Bound states for a half harmonic oscillator

[praharmitra]

We have a potential that is (1/2)kx^2 for x>0 and is infinity for x<0 ( half harmonic oscillator.

Now i want to calculate the bound states of the system for given E. My question is this:

Do we apply

1. $$\int p(x) dx = (n - \frac{1}{4} ) h$$ ( Since there is only one turning point that can have a connection formula. This is what is given in my book

2. $$\int p(x) dx = (n - \frac{1}{2} ) h$$ (This is what my teacher mentioned. I am not too sure about this.)

In both cases the integral is take over the entire classical path of the particle.

 

[Avodyne]

I'm not sure why you're messing around with semiclassical formulae when it's easy to solve this problem exactly ... Hint: on the right, an energy eigenfunction must be the same as an energy eigenfunction for the full oscillator (because they obey the same time-independent Schrödinger equation in this region), but the wall at x=0 means that the only allowed eigenfunctions must also have the property that ____________ (fill in the blank!). Only some of the eignefunctions of the full oscillator have this property ...

 

[Count Iblis]

Do what Avodyne says. But then you must also show that the set of eigenfunctions you get this way are all the eigenfunctions there are. You can do that by showing that any arbitrary eigenfunction for the half harmonic oscillator can be used to definine an eigenfunction for the full harmonic oscillator.