Chap 4. Quantum Mechanics In Three Dimensions

1
Chap 4. Quantum Mechanics In Three Dimensions

1. Schrodinger Equation in Spherical Coordinates
2. The Hydrogen Atom
3. Angular Momentum
4. Spin

2
4.1. Schrodinger Equation in Spherical
Coordinates

1. Separation of Variables
2. The Angular Equation
3. 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
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4.1.2. 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
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Associated Legendre Functions
Solutions to the m ? 0 case
are called associated Legendre functions defined
by
where
Griffiths
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
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Example 4.1. Infinite Spherical Well
Find the wave functions and the allowed energies.
Let
Ans
?
jl spherical Bessel function nl spherical
Neumann function
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Spherical Bessel Neumann Functions
jl (0) is finite
nl (0) ? ?
?
Let ? n l be the nth zero of jl .
?
(2l1)-fold degeneracy in m.
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1st Few Spherical Bessel Neumann Functions
Do Prob 4.9
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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 lt 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 ? ?
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4.2. The Hydrogen Atom
?

1. The Radial Wave Function
2. The Spectrum of Hydrogen

23
Bohrs Model
Circular orbit
?
?
Quantization of angular momentum
?
?
Bohr radius
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4.2.1. The Radial Wave Function
Bound States ( E lt 0 ) Set
?
Set
?
25
Asymptotic Behavior
? ? ?
?
u finite everywhere ?
? ? 0
?
u finite everywhere ?
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
?
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Eigenfunctions
Eigenfunction belonging to eigenenergy
is
where
with
31
Ground State
n ? 1, l ? 0.
Normalization
?
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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/(np)! Griffiths
value.
Differential eqs.
?
34
1st Few Laguerre Associated LaguerrePolynomials

Ln Arfken Mathematica convention ?
Griffiths / n!
Lna Arfken Mathematica convention ?
Griffiths / (na)!
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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 / (2n1)
Ln ( 0, ? ) 1 e? x x 1
Lnp ( 0, ? ) (?)p e? x xp x (pn)! / n!
Hn ( ??, ? ) (?)n exp(?x2/2) 1 en (?1)n n! ?2?
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Hydrogen Wave Functions
Griffiths convention
Arfken convention 3rd ed., eq(13.60)
Orthonormality
Arfken
37
First Few Rnl (r)
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Rnl Plots
(n 0) has n?1 nodes (n, n?1) has no node
Note Griffiths R31 plot is wrong.
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Density Plots of ?4 l 0
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)
Do Prob 4.13, 4.15.
(321)
Warning These are plots of ? , NOT ? 2 .
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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 gt Ef )
Plancks formula
?
Rydberg formula
where
Rydberg constant
42
H Spectrum
nf ? 1 nf ? 2 nf ? 3
Series Lyman Balmer Paschen
Radiation UV Visible IR
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4.3. Angular Momentum
CM
QM

1. Eigenvalues
2. Eigenfunctions

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Commutator Manipulation
distributive
?
?
Similarly
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Li , Lj
?
?
Cyclic permutation
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Uncertainty Principle
?
Only one component of L is determinate.
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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
?
?
?
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Since
?
N integer
Let
?max must be integer or half integer
?
or
Let
?
where
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Diagram Representation of L
l ? 2
L cant be represented by a vector fixed in space
since only ONE of it components can be
determinate.
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4.3.2. Eigenfunctions
Gradient in spherical coordinates
?
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Do Prob. 4.21
Read Prob 4.18, 4.19, 4.20
Do Prob 4.24
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4.4. Spin

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

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Spin
Spin is an intrinsic angular momentum satisfying
with
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4.4.1. Spin 1/2
(spin down )
2-D state space spanned by
(spin up )
In matrix form (spinors)
General state
Operators are 2?2 matrices.
?
?
?
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Pauli Matrices
?
?
?

Pauli matrices
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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
?
?
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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
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4.4.2. Electrons in a Magnetic Field
Amperes 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 ? ? cant be
aligned to B ? Larmor precession
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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
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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 )
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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.
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Alternative Description
In frame moving with particle
x component dropped for convenience
for t ? 0
Let
for 0 ? t ? T
?
where
?
for t gt T
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for t gt T
?
spin up particles move upward
spin up particles move downward
S-G apparatus can be used to prepare particles in
particular spin state.
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4.4.3. Addition of Angular Momenta
The angular momentum of a system in state ? can
be found by writing
is proportional to the probability of measuring
where
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
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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 S1? S2
States ?? ? ? ? ? ? ? 1 1 ? 1 0 ? 1 ?1 ? 0 0 ?
States ?? ? ? ? ? ? ? 1 1 ? 1 0 ? 1 ?1 ? 0 0 ?
States ? ? ?? ? ? ? ? 1 1 ? 1 0 ? 1 ?1 ? 0 0 ?
States ? ? ? ? ? ? ? 1 1 ? 1 0 ? 1 ?1 ? 0 0 ?
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Top state for S 1
?
?
Top state for S 0
must be orthogonal to 10 ?.
Normalization then gives
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J1 J2 ...
M m1 m2 M m1 m2 M m1 m2
m1 , m2
M ?
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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