Angular Momentum in Quantum Mechanics

Abstract

One of the lectures addressed to undergraduate students at Wright State University with integrated computer lab. My main objectives were: (1) To present Legendre polynomials and the associated Legendre functions as built-in within “Mathematica” package. (2) To calculate commutation relations between components of angular momentum operators. (3) To solve Angular Wave Equation by using the method of separation of variables, where azimuthal and polar parts are treated independently. (4) To visualize angular dependence of spherical harmonics and plot equal-probability surfaces.

Full text

Department of Physics, Wright State University
3640 Colonel Glenn Highway, Dayton, OH 45435
Dr. Kamil Walczak
PHY 4610: Lecture 1
“Angular Momentum in Quantum Mechanics”
To calculate commutation relations between
components of angular momentum operators
and to solve Angular Wave Equation.
To present Legendre polynomials and the associated
Legendre functions as built-in to “Mathematica”.
OBJECTIVES
To visualize angular dependence of spherical
harmonics and plot equal-probability surfaces.

Legendre Polynomials (1782)
01 / 08 / 2013Kamil Walczak
Adrien-Marie Legendre
French Mathematician
(1752-1833)
Legendre differential equation (second-order):
 
 
0)x(f1
dx )x(df
x2
dx )x(fd
x1 2
2
2 








x1 x1
ln)x(P
2
1
)x(Q 
(2) Legendre polynomial of the second kind:
(1) Legendre polynomial of the first kind
(Rodrigues formula/generator):



dx ])1x[(d
!21
)x(P 2

The general solution to the Legendre equation:
)x(QB)x(PA)x(f  
,...3,2,1,0

Legendre Polynomials: Properties
Kamil Walczak 01 / 08 / 2013
Legendre polynomials are orthogonal on the interval -1 ≤ x ≤ +1:
 k
1
1k1k2 2
dx)x(P)x(P 





Legendre polynomials at one point can be expressed by neighboring
Legendre polynomials at the same point via recurrence relations:
   
)x(Pn)x(P12x)x(P1 11    
 
)x(P)x(Px
1xdx )x(dP 1
2








 0
2y)x(P
yxy21
1



Legendre Polynomials are coefficients in Taylor series expansion (for |y| < 1):
(alternative definition)

Legendre Polynomials of the First Kind
Kamil Walczak 01 / 08 / 2013
Built-in Legendre polynomials
in “Mathematica” package:
}]1,1,x{],x,0[LegendreP[Plot 
}]1,1,x{],x,1[LegendreP[Plot 
}]1,1,x{],x,n[LegendreP[Plot 
..........
)x(P0
)x(P1
)x(P2
)x(P3
)x(P4
)x(P5
1P0
xP1
2
1
x
2
3
P2
2
x
2
3
x
2
5
P3
3
8
3
x
4
15
x
8
35
P24
4
x
8
15
x
4
35
x
8
63
P35
5
Analytical expressions:
..........

Kamil Walczak 01 / 08 / 2013
Legendre Polynomials of the Second Kind
Built-in Legendre polynomials
in “Mathematica” package:
}]1,1,x{],x,0[LegendreQ[Plot 
}]1,1,x{],x,1[LegendreQ[Plot 
}]1,1,x{],x,n[LegendreQ[Plot 
..........
)x(Q0
)x(Q1
)x(Q2
)x(Q3
)x(Q4
)x(Q5








x1 x1
ln
2
1
Q0
Analytical expressions:
..........
011 QPQ 
022 QPQ 

Associated Legendre Functions
Kamil Walczak 01 / 08 / 2013
Generalized Legendre’s differential equation (second-order):
 
0)x(f
x1m
)1(
dx )x(df
x2
dx )x(fd
x1 2
2
2
2
2







 
 
 
 
 
m
2m
2/m
2
m
m
m
2/m
2
m
mdx ])1x[(d
x1
!21
dx )x(Pd
x11)x(P 



 




