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Spin System - Quantum Mechanics - Exam, Exams of Quantum Mechanics

Dr. Bhim Rao Ambedkar UniversityQuantum Mechanics

This is the Past Exam of Quantum Mechanics which includes Wavefunction for Particle, Valid Wavefunction, Stern Gehrlach Device, Square Barrier, Spin System etc. Key important points are: Spin System, Energy Eigenstates and Eigenvectors, Hamiltonian Transform, Eigenstate Transform, Rotationally Invariant, Half-Integer Value, Hilbert Space, Clebsch Gordan Coefficients, Harmonic Oscillator

Typology: Exams

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Uploaded on 02/26/2013

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CTH/GU Fysik FKA081/FYN190
Quantum mechanics II Final exam
Time: Tuesday 24 oktober, 2006, 8:30
Place: V-huset
Examiner: Stellan ¨
Ostlund (0708-723201) (on travel)
Ling Bao (031-7723184)
Additional material: Your course text. You may not have extensive handwrit-
ten notes in the book. If you do, you will be asked to ex-
change the book for one without notes. You may also borrow
BETA , if necessary from each other if absolutely necessary.
Instructions:
I strongly suggest you go directly to the problems you know how to do and get them out
of the way. Think about substituting a symmetry or physical argument instead of messy
algebra; the algebra can thereby be drastically reduced in most of these problems. Please
choose one of Problem 6-alt 1 or Problem 6-alt 2 and indicate on the front which of those
two you want us to correct.
Problem 1 (10pt) - Spin system
The Hamiltonian for a spin-1 system is given by
H=AS2
z+B(S2
x−S2
y)
•(6 pt) Solve this problem exactly to find the energy eigenstates and eigenvectors.
•(2 pt) How does this Hamiltonian transform under time reversal
•(2 pt) How does each eigenstate transform under time reversal
Problem 2 (10pt) - Rotationally invariant H
The Hamiltonian for three spins is given by
H=SA·SB+SB·SC+SC·SA
For arbitrary value integer and half-integer value of the spin S,
•(2 ×1 pt )What is the size of the Hilbert space
•(2 ×2 pt ) What is largest eigenvalue of Hand what is the degeneracy
•(2 ×2 pt ) What is the smallest eigenvalues of Hand what is the degeneracy
1
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CTH/GU Fysik FKA081/FYN

Quantum mechanics II Final exam

Time: Tuesday 24 oktober, 2006, 8: Place: V-huset Examiner: Stellan Ostlund (0708-723201) (on travel)¨ Ling Bao (031-7723184) Additional material: Your course text. You may not have extensive handwrit- ten notes in the book. If you do, you will be asked to ex- change the book for one without notes. You may also borrow BETA , if necessary from each other if absolutely necessary.

Instructions:

I strongly suggest you go directly to the problems you know how to do and get them out of the way. Think about substituting a symmetry or physical argument instead of messy algebra; the algebra can thereby be drastically reduced in most of these problems. Please choose one of Problem 6-alt 1 or Problem 6-alt 2 and indicate on the front which of those two you want us to correct.

Problem 1 (10pt) - Spin system

The Hamiltonian for a spin-1 system is given by

H = AS z^2 + B(S x^2 − S^2 y )

  • (6 pt) Solve this problem exactly to find the energy eigenstates and eigenvectors.
  • (2 pt) How does this Hamiltonian transform under time reversal
  • (2 pt) How does each eigenstate transform under time reversal

Problem 2 (10pt) - Rotationally invariant H

The Hamiltonian for three spins is given by

H = SA · SB + SB · SC + SC · SA

For arbitrary value integer and half-integer value of the spin S,

  • (2 × 1 pt )What is the size of the Hilbert space
  • (2 × 2 pt ) What is largest eigenvalue of H and what is the degeneracy
  • (2 × 2 pt ) What is the smallest eigenvalues of H and what is the degeneracy

Problem 3 (10 pt) - Clebsch Gordan Coefficients

For the Zeeman-like Hamiltonian

HZ = α L · S + β(Lz + Sz )

write down exactly the eigenvalues and eigenfunctions in the | lz , sz 〉 basis for the case l = 1 and s = 1/2. The prefactor in front of Sz should really be 2, but I have made it 1 to keep the problem simple.

Problem 4 - Harmonic oscillator (10 pt )

Compute exactly the eigenvalues of the following Hamiltonian

H = a†a + λ(a + a†)

where [ a, a†^ ] = 1.

Problem 5 - Density Matrices ( 14 pt )

Unpolarized spin 1/2 atoms are coming out of an oven in three identical distinct beams. Each beam A,B and C is passed through a filter, each selecting atoms with polarization 120 degrees from each other, resulting in density matrices ρA , ρB and ρC. Assume that beam A is polarized along z. Recall the formula for entropy of a mixture S = −k T r ρ ln( ρ ).

  • (1 pt ) Consider doing an experiment on beam A only; Compute the density matrix. Is this a mixture or a pure state. What is the entropy?
  • (3 pt )Consider doing an experiment after having remixed A, B and C. What is the density matrix? Is this a mixture or pure state? What is the entropy?
  • (3 pt ) Consider mixing the output of B and C. Compute the density matrix. Is this a mixture or a pure state? Compute the entropy.

Now insert the mixture of B and C into a magnetic field B ˆx corresponding to a Hamiltonian H = μBSx.

  • (3 pt ) Compute the time evolution of the density matrix.
  • (3 pt ) Compute the expectation value of the spin vector in this mixture as a function of time.
  • (1 pt ) Physically interpret the time evolution of the spin and make sure you got the sign right in the time evolution equation.