Quantum Mechanics: Intermediate Flashcards
Quantum simple harmonic oscillator hamiltonian
Commutator of raising and lowering operators
Raising and lowering operators
De Broglie relation for wavenumber
Infinite square well wavefunctions
Free particle wavefunction
Angular momentum and z compontent of angular momentum operators
Pauli spin matrices
Spin eigenvectors
Spin raising and lowering operators
How does the spin part of the wavefunction interact with the spatial part of the wavefunction?
It (essentially) doesn’t. The two are separated by a tensor product, and spin operators always commute with spatial operators.
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?
How do the z components of the spins of two particles add?
In a quantum mechanical sense, what is the difference between bosons and fermions?
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.
What is the first order energy shift to the nth energy level?
How does the variational principle work?
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.
Adiabatic theorem in quantum mechanics
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’.
Probability current density
