Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

[thegirl]

Hi,

why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity.

I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and therefore the ground state energy level for the half harmonic oscillator is 3(h bar omega)/2.

I don't get why there wouldn't be even eigenfunctions and energy levels for n=2,4,6 etc.

 

[Nugatory]

I don't get why there wouldn't be even eigenfunctions and energy levels for n=2,4,6 etc.

The infinite potential at 0 implies that $\psi(0)=0$. The even solutions are all non-zero at that point, so they do not satisfy that boundary condition.

 

[thegirl]

Thank You!