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Coherent States in Quantum Mechanics
Kharanshu Solanki†and Abhiram Srikanth∗
School of Arts and Sciences, Ahmedabad University, Ahmedabad, India, 380009
†kharanshu.s@ahduni.edu.in∗abhiram.s@ahduni.edu.in
[June 2, 2022]
Although the title of this paper seems to suggest a comprehensive review of coherent states
in quantum mechanics, the actual contents of this paper will be far from comprehensive. The
concept of coherent states, like (really) any other concept in physics, is humongous, and it will
be impossible to emphasize all its details and richness in a short summary. Rather, I intend to
examine two fundamental properties of the coherent states of a quantum harmonic oscillator
- their representation in the energy basis, and their evolution over time. Upon understanding
these properties, one can then proceed to dive into the hugely important role that coherent states
play in quantum theory.
1. The Quantum Harmonic Oscillator Revisited
Our focus, throughout this paper, will be oriented towards the coherent states of a quantum
harmonic oscillator. Therefore, let us begin with a quick refresher of this rather important
system. The Hamiltonian of the harmonic oscillator consists of the usual kinetic energy term
and a potential term that is quadratic in the position operator.
ˆ
H=ˆp2
2m+1
2mω2ˆx2(1.1)
The procedure of factorizing this Hamiltonian in terms of the raising and lowering operators
(ˆa†and ˆarespectively) is well-known and leads to a modification of the form,
ˆ
H=¯
hωˆ
N+1
2(1.2)
where ˆ
N=ˆa†ˆais appropriately called the number operator. This factorization allows for
an algebraic solution of the oscillator. The energy eigenvalues Enand eigenstates |n⟩of the
Hamiltonian turn out to be,
En=¯
hωn+1
2
|n⟩=ˆa†n
√n!|0⟩
(1.3)
where, n=0,1,2,3,... are the eigenvalues of the number operator, and |0⟩is the ground state
of the harmonic oscillator. The basic idea that one forms of the quantum harmonic oscillator
is based on the action of the raising and lowering operators on an given eigenstate |n⟩. These
operations are,
.
2KHARANSHU SOLANKI
ˆa†|n⟩=√n+1|n+1⟩
ˆa|n⟩=√n|n+1⟩(1.4)
This means that the raising and lowering operators perform the task of raising and lowering
the energies (respectively) of the states that they act on. Figure 1 summarizes the basic idea of
the quantum harmonic oscillator.
ℏω (n - 1/2) ℏω (n + 1/2) ℏω (n + 3/2)
|n - 1⟩|n + 1⟩
|n⟩
ââ
✝
Fig.1. The harmonic oscillator can be summarized by this diagram. We place the allowed energy eigenvalues along
the real axis. Each of these is associated with a corresponding energy eigenstate. We can then employ the raising
and lowering operators to generate transitions between these states, by either raising or lowering their energy by one
quanta.
Of course, I have not explicitly derived any of the results that I claim here (and I do not intend
to do so), but the interested reader is directed towards the vast literature on the same [1,2].
2. What are Coherent States?
Following our refresher of the harmonic oscillator, we now find ourselves in a position to define
coherent states. Coherent states feature in a variety of different systems in quantum mechanics,
and the ones associated with the harmonic oscillator are sometimes called canonical coherent
states to distinguish them from the others1.
A coherent state is defined as an eigenstate of the lowering operator. More precisely, if a state |α⟩is
constructed such that,
ˆa|α⟩=α|α⟩(2.1)
then |α⟩is called a coherent state with eigenvalue α. As a consequence of the fact that ˆais
not Hermitian, we find that in general, α∈C. At the outset, this may seem like a sound
mathematical definition. But what does it conceptually mean to be an eigenstate of the lowering
operator? Equation (1.4) makes it difficult to believe whether the lowering operator could
even have eigenstates in the first place, since lowering the energy of a harmonic oscillator state
must definitely change the state itself. Well, we shall now build an explicit expression for the
coherent state, that will help us understand this definition better.
1For the sake of brevity, I shall treat the terms coherent and canonically coherent as being synonymous.
