ.
If the parameter λis varying slowly in time we use
equation (258)
dU
dλ =h∂H
∂λ i,(313)
but now the average on the right-hand side is taken over
the canonical distribution ρ,
h∂H
∂λ i=Z∂H
∂λ ρdqdp. (314)
The equation (313) determines the relation between β
and λalong a parametric slow action. The invariant Ψ
that we wish to find will then be a function of βand λ.
The right-hand side of (314) is written in a more con-
venient form as
1
ZZ∂H
∂λ δ(E− H)e−β E dqdpdE, (315)
which is equal to
−1
ZZ∂Φ
∂λ e−β E dE, (316)
where
Φ = Zϑ(E− H)dqdp, (317)
and depends on Eand λ.
The average energy Uis written as
U=1
ZZEe−β E ΩdE, (318)
and can be obtained from Zby
U=−1
Z
∂Z
∂β .(319)
Using the above results, the equation (313) becomes
dU
dλ =−1
ZZ∂Φ
∂λ e−β E dE. (320)
The invariant Ψ, which depends on βand λ, is such
that if Ψ(β, λ) is held constant then βis connected to λ
in such a way that (320) is fulfilled. It is equivalent to
say that, in equation (320),
dβ
dλ =−∂Ψ/∂λ
∂Ψ/∂β .(321)
For comparison with equation (320), we write this equa-
tion in the form
∂Ψ
∂β
dβ
dλ =−∂Ψ
∂λ .(322)
We show now that the following form,
Ψ = eβU Z, (323)
is invariant, where Zis given by (312), and is equivalent
to the expression
Z=βZΦe−βE dE, (324)
obtained by an integration by parts. To this end we cal-
culated its derivative with respect to βand λ. Recalling
that Φ does not depend on β, and using the result (319),
we find
∂Ψ
∂β =β∂U
∂β Ψ.(325)
The other derivative is
∂Ψ
∂λ =β∂U
∂λ +β
ZZ∂Φ
∂λ e−β E dEΨ.(326)
Replacing the two derivatives in equation (322), we get
∂U
∂β
dβ
dλ =−∂U
∂λ −1
ZZ∂Φ
∂λ e−β E dE, (327)
which is equivalent to equation (320), and shows that Ψ
is an invariant.
For a system of Nnoninteracting particles we replace
the result (307) in equation (324) to find, after integra-
tion in E,
Z=VN
N!(2πm
βh2)3N/2.(328)
Taking into account that U= 3N/2βwe obtain the in-
variant Ψ,
Ψ = e3N/2VN
N!2πm
βh23N/2
.(329)
IX. THERMODYNAMICS
A. Adiabatic invariants
An adiabatic process is understood as a process which
does not involve the exchange of heat. This type of

23
process played an important role in the development of
the theory of heat [89, 90]. Laplace for instance pro-
posed that the variations of pressure in the propagation of
sound in air are adiabatic processes. Carnot announced
his fundamental principle by means of a cyclic process in-
volving two isothermal and two adiabatic subprocesses.
Such a cyclic process was also used by Clausius when he
introduced the laws of thermodynamics. The term adia-
batic was coined by Rankine in 1859 to refer to processes
without the intervention of heat.
An adiabatic process may be a slow process or a very
fast one as long as the system is enclosed by adiabatic
walls or is in isolation. If the walls are not perfectly adi-
abatic, the process can still be adiabatic if it occurs very
rapidly. If an air pump is compressed quickly, there is
no time for heat to be exchanged with the surroundings
and the process is adiabatic. In this case, the temper-
ature of the air inside the pump increases. Just after
compression, considering that the walls of the pump are
not perfectly adiabatic, heat is release to the surround-
ing, which is felt by the pump handler. Thus, neither the
slowness nor the quickness are distinguishing features of
an adiabatic process.
Along an adiabatic process we may ask whether there
are state functions that are invariant along this process.
If the process is slow enough, this question was answered
by Clausius when he introduced in 1854 a quantity, which
in 1865 he called entropy that remains constant along
a slow adiabatic process [91–93]. But before Clausius,
it was known that the quantity pV γ, remains constant
when an ideal gas undergoes a slow adiabatic process.
