Homework Help: Zero point energy of particle

1. Oct 30, 2007

physgirl

1. The problem statement, all variables and given/known data

find the zero point energy of particle of mass m in one dimension under:
V(x)=(1/2)kx^2 for x>0; infinity for x<0

2. Relevant equations
??

3. The attempt at a solution

I'm completely lost :( I took a similar approach to deriving eigenvalues for a particle in a box but that didn't make sense to me... Any starters?

2. Oct 30, 2007

George Jones

Staff Emeritus
Do you recognize the potential for x>0?

3. Oct 30, 2007

physgirl

its the potential for a harmonic oscillator?

4. Oct 30, 2007

physgirl

hmm its a harmonic oscillator in a box with infinite length ?!

5. Oct 30, 2007

George Jones

Staff Emeritus
Yes, so, for x>0, the wavefunction has to satisfy the same time-independent Schrodinger that the wavefunction for a harmonic oscillator satisfies. But now there is an extra boundary condition.

6. Oct 30, 2007

physgirl

so it's a harmonic oscillator that has no access to x values less than 0?

7. Oct 30, 2007

physgirl

So you would just use the operator H=-(hbar)/(2m)[d^2/dx^2]+(1/2)kx^2
Plug that into the Schrodinger equation... but when do you impose the new boundary condition....?

8. Oct 30, 2007

George Jones

Staff Emeritus
Exactly.

Yes, and you know the solution to $H \psi = E \psi$ for this potential.

Think about solving the Schrodinger equation for a particle in a box. For what region (in terms of the potential) is this equation solved? Why are boundary conditions imposed for a particle in a box?

9. Oct 30, 2007

physgirl

I thought for a box particle, really the only significant boundary condition was that the wavefunction has to be continuous. Therefore the cosine term has to disappear to leave with just the sine term that WILL go to 0 when x=0... I'm looking at the solution for the oscillator SE and I just see: (a_n)(H_n)(e^-(1/2)(alpha*x)^2)
I'm not really sure how to impose a boundary condition that says x cannot be below 0...?

10. Oct 30, 2007

George Jones

Staff Emeritus
For the particle in a box, the wavefuntion is set to zero everywhere the potential in infinite, since the particle has zero probability of being in this region. The Schrodinger is sovled where the potential is not inifinite (V = 0 in the case of the Harmonic oscillator), giving a linear combination of sin and cos. These solutions are joined by the boundary condition requirement that the the wavefunction is continuous at the transition from V non-infinite to V infinite, which means that wavefunction has to be zero where the transition in the potential occurs.