COHERENT STATES IN QUANTUM MECHANICS 3
3. Representation of Coherent States in the Energy Basis
In order to make sense of the definition of coherent states, we must find an expression for them
in terms of the eigenstates of the harmonic oscillator Hamiltonian. We have,
ˆ
H|n⟩=En|n⟩(3.1)
The states |n⟩form a complete orthonormal basis (called the energy basis) for the associated
vector space. Since the coherent state |α⟩is also a vector in the same vector space, we may
expand it in the energy basis as,
|α⟩=
∞
∑
n=0
cn|n⟩(3.2)
where the coefficients are given as the overlap between |α⟩and |n⟩, i.e., cn=⟨α|n⟩. These
coefficients may be different for different coherent states. Now, consider the action of the
lowering operator on the coherent state.
ˆa|α⟩=ˆa
∞
∑
n=0
cn|n⟩
=
∞
∑
n=0
cnˆa|n⟩
⇒ˆa|α⟩=
∞
∑
n=1
cn√n|n−1⟩
(3.3)
Notice that in the final step, the sum starts from n=1because for n=0, the lowering operator
simply annihilates the ground state, and hence the term does not contribute to the expansion.
Since nis just a summing index, we can relabel (n→n+1), and rewrite equation (3.3) as,
ˆa|α⟩=
∞
∑
n=0
cn+1√n+1|n⟩(3.4)
Using equations (2.1), (3.2) and (3.4), we may write,
∞
∑
n=0
cn+1√n+1|n⟩=α
∞
∑
n=0
cn|n⟩
⇒
∞
∑
n=0hcn+1√n+1−αcni|n⟩=0
(3.5)
This is true only when the term inside the square parentheses is zero. Upon doing so, we arrive
at the following recursion relation,
4KHARANSHU SOLANKI
cn+1=α
√n+1cn=α
√n+1
α
√ncn−1=.. .
⇒cn=α
√n
α
√n−1··· α
√2
α
√1c0
⇒cn=αn
√n!c0
(3.6)
Substituting this result into equation (3.2) yields,
|α⟩=
∞
∑
n=0
αn
√n!c0|n⟩(3.7)
Now, in order to arrive at a final expression, we only need to find the coefficient c0, which can
be done by making use of the normalization condition.
∞
∑
n=0|cn|2=1
⇒
∞
∑
n=0
αn
√n!c0
2
=1
⇒
∞
∑
n=0
|α|2n
n!|c0|2=1
(3.8)
On the LHS, we find that the sum is just a Taylor expansion of e|α|2. Hence, we find that,
|c0|2e|α|2=1
⇒|c0|2=e−|α|2
⇒c0=e−|α|2/2
(3.9)
Substituting this into equation (3.7), we get the final expression for the coherent states of a
quantum harmonic oscillator as,
|α⟩=e−|α|2/2
∞
∑
n=0
αn
√n!|n⟩(3.10)
Equation (3.10) conveys an explicit representation of coherent states in the energy basis. As
a cautionary step, let us prove that this expression satisfies the eigenvalue equation of the
lowering operator.
COHERENT STATES IN QUANTUM MECHANICS 5
We have,
ˆa|α⟩=e−|α|2/2
∞
∑
n=0
αn
√n!ˆa|n⟩
=e−|α|2/2
∞
∑
n=1
αn
√n!√n|n−1⟩
=e−|α|2/2
∞
∑
n=1
αn
√n−1! |n−1⟩
=e−|α|2/2
∞
∑
n=0
αn+1
√n!|n⟩
=e−|α|2/2
∞
∑
n=0
αnα
√n!|n⟩
=αe−|α|2/2
∞
∑
n=0
αn
√n!|n⟩
⇒ˆa|α⟩=α|α⟩
(3.11)
Hence we find that the coherent states are indeed eigenstates of the lowering operator, as a
consequence of the fact that we are operating on a superposition of an infinite number of energy
eigenstates. Another interesting aspect of coherent states is that they are associated with the
Poisson distribution. If we have the quantum harmonic oscillator in a coherent state, then the
probability of measuring its energy in that state to be En, is,
P(En) = |cn|2
=
e−|α|2αn
√n!
2
⇒P(En) = |α|2n
n!e−|α|2
(3.12)
This outcome indeed has the form of a Poisson distribution [3]. This particular property of
coherent states has important implications. For instance, in the context of quantum optics, it
implies that the coherent states of photons can be split into other independent coherent states
- something that can only happen with Poisson statistics.
