1. Chap 4. Quantum Mechanics In Three Dimensions
Schrodinger Equation in Spherical Coordinates
The Hydrogen Atom
Angular Momentum
Spin

2. 4.1. Schrodinger Equation in Spherical Coordinates
Separation of Variables
The Angular Equation
The Radial Equation
Read Prob 4.1

3. Orthogonal Curvilinear Coordinates
Ref: G.Arfken, “Mathematical Methods for Physicists”, 3rd ed., Chap. 2.
B.Schutz, “Geometrical Methods of Mathematical Physics”, p.148.
Spherical coordinates :

4. 4.1.1. Separation of Variables
V  V(r)
 Spherical coordinates : 
Ansatz : 

5. Set
  dimensionless constant Or

6. Mnemonics 
Do Prob 4.2

7. The Angular Equation
Ansatz : 
Set 

8. Azimuthal Solutions 
 single-valued, i.e.,  

9. Legendre Polynomials  Setting m  0 gives
Frobenius method shows that convergence requires
See Arfken (3rd ed) Ex 8.5.5
The corresponding solutions are called the Legendre polynomials, which can also be defined by the Rodrigues formula :

10. 1st few Legendre Polynomials
Normalization:
Pl (1) = 1

11. Associated Legendre Functions
Solutions to the m  0 case :
are called associated Legendre functions defined by
Griffiths:
where
Arfken,
Mathenmatica:
Thus
while m takes on 2l + 1 values :
Note : Another independent solution exists but is not physically acceptable ( see Prob. 4.4 ).

12. 1st few Associated Legendre Polynomials

13. Normalization Spherical Harmonics Normalization : Griffiths : where
Arfken,
Mathenmatica:
Note :
Orthonormality :

14. 1st few Spherical Harmonics
Do Prob 4.3
Read Prob 4.4, 4.6

15. 4.1.3. The Radial Equation Set    Effective potential
Centrifugal term
Normalization :

16. Example 4.1. Infinite Spherical Well
Let
Find the wave functions and the allowed energies.
Ans : 
jl = spherical Bessel function
nl = spherical Neumann function

17. Spherical Bessel & Neumann Functions
jl (0) is finite
nl (0)   
Let  n l be the nth zero of jl . 
(2l+1)-fold degeneracy in m.

18. 1st Few Spherical Bessel & Neumann Functions
Do Prob 4.9

19. Bessel & Neumann Functions
The Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in cylindrical coordinates :    
Bessel Neumann functions
Modified Bessel functions
( for 2 < 0 )

20. Spherical Bessel & Neumann Functions
The spherical Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in spherical coordinates :     
Spherical Bessel Neumann functions

21. Asymptotic Forms
for x  0
for x  

22. The Radial Wave Function The Spectrum of Hydrogen
4.2. The Hydrogen Atom 
The Radial Wave Function
The Spectrum of Hydrogen

23. Bohr’s Model Circular orbit :   Quantization of angular momentum : 
Bohr radius

24. 4.2.1. The Radial Wave Function
Bound States ( E < 0 ) : Set 
Set 

25. Asymptotic Behavior    :  u finite everywhere    0 : 
Set

26. Factor-Out Asympototic Behavior

27. Frobenius Method   

28. Series Termination j   :    ( unacceptable for large  )
 Series must terminate :

29. Eigenenergies Let = principal quantum number   n  1, 2, 3, ...
Bohr radius : 

30. Eigenfunctions
Eigenfunction belonging to eigenenergy is
where
with

31. Ground State
n  1, l  0.
Normalization : 

32. 1st Excited States n  2, l  0, 1. m  1, 0, 1
Normalization : see Prob. 4.11
Degeneracy of nth excited state :

33. Associated Laguerre Polynomials
qth Laguerre polynomial ; Used by Griffiths.
Used by Arfken & Mathematica.
1/n! of Griffiths’ value.
Associated Laguerre polynomial
Used by Arfken & Mathematica.
1/(n+p)! Griffiths’ value.
Differential eqs. : 

34. 1st Few Laguerre & Associated LaguerrePolynomials
Ln : Arfken & Mathematica convention
 Griffiths’ / n!
Lna : Arfken & Mathematica convention
 Griffiths’ / (n+a)!

35. Orthogonal Polynomials
Ref: M.Abramowitz, I.A.Stegun, “Handbook of Mathematical Functions”, Chap 22.
Orthogonality:
w  weight function
Differential eq.:
Recurrence relations:
Rodrigues’ formula: fn
(a,b) en w g
Standard An Pn
(1,1)
()n n! 2n 1
x2  1
Pn(1) = 1
2 / (2n+1) Ln
( 0,  )
e x x
Lnp
()p
e x xp
(p+n)! / n! Hn
( ,  )
()n
exp(x2/2)
en = (1)n
n! 2

36. Hydrogen Wave Functions
Griffiths
convention
Arfken
convention :
[3rd ed., eq(13.60)]
Orthonormality :
Arfken

37. First Few Rnl (r)

38. Rnl Plots (n 0) has n1 nodes (n, n1) has no node
Note: Griffiths’ R31 plot is wrong.

39. Density Plots of 4 l 0 (400) (410) (430) (420) (n 0 m) has n1 nodes
White = Off-scale
(400)
(410)
(n 0 m) has n1 nodes
(n, n1, m) has no node
(430)
(420)
White = Off-scale

40. Surfaces of constant | 3 l m |
(300)
(322)
(320)
(310)
Warning : These are plots of | |, NOT | |2 .
(321)
Do Prob 4.13,
4.15.

