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 Schrödinger 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 $\lambda=-1$.