(1) Associated Legendre function of the first kind:
(2) Associated Legendre function of the second kind:
 
m
m
2/m
2m dx )x(Qd
x1)x(Q 

The general solution to the generalized Legendre’s equation:
)x(QB)x(PA)x(f mm
m 

Legendre Functions: Properties
Kamil Walczak 01 / 08 / 2013
Associated Legendre functions are orthogonal on the interval -1 ≤ x ≤ +1:
 k
1
1
mm
k)!mk( )!mk(
)1k2( 2
dx)x(P)x(P 







Associated Legendre functions at one point can be expressed by neighboring
Legendre functions at the same point via recurrence relations:
     
)x(P1m)x(P12x)x(Pm m2
m1
m   
)x(P)x(P
x1
x]1m2[
)x(P]1m[]m[ 1m1m
2
m 


 







m
m
2/1m2m
m2/m2 y)x(P
)yxy1(!m2 y)x1()!m2(



Associated Legendre functions are coefficients in Taylor series expansion:
(alternative definition)

Associated Legendre Functions: Plots
Kamil Walczak 01 / 08 / 2013
Built-in Legendre Polynomials
in “Mathematica” package:
}]1,1,x{],x,0,3[LegendreP[Plot 
}]1,1,x{],x,1,3[LegendreP[Plot 
}]1,1,x{],x,m,[LegendreP[Plot  
..........
)x(P1
3
)x(P2
3
)x(P3
3
)x(P0
3
x
2
3
x
2
5
P30
3
 
221
3x51x1
2
3
P
 
22
3x1x15P
Analytical expressions:
 
2/3
23
3x115P

Momentum Operator
Kamil Walczak 01 / 08 / 2013
)r(up)r(u
x
ipxp





)r(up)r(up
ˆpp
 
The eigenvalue problem for momentum operator (position representation):
)r(up)r(u
y
ipyp





)r(up)r(u
z
ipzp





)r(up)r(ui pp



The wavefunction associated with free particle:





 rp
i
exp)r(up



















 z
ip
ˆ
,
y
ip
ˆ
,
x
ip
ˆzyx 
 
ip
ˆ

Orbital Angular Momentum
Kamil Walczak 01 / 08 / 2013
In classical mechanics, the angular momentum is defined as follows:
 
 
 
xyzzxyyzx
zyx
zyx ypxp1xpzp1zpyp1
ppp
zyx
111
detprL 










 



unit vectors (versors) along the particular axes (x,y,z)

zyx 1,1,1 
x
L
y
L
z
L
In quantum mechanics, the angular momentum is mathematically defined as
the cross product of the wavefunction’s position and momentum operator:
 

rip
ˆ
r
ˆ
L
ˆ
Representation in the form of components:
)L
ˆ
,L
ˆ
,L
ˆ
(L
ˆ
1L
ˆ
1L
ˆ
1L
ˆzyxzzyyxx  

Angular Momentum: Components
Kamil Walczak 01 / 08 / 2013
Particular components of angular momentum operator
(position representation):













 z
y
y
zip
ˆ
zp
ˆ
yL
ˆyzx 











 x
z
z
xip
ˆ
xp
ˆ
zL
ˆzxy 













 y
x
x
yip
ˆ
yp
ˆ
xL
ˆxyz 
The square of the amplitude of angular momentum vector:
2
z
2
y
2
x
2L
ˆ
L
ˆ
L
ˆ
L
ˆ

Angular Momentum: Commutators
Kamil Walczak 01 / 08 / 2013
]p
ˆ
xp
ˆ
z,p
ˆ
zp
ˆ
y[]L
ˆ
,L
ˆ
[zxyzyx 
 