Here, pis the pressure, Vis the volume and γis the
ratio of the two types of specific heats.
The result that pV γis a constant was obtained theo-
retically by Poisson in 1823 [94] by using the equation of
state of an ideal gas, pV proportional to the temperature
T, and the assumption that γis constant. The starting
point of his derivation is the equation
γdV
∂V /∂T +dp
∂p/∂ T = 0,(330)
valid along an adiabatic process, which for an ideal gas
becomes pdV +V dp. Considering that γis constant, the
integration of this equation gives pV γequal to a constant.
Although the equation (330) was derived by Poisson by
assuming that heat is a state function, nevertheless it
remains valid within thermodynamics. In fact, the in-
variance of pV γwas derived by Clausius in 1850 within
the realm of thermodynamics [93, 95].
The derivation of the adiabatic invariant pV γ, either
by Poisson or by Clausius, involved thermal properties
that depended explicitly on the temperature. However,
as an adiabatic process involves no heat we may presume
that any adiabatic invariant could be derived without
referring to thermal properties. That is, a derivation
carried out within the realm of mechanics by considering
a parametric slow process. To show that this is indeed
the case for the Poisson adiabatic invariant, we follow the
reasoning of Rayleigh, which is summarized by equation
(29).
If the volume Vof a gas enclosed in a vessel is varied
slowly, then in accordance with the equation (29), the
variation of the energy with the volume is
dE
dV =−p, (331)
where pis the pressure of the gas. For a system of nonin-
teracting molecules, it follows from the laws of mechanics
that the pressure pis two-thirds of the kinetic energy E,
p=2E
3V,(332)
a result derived by Kr¨onig [96], by Clausius [97, 98], and
by Maxwell [99] within the kinetic theory. Considering a
simple gas with only translational degrees of freedom, E
is the total energy. Replacing the result (332) into (331),
we find by integration that EV 2/3is constant, from which
follows that pV 5/3is an adiabatic invariant.
A similar invariant is obtained for the radiation. In
his treatise on electricity and magnetism of 1873 [100],
Maxwell showed that the pressure of radiation is one-
third of the density of energy, or
p=E
3V.(333)
Replacing this result into (331), we find by integration
that EV 1/3from which follows that pV 4/3is an adiabatic
invariant.
It is worth mentioning that in the derivation of the
Stefan-Boltzmann law carried out by Boltzmann in a pa-
per of 1884 [101], he made use of the Maxwell relation
(333) along with thermodynamic reasoning. A derivation
of this law by the use of the invariant just obtained is as
follows. As the entropy Sis an invariant we may con-
sider it as a function of the invariant EV 1/3. Assuming
that it is a homogeneous function of Eand V, it follows
that Sis proportional to V1/4E3/4. The temperature
is obtained by 1/T =∂S/∂E from which follows that
E=aV T 4. Replacing this result in equation (333), we
find pto be proportional to T4, which is a statement of
the Stefan-Boltzmann law.
B. Boltzmann and Gibbs entropy
The concept of entropy was introduced by Clausius as
a quantity that is constant along a slow adiabatic pro-
cess and that increases in an irreversible process. These
properties of the entropy are a brief statement of the sec-
ond law of thermodynamics introduced by Clausius. The
definition of entropy that emerges from the Boltzmann
writings is laid down on the second property, related to
the increase of entropy, and can be found in his book on
the theory of gases [102, 103].

24
In a paper of 1872 Boltzmann proved that the quantity
[104]
H=Zfln fd3rd3v(334)
never increases, where fis the one-particle probabil-
ity distribution. The negative of Hwas understood by
Boltzmann as proportional to the entropy. To demon-
strate this result, which comprises the Boltzmann H-
theorem, Boltzmann used the transport equation that
he introduced in the same paper. Replacing the Maxwell
distribution in the expression for H, he obtained the fol-
lowing expression for the entropy of an ideal gas [104]
Nln V4πT
3m3/2
+3
2N, (335)
where Nis the number of molecules, Vthe volume, m
the mass of a molecule, and Tis the mean kinetic energy.
Later on, in 1877, Boltzmann related entropy to prob-
ability, which he stated in the following terms [105, 106].
