13.1 - Revision QP

Question 1

What is an observable?

Answer 1

An observable A is a Hermitian operator which represents a physical quantity that can be measured. The real eigenvalues of A are the possible outcome values of measurement of this observable. Its eigenvectors represent the state of the system after measurement.

Question 2

What is spin precession?

Answer 2

A phenomenon in quantum physics wherein the probability of measuring a particular spin component changes with time.

Question 3

What is the orthonormality of the basis vectors |+> and |->?

Answer 3

= = 0

= = 1

Question 4

What is the expectation value of an operator, A, corresponding to an observable?

Answer 4

The expectation value of the operator, A, is the average result of repeated measurement. For a quantum state |Ψ> the expectation value may be calculated using:

<a> = </a>

Question 5

What are energy eigenvalues and eigenstates?

Answer 5

For a quantum system the energy eigenvalues E_n & the corresponding energy eigenstates |E_n> are defined by the eigenvalue equation.

H |E_n> = E_n |E_n>

Question 6

What is a stationary state?

Answer 6

A stationary state is an energy eigenstate. The time evolution of this state at time t=0 is

|Ψ(t)> = e^-iE_nt/ℏ * |E_n>

The expectation value of energy (or any other observable) is independent of time:

= E_n

Question 7

Are the energies in wave functions of a finite square well greater or lesser than those in an infinite square well?

Answer 7

Inside the well, the wave functions are sinusoidal, but they have somewhat larger wavelengths than the infinite square well.

This is required for the wave function to match smoothly to the decaying (non-zero) wave function outside the well.

The wavelength is the de Broglie wavelength λ = ħ / ρ. We have E = ρ^2/2m, so a longer wavelength implies smaller energies. The finite square well wave functions have smaller energies compared with the infinite case.

Question 8

What is the Born Rule?

Answer 8

The Born Rule gives a formula for the probabilities of the outcomes of quantum measurements. If a measurement outcome is represented by
|Φ> and the state is represented by |Ψ>. The probability to observe it is ||^2.

Question 9

What is the Schrodinger equation?

Answer 9

The Schrodinger equation is

iħd/dt|Ψ> = H|Ψ>

Where H is the Hamiltonian of the system, which is the observable representing total energy. It represents the time evolution of a quantum state.