   
 
yzzxzxyz p
ˆ
zp
ˆ
yp
ˆ
xp
ˆ
zp
ˆ
xp
ˆ
zp
ˆ
zp
ˆ
y
]L
ˆ
,L
ˆ
[]L
ˆ
,L
ˆ
[]LL
ˆ
L
ˆ
,L
ˆ
[]L
ˆ
,L
ˆ
[2
zx
2
yx
2
z
2
y
2
xx
2
x
]L
ˆ
,L
ˆ
[L
ˆ
L
ˆ
]L
ˆ
,L
ˆ
[]L
ˆ
,L
ˆ
[L
ˆ
L
ˆ
]L
ˆ
,L
ˆ
[zxzzzxyxyyyx 
 
zxy L
ˆ
ip
ˆ
yp
ˆ
xi  
Simultaneous knowledge of two components of angular momentum is impossible!
zyx L
ˆ
i]L
ˆ
,L
ˆ
[
xzy L
ˆ
i]L
ˆ
,L
ˆ
[
yxz L
ˆ
i]L
ˆ
,L
ˆ
[
Simultaneous knowledge of the length of the angular momentum vector
and one of its components is possible!
0L
ˆ
L
ˆ
iL
ˆ
L
ˆ
iL
ˆ
L
ˆ
iL
ˆ
L
ˆ
iyzzyzyyz  
0]L
ˆ
,L
ˆ
[2
x
0]L
ˆ
,L
ˆ
[2
y
0]L
ˆ
,L
ˆ
[2
z

Spherical Coordinates
Kamil Walczak 01 / 08 / 2013
)cos()sin(rx 
)sin()sin(ry 
)cos(rz 
222 zyxr 
)r/zarccos(
 
x/yarctan
Forward and reverse coordinate transformations:
Orbital angular momentum operators in terms of spherical coordinates:













 coscotsiniL
ˆx













 sincotcosiL
ˆy


 iL
ˆz
























 2
2
2
22
z
2
y
2
x
2sin
1
sin
sin
1
L
ˆ
L
ˆ
L
ˆ
L
ˆ

Scheme of Derivation
Kamil Walczak 01 / 08 / 2013


cossin
x
r


sinsin
y
r
rcoscos
x




rsincos
y









sinrsin
x





sinr
cos
y


cos
z
r
r
sin
z




0
z


Derivatives of spherical coordinates with respect to Cartesian coordinates:






































xxrx
r
sinsinri
y
x
x
yiL
ˆz
Exemplary computation of z-component of angular momentum:




























 i
yyry
r
cossinr

Solution for Z-Component
Kamil Walczak 01 / 08 / 2013
The eigenvalue problem for the z-component of angular momentum:
z
Li  
 /)2(iL/iL zz AeAe  

 Ae
The wavefunction should be single valued:
 zz LL
ˆ
After substitution we obtain:
Let us assume the solution in the form:
)(L
d)(d
iz






Ae
d
d

z
iL

)2()( 
1e /iL2z



 im
me
2
1
)(
,...2,1,0m 
with
mLz
and

Angular Wave Equation
Kamil Walczak 01 / 08 / 2013
The eigenvalue problem for the square of angular momentum:
),(Y)1(),(Y
)(sin1
)sin(
)sin(
12
2
2
2
2























2
2
2),(Y
),(Y)(sin)1()sin()sin( 




















Y)1(YL
ˆ22  
Multiplying both sides of above equation by and regrouping:
YLYL
ˆ22 
Expressing operator in spherical coordinates, we obtain Angular Wave Equation:
Let us express the eigenvalues of the square of angular momentum as:








2
2)(sin


Separation of Variables
Kamil Walczak 01 / 08 / 2013
The eigenvalue problem for the square of angular momentum:
2
2
2d)(d
)(
1
sin)1(
d)(d
sin
d
d
sin
)(
1




















 
2
m
(function of polar angle) (function of azimuthal angle)
The general solution is a product of two functions (separation of variables method):
)()(),(Y 
 
0)(msin)1(
d)(d
sin
d
d
sin 22 
















0)(m
d)(d 2
2
2


Polar part:
Azimuthal part:

Solution for Azimuthal Part
Kamil Walczak 01 / 08 / 2013
The second equation for azimuthal angle:
0)(m
d)(d 2
2
2




 im
me
2
1
)(
Normalized solution for the azimuthal part of Angular Wave Equation:
,...2,1,0m 
with
The first equation for polar angle:
 
0)(msin)1(
d)(d
sin
d
d
sin 22 
















 
0)(msin)1(
d)(d
cossin
d)(d
sin 22
2
2
2







Introducing New Variable
Kamil Walczak 01 / 08 / 2013
Dividing the above equation by we obtain:
0)(
)(cos1 m
)1(
d)(d
)cot(
d)(d 2
2
2
2