In most cases the initial state of a system is a very im-
probable one and the system has the tendency to reach
more probable states, those of thermal equilibrium. If we
apply this to the second law, we can identify that quan-
tity, which is usually called entropy, with the probability
of the state in question. For an ideal gas of nmolecules,
Boltzmann finds the relation between entropy and prob-
ability as follows [105, 106]. The number of complexions
that corresponds to a given repartition n0,n1,n2, and
so on, of the nmolecules into the kinetic energies 0, ε,
2ε, and so on, is the permutation number
P=n!
n0!n1!n2!. . . .(336)
The most probable state corresponds to the maximum of
P, or equivalently to the maximum of ln P. Considering
that niare large numbers we may use the approximation
ln n! = nln n−nto find
ln P=nln n−X
i
niln ni.(337)
Taking the continuous limit of the energy, the second
term of this expression yields the quantity −H, which is
identified as the entropy. Boltzmann formulates a gen-
eral principle relating Pand entropy in following terms
[105, 106]. The measure of permutability of all bodies will
always grow in the course of the state changes and can at
most remain constant as long as all bodies are in thermal
equilibrium.
In a paper of 1901 on the radiation formula, Planck
translated the Boltzmann relation between entropy and
probability in the following terms [107]
S=kln W, (338)
with the exception of an additive constant, where Wis
the probability (Wahrscheinlichkeit in the original paper)
that the Nresonators have a total energy U, and kis one
of the two constants of nature introduced by Planck [108],
which is the Boltzmann constant. Following Boltzmann,
Planck determines the number of complexions which he
says is proportional to W. Although, Planck speaks of
Was a probability, the quantity is in fact understood in
formula (338) as the reciprocal of the probability.
Gibbs considered two types of entropy, given by (253)
and (254). The first form is similar to (338) in the sense
that both are proportional to the logarithm of the num-
ber of states of a system with a fixed energy. Of the two
types of entropy, Gibbs opted for the first. These two
types of entropy considered by Gibbs refer to the micro-
canonical distribution. He also introduced a form of the
entropy for systems described by the canonical distribu-
tion. In this case Gibbs defined entropy as the average
of the index of probability with the sign reversed. As the
index of probability is the logarithm of the probability,
the Gibbs canonical entropy is
SG=−kZρln ρdqdp. (339)
Although Gibbs speaks of average, in fact the entropy
given by (339) is not an average of a state function, as
ln ρis not properly a state function.
It is worth comparing the expression (339) with that
related to Hgiven by equation (334). If we consider a
system of Nparticles, the formula (334) gives the expres-
sion
SB=−NZfln fd3rd3v. (340)
If the particles do not interact, both expression (339) and
(340) give the same value. If the particles interact, SBis
distinct from SG.
As we have seen above, the invariance of the second
form of the microcanonical entropy, given (254), was the
concern of Paul Hertz, who demonstrated the invariance
in a paper published in 1910 [82]. The invariance of the
entropy was also the concern of Einstein. In a paper of
1914, he considered the entropy of a quantum system
with a discretized spectrum of energies which depended
on a parameter [11]. He used the Planck formula (338),
relating the entropy with the number Zof possible quan-
tum states, and asked whether the entropy remains valid
when the states of the system varies under the change
of the parameter. Using the Ehrenfest principle he con-
cluded that Zis indeed an invariant under the slow para-
metric action and so is the entropy.
C. Work and heat
The law of conservation of energy tells that the in-
crease of the energy of a system equals the work done
on the system. Thermodynamics distinguishes two types
of work. One of them is the heat Q, sometimes called

25
internal work, and the other is the external work W,
of simply work. The variation ∆Uof the energy is thus
∆U=Q+W. In thermodynamics, the distinction is pro-
vided by the introduction of adiabatic walls. If a system
is enclosed by adiabatic walls, there is no heat involved
and the increase in energy is the external work. However,
we are faced here with a circular reasoning and another
way of distinguish heat and work is necessary [109].
The distinction between the two types of work is ob-
tained by considering the connection of a system with
the environment. One of them is the ordinary interac-
tion, that we call dynamic connection, in which the state
of the system varies by virtue of the connection of the dy-
namic variables of the system with those of the environ-
ment. The other is the parametric connection in which
the state of the system varies by virtue of the variation
of a parameter. A more precise distinction is provided by
considering the external forces acting on the system. In
the dynamic connection, the external forces, depend on
the dynamic variables of the environment but not on the
parameters. In the parametric connection, the external
forces depend on the parameter but not on the dynamic
variables of the environment.
