Simple Harmonic Oscillator Problem with Slight Variation

[Kvm90]

Homework Statement

A particle is moving in a simple harmonic oscillator potential V(x)=1/2*K*x^2 for x$$\geq0$$, but with an infinite potential barrier at x=0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies.

Homework Equations

I am thinking that the wave function would be $$\Psi=Ae^(i*x*sqrt{2m(E-1/2*k*x^2)}/hbar)+Be^(-i*x*sqrt{(2mE-1/2*K*x^2)}/hbar)<br /> <br /> Is this the right wave function? How do I go about finding the energies? Thanks for any help you can afford!<br /> <br /> (Sorry for the incompetency with Latex)$$

 

[kreil]

The systematic way to solve these problems is to plug the potentials into the Schrödinger equation, solve the resulting differential equation for psi, impose boundary criteria, then normalize the wavefunction

 

[Kvm90]

yea then you get

d^2Psi/dx^2 + [2m(E-1/2Kx^2]/hbar^2*psi = 0

I do not know how to solve this differential equation.

I'm pretty sure there's a lengthy integration and manipulation of the differential equation to yield psi(s) = f(s)e^[s^2/2] and then some more manipulation to lead to values for this polynomial f(s).

NOTE: s=x*(Km)^(1/4)/hbar^(1/2)
lambda = [2*sqrt(m)*E]/(hbar*sqrt(K))

in the initial substitution into the Schrödinger equation to yield:
d^psi/dx^2 + (lambda - s^2)*psi = 0

Anyone know of this integration technique (Gaussian)? Anyone know whether the technique still applies if x can only be greater than 0?

 

[kreil]

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/SimpleHarmOsc.htm

See the section entitled, "Schrödinger’s Equation and the Ground State Wave Function"

 

[Kvm90]

wow great site in general, thanks a lot