Examples to practice

Question 1

Infinite square well

Answer 1

Potential is zero between 0 and a, and infinite outside that range.
ψ(0) = 0 and ψ(a) = 0
use TISE with ψ(x) = Asin(kx) + Bcos(kx), with k^2 = 2mE/ħ^2
quantised energy should be Eₙ = 1/2m (nπħ/a)^2, n >= 1

Question 2

Find ground state wavefunction by applying destruction operator

Answer 2

Apply lowering operator to ground state ψ₀.
Realise that âψ₀ must equal zero, as no state can hae less energy than ψ₀.
Solve resulting first-order linear differential equation.
Should get ψ₀ = Aexp(-x² / 2a²) where a=root(ħ/mω)
All other wavefunctions can be constucted from this one.