 
Let us change the variable:
)cos(x 


d)sin(d
d
dx
dx

 d
d
)sin(
1
dx
d
dx
d
)sin(
d
d

dx
d
)cos(
dxdd
)sin(
dx
d
)sin(
d
d
d
d
d
d
d
d2
2
2



















dx
d
)cos(
dx
d
)(sin
d
d2
2
2
2
2

)(sin2

Solution for Polar Part
Kamil Walczak 01 / 08 / 2013
Angular Wave Equation becomes generalized Legendre equation:
 
0)x(
x1m
)1(
dx )x(d
x2
dx )x(d
x1 2
2
2
2
2












General solution is a linear combination of associated Legendre functions:
)(cosQB)(cosPA)(cos mmm  
(non-normalizable
within the interval
-1 ≤ x ≤ +1)
1dx)x(P)x(PA 1
1
mm2 




)!m( )!m(
)12( 2





)!m( )!m(
2)12(
A



Normalization condition:
(orthogonality
condition)
Normalization factor:

Spherical Harmonics
Kamil Walczak 01 / 08 / 2013
)(cosP
)!m( )!m(
2)12(
)(cos mm 


  

Normalized solution for the polar part of Angular Wave Equation (AWE):
,...3,2,1,0
with
}]2,0,{},,0,{]],,,m,[armonicYSphericalH[Re[D3Plot  
Built-in Spherical Harmonics in “Mathematica” package:
The general solutions of AWE are Spherical Harmonics:
)(cosPe
)!m( )!m(
4)12(
)1(),(Y mimm
m




 
 

with
,...3,2,1,0
and
,...,2,1,0m 
),(Y)1(),(Y *
m
m
m, 


Spherical Harmonics: Angular Dependence
Kamil Walczak 01 / 08 / 2013
 
10
Ye


 
10
Ym




2
10
Y

Kamil Walczak 01 / 08 / 2013
Spherical Harmonics: Angular Dependence
 
11
Ye


 
11
Ym




2
11
Y

Kamil Walczak 01 / 08 / 2013
Spherical Harmonics: Angular Dependence
 
31
Ye






2
31
Y
 
31
Ym

Kamil Walczak 01 / 08 / 2013
Spherical Harmonics: Angular Dependence
 
32
Ye






2
32
Y
 
32
Ym

Angular Momentum: Visualization
Kamil Walczak 01 / 08 / 2013
2Lz

z
L
0Lz

z
L
2Lz
B

2Lz

z
L
0Lz

z
L
2Lz
z
z
6|L| 
2
Eigenvalue Equations:
m
2
m
2Y)1(YL
ˆ  
mmz YmYL
ˆ 

Equal-Probability Surfaces
Kamil Walczak 01 / 08 / 2013
2
00 |Y|
2
10 |Y|
2
11 |Y|
2
20 |Y|
2
21 |Y|
2
22 |Y|
z
z
z
z
z
z

Equal-Probability Surfaces
Kamil Walczak 01 / 08 / 2013
2
30 |Y|
2
31 |Y|
2
33 |Y|
2
32 |Y|
z
z
z
z

Concluding Remarks
Kamil Walczak 01 / 08 / 2013
Angular momentum is quantized; it is not varying continuously, but via “quantum
leaps” (or some abrupt changes) between certain allowed values!
Angular momentum is characterized by two quantum numbers (orbital and
magnetic), while its natural unit is reduced Planck’s constant (quantum of action)!
Two orthogonal components of angular momentum can not be simultaneously
known or measured, except of the trivial case when all of them are zero!
Total angular momentum is always conserved (due to isotropy of space), while
orbital angular momentum may be not (spin-orbit coupling may transfer angular
momentum between orbital and spin degrees of freedom)!
Angular momentum quantization rule is applicable even to macroscopic systems;
however, its discrete steps at macroscale are too small to be distinguished!
Simultaneous knowledge of the length of the angular momentum vector
and one of its components is possible!