Section 2

Question 1

Provide motivation for TDSE by considering classical wave equation and modifications that would have to be made to it to satisfy equations E = ħω and E = p^2/(2m).

Answer 1

See pages 5-10 of section 2.

Question 2

Write down TDSE.

Answer 2

See page 10 of section 2.

Question 3

When is TDSE separable?

Answer 3

When V(x, t) := V(x)

corresponds to a closed system

Question 4

Use separation of variables to produce TISE from TDSE.

Answer 4

See pages 12-16 of section 2.

Question 5

Write down TISE.

Answer 5

See page 16 of section 2.

Question 6

Derive time dependence part of TDSE (time dependence for a closed system).

Answer 6

See pages 12-15 of section 2.

Question 7

If ψ(x) is a solution to TISE, then what is the time dependent solution to TDSE?

Answer 7

Ψ(x, t) = ψ(x)*exp(-iωt)

Question 8

Solve TISE in a region of constant potential, V(x) = V_0, for E > V_0, and show that the solutions are ψ(x) = exp(±ikx), where k = sqrt(2m * (E - V_0))/ħ.

Answer 8

See pages 19-21 of section 2.

Question 9

What is the general solution to TISE in terms of k and x?

Answer 9

Linear combination of individual solutions:

ψ(x) = Aexp(ikx) + Bexp(-ikx)

Question 10

What is an alternate form for the general solution ψ(x) = Aexp(ikx) + Bexp(-ikx)?

Answer 10

ψ(x) = Acos(kx) + Bsin(kx)