41. 4.2.2. The Spectrum of Hydrogen
H under perturbation  transition between “stationary” levels:
energy absorbed : to higher excited state
energy released : to lower state
H emitting light ( Ei > Ef ) :
Planck’s formula : 
Rydberg formula
where
Rydberg constant

42. H Spectrum nf  1 nf  2 nf  3 Series Lyman Balmer Paschen RadiationUV
Visible IR

43. Eigenvalues Eigenfunctions
4.3. Angular Momentum
CM :
QM :
Eigenvalues
Eigenfunctions

44. Commutator Manipulation
distributive 
Similarly

45. [ Li , Lj ]  
Cyclic permutation :

46. Uncertainty Principle
Only one component of L is determinate.

47. [ L2, L ] Similarly for i  x, y, z i.e.
 L2 & Lz share the same eigenfunction :

48. 4.3.1. Eigenvalues Ladder operators :  Let 
L f is an eigenfunction of L2.
L f is an eigenfunction of Lz.
L raiseslowers eigenvalue of Lz by .

49. Lz finite   max  max().
Let 
Also
Now   
Lz finite   min  min().
Let 
Also    

50. Since 
Let
N = integer
max must be integer or half integer  or
Let 
where

51. Diagram Representation of L
L can’t be represented by a vector fixed in space since only ONE of it components can be determinate.

52. Eigenfunctions
Gradient in spherical coordinates : 

53. Do Prob. 4.21
Read Prob 4.18, 4.19, 4.20
Do Prob 4.24

54. Spin 1/2 Electrons in a Magnetic Field Addition of Angular Momenta

55. Spin
Spin is an intrinsic angular momentum satisfying
with

56. 4.4.1. Spin 1/2 2-D state space spanned by (spin up ) (spin down )
In matrix form (spinors) :
General state :
Operators are 22 matrices.   

57. Pauli Matrices   
Pauli matrices

58. Spin Measurements Let particle be in normalized state : with
Measuring Sz then has a probability | a |2 of getting /2,
and probability | b |2 of getting /2,
Characteristic equation is
Eigenvalues : 
Eigenvectors :  

59. Writing  in terms of (x) : 
Measuring Sx then has a probability |  |2 of getting /2,
and probability |  |2 of getting /2,
Read last paragraph on p.176.
Do Prob 4.26, 27
Read Prob 4.30

60. 4.4.2. Electrons in a Magnetic Field
Ampere’s law : Current loop  magnetic moment .
Likewise charge particle with angular momentum. 
  gyromagnetic ratio
QM : Spin is an angular momentum :
 experiences a torque when placed in a magnetic field B :
 tends to align  with B,
i.e.,  // B is the ground state with   0 
QM : L has no fixed direction   can’t be aligned to B
 Larmor precession

61. Example 4.3. Larmor Precession
Consider a spin ½ particle at rest in uniform
The Hamiltonian is +  H
E+   B0  / 2
E  + B0  / 2 Sz
 / 2
 / 2
H & S share the same eigenstates :
Time evolution of is

62. 
if a, b are real
Sinilarly

63. Set 
i.e.,  S  is tilted a constant angle  from the z-axis,
and precesses with the Larmor frequency
( same as the classical law )

64. Example 4.4. The Stern-Gerlach Experiment
Force on  in inhomogeneous B :
Consider a particle moving in the y-direction in a field
Note : The  x term is to make sure
Due to precession about B0 , Sx oscillates rapidly & averages to zero.
 Net force is
 incident beam splits into two.
In contrast, CM expects a continuous spread-out.

65. Alternative Description
In frame moving with particle :
x component dropped for convenience
Let
for t  0 
for 0  t  T
where 
for t > T

66. for t > T 
spin up particles move upward
spin up particles move downward
S-G apparatus can be used to prepare particles in particular spin state.

67. 4.4.3. Addition of Angular Momenta
The angular momentum of a system in state  can be found by writing
where
is proportional to the probability of measuring
If the system has two types of angular momenta j1 and j2 , its state can be written as
where
The total angular momentum of the system is therefore described by the quantities

68. Since either set of basis is complete, we have
The transformation coefficients
are called Clebsch-Gordan coeffiecients (CGCs).
The problem is equivalent to writing the direct product space
as a direct sum of irreducible spaces

69. 4.4.3. Addition of Angular Momenta
Rules for adding two angular momenta : j1 j2
j1j2
J1J2...Jn
State
| j1 m1 
| j2 m2 
| j1 m1  | j2 m2 
| J1 M1  ...| Jn Mn 
Possible values of S are
j1 + j2 , j1 + j2 1, ..., | j1  j2 | + 1, | j1  j2 |
Only states with M = m1 + m2 are related.
Linear transformation between | j1 m1  | j2 m2  and | Jk Mk  can be obtained by applying the lowering operator to the relation between the “top” states.
Coefficients of these linear transformation are called the Clebsch-Gordan ( C-G ) coefficients.

70. Example: Two Spin ½ Particles
s1 = ½ , s2 = ½  s1 + s2 = 1, s1  s2 = 0
 Possible total S  1, 0 s1 s2
s1s2
S1 S2
States
|  
|  
|   
| 1 1 
| 1 0 
| 1 1 
| 0 0 
|  
|   
|   
|   

71. “Top” state for S = 1 :  
“Top” state for S = 0 :
must be orthogonal to | 10 .
Normalization then gives

72. J1 J2
...
M = m1 + m2
m1 , m2
M 

73. Clebsch-Gordan ( C-G ) coefficients
Shaded column gives
Shaded row gives
Sum of the squares of each row or column is 1.
Do Prob 4.36