Topic 3

Question 1

The full wavefunction solution when V=0?

Answer 1

[-ℏ/2m∂^2∂x^2Ψ(x, t) =EΨ(x, t) ⇒ ψ’‘(x) = −k^2ψ(x) where k^2 =2mE/ℏ > 0

• Solutions: plane waves ψ(x) = exp (±ikx) = exp(±ipx/ℏ)
representing a plane wave moving in the +ve (-ve) x direction with momentum p = ℏk > 0 and energy E = ℏ^2k^2/2m (JUST POSTION)

full
ψ(x,t) = exp( i/ℏ(Et±px) either +ve or -ve direction

one expression
ψ(x,t) = exp(- i/ℏ(Et-px)

Question 2

What is the wave equation solution for an IPW?

Answer 2

• Find the general solution
• Apply general conditions on the wave function:
  • ψ must be continuous
  • solutions are discrete (quantized) state
  • find momentum and then the energy
  • determine the normalisation constant A by requiring that the total probability must equal 1
• Write the function in full

Question 3

Key point on the symmetric Infinite Potential Well?

Answer 3

The wavefunctions can be obtained by taking the solutions of the IPW at 0 ≤ x ≤ L and making
a substitution x → x +L/2

The states with n odd are
symmetric w.r.t. the reflection x → −x, and the states
with n even are antisymmetric.
• The symmetric states have
even parity and antisymmetric states have odd parity