These two types of connection with the environment,
allows us to mechanically distinguish Qand W. If the
system is only parametrically connected with the envi-
ronment, there is no heat involved, Q= 0, and the change
of energy of the system is the work Wcaused by the vari-
ation of the parameter. If the parameter is held constant
and the system is dynamically connected with the en-
vironment, there is no work related to the variation of
the parameter, W= 0, and the variation in energy is
the work Qrelated to the dynamic variables, which is
understood as heat.
If the system is connected to the surrounding only
through the parametric action, the resulting process is
thus an adiabatic process, in the thermodynamic sense.
The idea of adiabatic process as a purely mechanical pro-
cess is implicit in the works of Clausius and Boltzmann on
the kinetic theory as they considered a thermodynamic
system as a mechanical system. According to Jammer
[2], the idea of an adiabatic process as related with the
slow variation of a parameter is to be found in the works
of Helmholtz and Heinrich Hertz, who tried to identify
a parameter of the system to a cyclic variable. As we
have seen above a precise understanding of a adiabatic
process as the result of a variation of a parameter was
implicit in the paper of Paul Hertz, who developed this
concept from a hint given by Gibbs. It appears that it
was on this understanding that Einstein called the Ehren-
fest hypothesis about parametric invariants the adiabatic
hypothesis, and that Ehrenfest called the parametric in-
variant the adiabatic invariant.
If the system is connected to the surrounding only
through the parametric action, the resulting process is
an adiabatic process no matter if the resulting process
is slow or not. The distinction between a slow and a
rapid process is that in the slow process the system re-
mains in equilibrium or rather near equilibrium as we
are speaking of a process evolving in time. Processes of
this type are usually called quasi-static processes, and
for that reason sometimes the slow parametric action is
named quasi-static. We think that this is also inappro-
priate due to the existence of quasi-static processes, such
as the isothermal process, which are not adiabatic. Thus
we may say that the slow parametric action results in a
quasi-static adiabatic process or a slow adiabatic process.
D. Equilibrium thermodynamics
The existence of parametric invariance in classical and
quantum mechanics is the crucial feature that allows
the construction of a continuous sequence of equilib-
rium thermodynamic states when a parameter is chan-
ged slowly in time, as may happen when work is done
upon a system. This is understood as follows. If a sys-
tem in equilibrium is perturbed during a finite interval
of time ∆t, it will be found in a nonequilibrium state at
the end of the perturbation, and we have to wait another
finite interval of time τfor the system to reach equi-
librium again. Suppose that the perturbation is carried
out by the variation of a parameter λwhich changes by
∆λ=c∆tduring the interval ∆t, where cis the rate in
which the parameter is changed. If cis small enough, the
adiabatic invariance guarantees that τbecomes negligi-
ble and the system remains in equilibrium after the finite
change ∆λof the parameter.
In accordance with the understanding concerning heat
and work, we write the variation of the energy during a
certain small interval of time when the parameters λiis
slowly varying in time as
dU =dQ +dW, (341)
where dW is the work performed by the system. As the
parameter are slowly varying in time, the resulting pro-
cess is adiabatic, dQ = 0 and dW =dU . Using the
expression (272) for dU , we write
dW =−X
i
Fidλ, (342)
where
Fi=−∂H
∂λ ,(343)
and His the Hamiltonian of the system. The time aver-
age is in turn replaced by the average
Fi=−h∂H
∂λ i,(344)
over a Gibbs probability distribution ρ. The energy Uis
understood as the average U=hHi.
If we consider a slow variation of the parameter, this
process will corresponds to lines that lay down on a sur-
face on the space spanned by the parameters λiand by

26
the energy E. Along any lines there is no exchange of
heat and the parametric invariant Φ is constant. That
is, each surface is characterized by a certain value of Φ.
