Section 4

Question 1

The time evolution of a quantum state of definite energy is

Answer 1

Ψ(t) = exp[-iEt/ℏ] Ψ(0)

Question 2

Unitary evolution

Answer 2

|Ψ(t)|^2 = |Ψ(0)|^2 : a stationary state

Question 3

The eigenvectors of stationary states are also

Answer 3

time-invariant

Question 4

Time-dependent state evolution

Answer 4

iℏ ∂Ψ(r,t)/∂t = H(hat)Ψ(r,t)

Question 5

eigenstate of the Hamiltonian

Answer 5

A stationary state has definite energy

Question 6

The Hamiltonian governs

Answer 6

the time-evolution of the state, and hence the expectation values

Question 7

Ehrenfest’s Theorem

Answer 7

is an example of the correspondence principle: QM limits to classical physics for large systems

Question 8

Newtonian momentum derivation

Answer 8

[x(hat),H(hat)] = [x(hat),p(hat)^2/2m + V(x)]

since x(hat) commutes with V(x(hat))

d<x>/dt = 1/2iℏm <[x(hat),p(hat)^2]></x>

=> <p> = m d<r>/dt</r>

Question 9

similarly derivation gives, [p(hat),H(hat)] =>

Answer 9

d<p>/dt = -<ΔV></ΔV>

Question 10

Stationary states are

Answer 10

super-important solutions of time-invarient probability density; time-evolution

Question 11

eigenvalue equation

Answer 11

H(hat) Ψi(x) = EiΨi(x)

Question 12

The time-independent schrodinger equation

Answer 12

[ -ℏ^2/2m d^2/dx^2 + V(x) ]Ψ(x) = EΨ(x)

Question 13

Probability conservation

Answer 13

dPtot(t)/dt = 0

Question 14

Schrodinger equation links

Answer 14

wave-function time evolution and spatial curvature

Question 15

The flux is a probability current

Answer 15

J(r,t) = ℏ/2im [Ψ∇Ψ - Ψ∇Ψ]

Question 16

Conservation equation

Answer 16

dP(r,t)/dt = -∇ . J(r,t)

Question 17

Any change in probability density at a point is

Answer 17

balanced by a flow of probability current into or out of that region

Question 18

∇Ψ(x) in definition of J

Answer 18

spatial wave-functions must be continuous, to avoid delta-function spikes in flux

Question 19

∇ . J in probability conservation

Answer 19

wave-functions must be differentiable everywhere, i.e. ∂Ψ/∂x must be continuous.