Quantum Mechanics: Intermediate

Question 1

Quantum simple harmonic oscillator hamiltonian

Answer 1

Question 2

Commutator of raising and lowering operators

Answer 2

Question 3

Raising and lowering operators

Answer 3

Question 4

De Broglie relation for wavenumber

Answer 4

Question 5

Infinite square well wavefunctions

Answer 5

Question 6

Free particle wavefunction

Answer 6

Question 7

Angular momentum and z compontent of angular momentum operators

Answer 7

Question 8

Pauli spin matrices

Answer 8

Question 9

Spin eigenvectors

Answer 9

Question 10

Spin raising and lowering operators

Answer 10

Question 11

How does the spin part of the wavefunction interact with the spatial part of the wavefunction?

Answer 11

It (essentially) doesn’t. The two are separated by a tensor product, and spin operators always commute with spatial operators.

Question 12

Suppose you have two particles, one of spin s, the other of spin s’. What are the possible total spins for the overall system of the two?

Answer 12

Question 13

How do the z components of the spins of two particles add?

Answer 13

Question 14

In a quantum mechanical sense, what is the difference between bosons and fermions?

Answer 14

Under the interchange of two identical particles, boson wavefunctions are symmetric, while fermion wavefunctions are antisymmetric. As such, no two identical fermions can occupy the exact same quantum state.

Question 15

What is the first order energy shift to the nth energy level?

Answer 15

Question 16

How does the variational principle work?

Answer 16

It is based on the observation that for any normalized wavefunction, the expectation of the energy is greater than or equal to the ground state energy. Thus, to find the ground state energy, we pick a “trial wavefunction” which can be adjusted by some parameter (say, a Gaussian of variable standard deviation), then minimize the expectation of the energy of this wavefunction with respect to the given parameter.

Question 17

Adiabatic theorem in quantum mechanics

Answer 17

Suppose a particle is in the nth eigenstate of a Hamiltonian H. If H is slowly changed to H’, then at the end of the process the particle will end up in the nth eigenstate of H’.

Question 18

Probability current density

Answer 18