Question

For the ground-level harmonic oscillator wave function given by ?(x)=Cexp(?mk??x22?), |?|^2 has a maximum at x=0.

a. Compute the ratio of |?|^2 at x=+A to |?|^2 at x=0, where A is given by 12/k?A^2=(n+1/2)?? with n=0 for the ground level.

b. Compute the ratio of |?|2 at x=+2A to |?|^2 at x=0.

Answer 1

The ground state wave function is

where

where c is the normalization constant

The turning point is given the point where the energy is equal to the potential energy

Hence the positive turning point is

a) Therefore the wave function at x = 0 is

and hence

Wavefunction at the turning point is

Therefore

Hence the ratio is

b) Note that

And hence

Therefore the ratio is