PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 2: Solutions
1. Consider the most general normalized spin-1/2 state: | = c+ |+ + c | , where Sz | = 2 | . a.) Compute Sx , Sy and Sz . b.) Compute the variances Sx , Sy , and Sz . c.) Prove tha
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 2
1. Consider the most general normalized spin-1/2 state: | = c+ |+ + c | , where Sz | = 2 | . a.) Compute Sx , Sy and Sz . b.) Compute the variances Sx , Sy , and Sz . c.) Prove that Sx = 2 |c
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 1: Solutions
1. [25 points] This problem shows you how to derive the matrix representations of spin operators from rst principles. a.) For spin 1/2, use the eigenvalue equation Sz |ms = ms |ms
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 1
1. [25 points] This problem shows you how to derive the matrix representations of spin operators from rst principles. a.) For spin 1/2, use the eigenvalue equation Sz |ms = ms |ms to nd the c
HOMEWORK ASSIGNMENT 10: Solutions
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: partial waves, scattering resonances.
1. Hard-sphere S-wave scattering: Consider S-wave scattering from a hard sphere of radius a. Make the ansatz eikr eikr (r
HOMEWORK ASSIGNMENT 10
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: partial waves, scattering resonances.
1. Hard-sphere S-wave scattering: Consider S-wave scattering from a hard sphere of radius a. Make the ansatz eikr eikr (r, , ) = (1
HOMEWORK ASSIGNMENT 9: Solutions
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Scattering amplitude, cross-section.
1. Use the Lippman-Schwinger equation, | = |0 + GV | , (1)
to solve the one-dimensional problem of resonant tunneling throu
HOMEWORK ASSIGNMENT 9
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Scattering amplitude, cross-section.
1. Use the Lippman-Schwinger equation, | = |0 + GV | , (1)
to solve the one-dimensional problem of resonant tunneling through two delt
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Greens functions, T-matrix.
1. The full Greens function: A system with hamiltonian H has a Greens function dened by GH (E ) = (E H + i)1 For case H = H0 + V , there is als
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Greens functions, T-matrix.
1. The full Greens function: A system with hamiltonian H has a Greens function dened by GH (E ) = (E H + i)1 For case H = H0 + V , there is als
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 7:
1. Spontaneous emission: In this problem, we will use the Fermi Golden Rule to estimate the spontaneous emission rate of an atom. We consider an atom which is initially excited, and therefor
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 7:
1. Spontaneous emission: In this problem, we will use the Fermi Golden Rule to estimate the spontaneous emission rate of an atom. We consider an atom which is initially excited, and therefor
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 6: SOLUTIONS
1. The ne structure consists of the relativistic mass correction, the spin-orbit interaction, and the Darwin term. Based on the fact that j = + 1/2 or j = 1/2, show that for all n,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 6:
1. The ne structure consists of the relativistic mass correction, the spin-orbit interaction, and the Darwin term. Based on the fact that j = + 1/2 or j = 1/2, show that for all n, , and j ,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 5: Solutions
1. [20 pts] The goal of this problem is to compute the Stark eect to rst-order for the n = 3 level of the hydrogen atoms. The stark shift is governed by the potential: VE = eE0 Z,
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 5:
1. [20 pts] The goal of this problem is to compute the Stark eect to rst-order for the n = 3 level of the hydrogen atoms. The stark shift is governed by the potential: VE = eE0 Z, so that yo
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 4:
1. [20 points] Consider the shifted harmonic oscillator: H= P2 1 + m 2 X 2 + aX. 2m 2
Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to rstorder.
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 4:
1. [20 points] Consider the shifted harmonic oscillator: H= P2 1 + m 2 X 2 + aX. 2m 2
Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to rst-order
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 3: Solutions
1. Cohen-Tannoudji problem 10.1, page 1086 Answer: a.) For the 1s ground state, we have = 0, s = 1/2, and i = 1. Thus the possible values of j run from jmin = | s| = 1/2 to jmax =
PHYS852 Quantum Mechanics II, Spring 2009
HOMEWORK ASSIGNMENT 3:
1. Cohen-Tannoudji problem 10.1, page 1086 2. Cohen-Tannoudji problem 10.5, page 1087 3. The exact normalized eigenvalues and eigenstates of the hamiltonian H = Sz + Sx are 1 2 + 2 = 2 2 + 2