4. The Time Evolution of Coherent States
In order to start thinking about how coherent states might evolve over time, we must first
realize that the Hamiltonian of the harmonic oscillator is time-independent. This means that
if the state at time 0 is equal to some coherent state |α0⟩, i.e.,
|ψ(0)⟩=|α0⟩=e−|α0|2/2
∞
∑
n=0
αn
0
√n!|n⟩(4.1)
then, the states at some later time tare given as,
6KHARANSHU SOLANKI
|ψ(t)⟩=e−|α0|2/2
∞
∑
n=0
αn
0
√n!e−iEnt/¯
h|n⟩(4.2)
For the harmonic oscillator, Enis given by equation (1.3). Therefore, we get,
|ψ(t)⟩=e−|α0|2/2
∞
∑
n=0
αn
0
√n!e−i¯
hω(n+1
2)t/¯
h|n⟩
⇒|ψ(t)⟩=e−iωt/2e−|α0|2/2
∞
∑
n=0α0e−iωtn
√n!|n⟩
(4.3)
Notice that if we define α=α0e−iωt, then |α|2=|α0|2, and hence we can write,
|ψ(t)⟩=e−iωt/2e−|α|2/2
∞
∑
n=0
αn
√n!|n⟩
⇒ |ψ(t)⟩=e−iωt/2|α⟩
(4.4)
This means that if the initial state of the harmonic oscillator is an eigenstate of the lowering
operator with eigenvalue α0, then the state of the harmonic oscillator at some later time twill
be an eigenstate of the lowering operator with the eigenvalue α0e−iωt. This leads to a key result
about the time dependence of coherent states. A coherent state remains coherent at all times.
5. Concluding Remarks
We have described the coherent states of a quantum harmonic oscillator - starting out by
defining them, and then proving that definition in the energy basis of a harmonic oscillator.
But a natural question that comes to the mind is that why did we not define coherent states as
the eigenstates of the raising operator instead? Well, it turns out [4] (and one can work this out
easily - the algebra is fairly similar to the one for the lowering operator) that if we try to expand
the eigenstates of the raising operator in the energy basis of the harmonic oscillator, then all the
expansion coefficients must necessarily vanish, and hence the only possible eigenstate is the
zero vector - which is a trivial solution (and hence not considered). In other words, the raising
operator has no eigenstates to begin with, and hence it is impossible to define coherent states in
terms of the raising operator.
Although we have defined coherent states only in the context of the harmonic oscillator, what
makes them really interesting is that they are the states that most closely resemble the classical
harmonic motion of the harmonic oscillator. In a sense, they provide some hints about the
relation between classical and quantum mechanics. More than anything else, they once again
convey the importance of the harmonic oscillator to quantum mechanics (and to physics in
general). For instance, the harmonic oscillator is used in modelling the motion of a charged
particle in a uniform magnetic field [5], in developing theory of angular momentum [6], and
also in more interesting topics like the bosonic, fermionic and supersymmetric oscillators. In
fact, in quantum field theory [7], one can define the so-called field operators, which also act
just like the harmonic oscillator.
COHERENT STATES IN QUANTUM MECHANICS 7
References
1. R.Shankar, Principles of quantum mechanics. New York, NY: Plenum, 1980. [Online]. Available:
https://cds.cern.ch/record/102017
2. Z.Nouredine, Quantum Mechanics: Concepts and Applications. Wiley, 2008.
3. M.Kardar, Statistical Physics of Particles. Cambridge University Press, 2007.
4. N.Wheeler, “Harmonic oscillator - revisited: Coherent states,” 12 2012. [Online].
Available: https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/
Miscellaneous%20Essays/Oscillator-Coherent%20States.pdf
5. S.Weinberg, Lectures on Quantum Theory. New York: Cambridge University Press, 2013.
6. .E.B. Schwinger, J., Quantum mechanics: Symbolism of atomic measurements. Berlin: Springer, 2001.
7. T.Lancaster and S.J. Blundell, Quantum field theory for the gifted amateur. Oxford: Oxford University
Press, Apr 2014. [Online]. Available: https://cds.cern.ch/record/1629337
