QM 1-D Harmonic Oscillator Eigenfunction Problem

[tis]

Homework Statement

A particle of mass m moves in a 1-D Harmonic oscillator potential with frequency \omega.
The second excited state is \psi_{2}(x) = C(2 \alpha^{2} x^{2} + \lambda) e^{-\frac{1}{2} a^{2} x^{2}} with energy eigenvalue E_{2} = \frac{5}{2} \hbar \omega.

C and \lambda are constants. Show that the constant \lambda is not arbitrary.

NOTE: \int_{-\infty}^{\infty} e^{-\frac{1}{2} a x^{2}} dx = \sqrt{2\pi / a}, a > 0.

Homework Equations

That I can think of: TISE, normalization condition for wavefunctions.

The Attempt at a Solution

I assume there's a solution in substituting into the time-independent Schrodinger Equation and solving for \lambda, but the equation seems very difficult. And I have no idea how to utilize the hint provided; everything I've tried (normalization condition, etc.) just gives a dependence on C. I've scoured my textbooks with no luck, their only relevant info is using ladder operators and other methods to produce eigenfunctions.

Just after a point in the right direction. Thanks in advance for your help!

 

[tman12321]

Your idea to substitute this into the TISE seems like a good start. All you will have to do is take derivatives and rearrange some terms, and you will end up with an equation which determines lambda. Give it a try.

 

[tis]

Figured it out. Take the ground state of the harmonic oscillator \psi_{0} = A e^{-\frac{1}{2}\alpha^{2} x^{2}} and use the orthogonality condition \int_{-\infty}^{\infty} \psi_{0}^{*} \psi_{2} \ dx = 0.

From there just expand, cancel out A* and C, and solve the integrals with integration by parts and the hint provided. Eventually you come to 0=2\alpha^{2}\frac{\sqrt{\pi}}{2\alpha^{3}}+\lambda\frac{\sqrt{\pi}}{ \alpha }, hence \lambda=-1.