The energy can then be considered as function of Φ and
of the parameters, so that
dU =AdΦ−X
i
Fidλi,(345)
which compared to (341) tell us that
dQ =AdΦ.(346)
The relation (346) is in accordance with the Clausius
relation dQ =T dS. However, it does not mean necessar-
ily that the entropy Sis equal or proportional do Φ. It
says that Sis a function of Φ. To determine the specific
function, we must introduce a condition. We may sup-
pose that Sscales as the size of the system. Taking into
account that Φ scales with the size of the system to the
n-th power, where nis the number of degrees of freedom,
we may place ln Φ as proportional to Sand write
S=kln Φ,(347)
where kis a constant with the same physical dimension
of the entropy.
Having established the relation between the entropy S
and the invariant Φ, we may determine the temperature
Tby its thermodynamic definition 1/T =∂ S/∂E. In the
case of a system described by the Gibbs microcanonical
distribution, Φ is given by (251), and we find
1
kT =Ω
Φ,(348)
where Ω is given by (252) and is related to Φ by Ω =
∂Φ/∂E.
E. Canonical distribution
If a system is in contact with the environment only
through a parametric connection, the Hamiltonian His a
function of the internal variables, which are the dynamic
variables qand pof the system. The external forces de-
pend on the parameters and on the internal variables but
not on the external variables, which are the variables of
the environment. In this case the energy of the system
is the average of Hwhich is a function of the internal
variables.
Now let us consider the case where the environment is
subject to the dynamic connection. In this case the exter-
nal forces are functions of both the internal and external
variables and the energy of the system will have a term
that is related to the external variables. If the system
is nearly equilibrium the treatment given by the statisti-
cal mechanics is to replace the dynamic description by a
description in terms of a probabilistic description given
for example by the Gibbs canonical description. In this
description the probability distribution is a function of
a Hamiltonian which is understood as the Hamiltonian
Hthat one obtains by removing the terms describing the
dynamic interaction with the environment. This does not
mean that the interaction has been neglected. In fact, it
is taken into account through the probability distribution
itself, such as the Gibbs canonical distribution which de-
scribes the interaction with a heat reservoir at a given
temperature. Therefore, the energy Uthat enters equa-
tion (341) is the average of Hwhich depends only on the
internal dynamic variables.
In the present case of the dynamic connection, in which
the system is described by the Gibbs canonical distri-
bution the parametric invariant is Ψ given by equation
(323),
Ψ = eβU Z, (349)
Again the usual choice is a logarithm relation
S=kln Ψ = kβU +kln Z, (350)
which gives an entropy that increases with the size of
the system. Deriving this relation with respect to βand
using relation (319), and comparing with the Clausius
relation dU =T dS, one finds ∂S/∂β =kβ∂U /∂β, which
gives the relation β= 1/kT between the parameter β
and T.
What is the relation between the invariant Ψ and the
Gibbs expression for the entropy
SG=−kZρln ρdqdp. (351)
If we replace the distribution (308) in this equation we
find that it equals kln Ψ, that is, it is an invariant and
coincides with the entropy (350) that scales with the size
of the system.
X. CONCLUSION
We have argued that the slow variation of a parame-
ter of a system results in a slow adiabatic process. Thus
a parametric invariant such as the Hertz invariant is an
adiabatic invariant. As the entropy is a thermodynamic
variable that characterizes a reversible adiabatic process,
understood as slow adiabatic process, it is constant along
this process and can be understood as a parametric in-
variant. As the entropy grows with the size of the system,
it is defined as the logarithm of the Hertz invariant.
The Hertz invariant Φ is the volume in phase space
of the region enclosed by the surface of constant energy.
It remains constant when a parameter is slowly changed
with time. We have extended this result to the case where
the number of particles varies with time with the conclu-
sion that Φ∗= Φ/N! is the invariant. We have also
considered the quantum version of the Hertz invariant
for the case of the variation of a parameter as well as the

27
variation of the number of particles. In this last case, the
classical limit results in the quantity Φ∗= Φ/N !
The parametric invariance allows the distinction be-
tween heat and work. This is provided by considering
the two types of connections of the system with the envi-
ronment that we have called dynamical and parametric
connection. If only the second is present and the param-
eter is varied, the resulting process is adiabatic. If in ad-
dition the variation of the parameter is slow, the entropy
characterizes this process and can thus be associated to
the parametric invariant.
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