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PHYS4115 Quantum MechanicsII Fall 2014
Assignment 4: Time-Independent Perturbation Theory
The due date for this assignment is Friday 14 November, 2014.
1. Non-degenerate time-independent perturbation theory: Consider a perturbed
harmonic oscillator
2
2
h d
H = H0 + H = , where H0 = 2m
+ 12 kx2 is the unperturbed harmonic oscillator
dx2
Hamiltonian, and H = = Bpx is the time-independent perturbation.
(a) Find the …rst-order correction of the energy , En1 , due to H = .HINT: write px
as a linear combination of a (equation 2.47)
(b) Find the second-order correction of the energy, En2 ,due to H = .
(c) Find the …rst order correction the the wavefunction
1
n
(equation 6.13)
2. Degenerate time-independent perturbation theory:
Consider the 2D harmonic oscillator Hamiltonian
0
H =
1
h2 @ 2
+ kx2
2
2m @x
2
h2 @ 2
1
+ ky 2 ;
2
2m @y
2
with the eigensolution
nx ny nz
=
nx
(x)
ny
(y) ;
where nx (x) and ny (z)are the 1D harmonic oscillator eigensolutions in coordinate
x and y. The corresponding eigenvalue of nx ny nz = nx (x) ny (y) is
En0x ny = h! (nx + ny + 1) , nx ; ny = 0; 1; 2:::;
where ! =
q
k
.
m
Consider the 3-fold degenerate states
Consider the 3-fold degenerate states
Wavefunction nx ny Enx ny
2 0 3h!
1 = 20
1 1 3h!
2 = 11
=
0 2 3h!
3
02
Consider the time-dependent perturbation
Hz= = xy
(a) Calculate the 3
3 Wij = hij H = jji matrix.
(b) Find the …rst-order correction to the energy of the 3-fold degenerate states.
1
(c) Find the good states (the good linear combinations) of the 3-fold degenerate
states. Calculate the expectation value hxyi and hxy 2 i in each of these good
states.
3. Pseudo-Fine Structure. Consider the Bohr hydrogen solution with spin
perturbed by a time-independent potential energy (A is a constant)
H= =
n`m`
,
! !
A L S
r
(a) We wish to use …nd the energy correction by perturbation theory. Brie‡y explain
why it is valid to use non-degenerate perturbation theory for this problem
even though the n`m` solution or the alternate representation jn`jmj i solution
is degenerate.
HINT: 1) You must use the correct choice of "reference" state representation
!
! !
either n`m` = jn`m` ms i or jn`jmj i; 2) Note that J = L + S and J 2 =
! !
! !
L+S
L + S ; 3) to choose the correct reprersentation consider the commutation relations L2 ; H = , J 2 ; H = ; Lz ; H = ; Sz ; H = ; Jz ; H = , but you don’t
have to prove them (unless you want to), just recall them.
(b) Before proceeding we will need the Feynman-Hellman theorem (FHT). Consider a quantum solution H ( ) ( ) = E ( ) ( ), where = e; m; h:: are the
various parameters in the Hamiltonian, which will also be present in E and .
Prove that @hHi
= @E
. Justify all steps.
@
@
(c) Use FHT to …nd the …rst-order correction for the energy E 1 , in terms of A, the
2
quantum number n; j; `; s = 1=2, h and the Bohr’s radius a = 4me0eh2 . Even if you
weren’t successful in part b) you can still use FHT.
HINT: For FHT note the Bohr Hydrogen Hamiltonian is H0 =
Look carefully and choose the right parameter!
4. Problem 6.36
2
h2
r2
2m
4
e2 1
.
0 r

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