Phys 852, Quantum mechanics II, Spring 2008
Scattering theory: the T-Matrix approach
2/25/2008 Prof. Michael G. Moore, Michigan State University
1
Statement of the Problem:
Scattering theory is essentially time-independent perturbation theory applied to t
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion.
1. Spherical Bessel functions: The spherical Bessel function j () is dened as j () = (1) 1d d
sin .
Show that
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion.
1. Spherical Bessel functions: The spherical Bessel function j () is dened as j () = (1) 1d d
sin .
Show that
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion.
1. Spherical Bessel functions: The spherical Bessel function j () is dened as j () = (1)
1d d
sin .
Show tha
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion.
1. Spherical Bessel functions: The spherical Bessel function j () is dened as j () = (1)
1d d
sin .
Show tha
HOMEWORK ASSIGNMENT 8
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion.
1. Spherical Bessel functions: The spherical Bessel function j () is dened as j () = (1)
1d d
sin .
Show tha
HOMEWORK ASSIGNMENT 9
PHYS852 Quantum Mechanics II, Spring 2008 1. Use the Lippman-Schwinger equation: | = |0 + GV | (1)
to solve the one-dimensional problem of resonant tunneling through two delta-potentials. Take 0 (x) = eikx and V (x) = g [ (x) + (x L)
HOMEWORK ASSIGNMENT 9
PHYS852 Quantum Mechanics II, Spring 2008
1. Use the Lippman-Schwinger equation: | = |0 + GV | (1)
to solve the one-dimensional problem of resonant tunneling through two delta-potentials. Take 0 (x) = eikx and V (x) = g [(x) + (x L)]
HOMEWORK ASSIGNMENT 10
PHYS852 Quantum Mechanics II, Spring 2008
1. Dark State Adiabatic Passage:[Should be fairly easy] An atomic -system consists of two groundstate hyperne sub-levels couple via an electronically excited state. Let |1 and |2 refer to th
Phys 852, Quantum mechanics II, Spring 2008
Non-Degenerate Time-Independent Perturbation Theory
1/14/2008 Prof. Michael G. Moore, Michigan State University
1
The central problem in time-independent perturbation theory:
Let H0 be the unperturbed (a.k.a. ba
Phys 852, Quantum mechanics II, Spring 2008
Scattering theory
2/25/2008 Prof. Michael G. Moore, Michigan State University
1
Statement of the Problem:
Scattering theory is essentially time-independent perturbation theory applied to the case of a continuous
Phys 852, Quantum mechanics II, Spring 2008
Scattering theory
2/25/2008 Prof. Michael G. Moore, Michigan State University
1
Statement of the Problem:
Scattering theory is essentially time-independent perturbation theory applied to the case of a continuous
Phys 852, Quantum mechanics II, Spring 2008
Second Quantization: Non-Relativistic Quantum Field Theory
4/21/2008 Prof. Michael G. Moore, Michigan State University
In standard one-body quantum mechanics, the state of the system | is completely determined b
Phys 852, Quantum mechanics II, Spring 2008
Time-Dependent Perturbation Theory
1/14/2008 Prof. Michael G. Moore, Michigan State University
1
The central problem in time-dependent perturbation theory:
In time-independent perturbation theory, the object was
Phys 852, Quantum mechanics II, Spring 2008
Time-Independent Perturbation Theory
1/14/2008 Prof. Michael G. Moore, Michigan State University
1
The central problem in time-independent perturbation theory:
Let H0 be the unperturbed (a.k.a. background, bare)
HOMEWORK ASSIGNMENT 7
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Greens functions, T-matrix.
1. Weak-eld Zeeman Eect: Consider a hydrogen atom in a uniform magnetic eld. Assume that the Zeeman shift is large compared to the hyperne spli
HOMEWORK ASSIGNMENT 7
PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Greens functions, T-matrix. 1. Weak-eld Zeeman Eect: Consider a hydrogen atom in a uniform magnetic eld. Assume that the Zeeman shift is large compared to the hyperne spli
HOMEWORK ASSIGNMENT 1
PHYS852 Quantum Mechanics I, Spring 2008 1. Let the states |1 and |2 form an orthonormal basis spanning a 2-d Hilbert space. Let H = a|1 1| + b|2 2| + c|1 2| + d|2 1| be the Hamiltonian of the system. What condition on c and d does t
HOMEWORK ASSIGNMENT 1
PHYS852 Quantum Mechanics I, Spring 2008 1. Let the states |1 and |2 form an orthonormal basis spanning a 2-d Hilbert space. Let H = a|1 1| + b|2 2| + c|1 2| + d|2 1| be the Hamiltonian of the system. What condition on c and d does t
HOMEWORK ASSIGNMENT 2
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: 1st order perturbation theory 1. The two-level Rabi model is ubiquitous in quantum mechanics. In order to compare the results of perturbation theory with the exact results, nd
HOMEWORK ASSIGNMENT 2 Solutions
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: 1st order perturbation theory 1. The two-level Rabi model is ubiquitous in quantum mechanics. In order to compare the results of perturbation theory with the exact re
HOMEWORK ASSIGNMENT 3
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Time-independent perturbation theory up to 2nd order and the degenerate case
1. Orthogonality: Start from equation (33) in the lecture notes, and prove that in 2nd order nondeg
HOMEWORK ASSIGNMENT 3 SOLUTIONS
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Time-independent perturbation theory up to 2nd order (and higher for 2-level system)and the degenerate case 1. Orthogonality: Start from equation (33) in the lecture
HOMEWORK ASSIGNMENT 4
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Atomic Physics applications: STARK EFFECT, Zeeman eect, spin-orbit coupling
1. [15 pts] Compute the Stark eect to lowest non-vanishing order for the n = 3 level of the hydrogen
HOMEWORK ASSIGNMENT 4
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Atomic Physics applications: STARK EFFECT, Zeeman eect, spin-orbit coupling
1. [15 pts] Compute the Stark eect to lowest non-vanishing order for the n = 3 level of the hydrogen
HOMEWORK ASSIGNMENT 5
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Fine structure, hyperne structure, time-dependent perturbation theory
1. Consider a hydrogen atom in the ground state n = 1, = 0. Using the classical kinetic energy 2 formula T
HOMEWORK ASSIGNMENT 5
PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Fine structure, hyperne structure, time-dependent perturbation theory 1. Consider a hydrogen atom in the ground state n = 1, = 0. Using the classical kinetic energy 2 formula T