Eigenfunction energy levels in a harmonic well

[Lazy Rat]

Homework Statement

If the first two energy eigenfunctions are
$\psi _0(x) = (\frac {1}{\sqrt \pi a})^ \frac{1}{2} e^\frac{-x^2}{2a^2}$,
$\psi _1(x) = (\frac {1}{2\sqrt \pi a})^ \frac{1}{2}\frac{2x}{a} e^\frac{-x^2}{2a^2}$

Homework Equations

The Attempt at a Solution

Would it then be correct to presume
$\psi _3(x) = (\frac {1}{4\sqrt \pi a})^ \frac{1}{2}\frac{4x}{a} e^\frac{-x^2}{2a^2}$

Thank you for your time in considering this.

 

[kuruman]

Would it then be correct to presume ...

It would not because it is not orthogonal to $\psi_1(x)$ but the same as $\psi_1(x)$. Also, you do not state the question that the problem asks.

 

[RedDelicious]

No. In terms of ladder operators, the nth eigenfunction is given by

$$<br /> |n \rangle \equiv \psi_{n}(x) = \frac{(a^\dagger)^n}{\sqrt{n!}} |0 \rangle<br />$$

 

[Lazy Rat]

the specific question goes as so

For this equation

$\Psi (x,0) = \frac {1}{\sqrt{2}}(\psi_1 (x)-\psi_3 (x))$

The system is undisturbed, obtain an expression for $\psi (x,t)$ that is valid for all t ≥ 0. Express in terms of the functions $\psi_1 (x)$, $\psi_3 (x)$ and $ω_0$, the classical angular frequency of the oscillator.

I am trying to approach this by simply inputting the eigenfunctions for

$\psi _1(x) = (\frac {1}{2\sqrt \pi a})^ \frac{1}{2}\frac{2x}{a} e^\frac{-x^2}{2a^2}$

And then for

$\psi _3(x)$ (which as yet I haven't understood)

And

$a = \sqrt{\frac {\hbar}{ω_0}}$

Would this be the correct approach to express in the terms as stated?

Thank you for assisting me with my problem.

 

[kuruman]

You might find this reference useful, especially the first two sections and Equation (15.23)
http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter15.pdf

 

[Lazy Rat]

So would i use the fact that $E_1 = \frac {3}{2} \hbar ω_0$ which would give $e^ \frac {- 3iω_0t}{2}$
And $E_3 = \frac {7}{2} \hbar ω_0$ which would give $e^ \frac {- 7iω_0t}{2}$

Am I on the right track?

 

[PeroK]

So would i use the fact that $E_1 = \frac {3}{2} \hbar ω_0$ which would give $e^ \frac {- 3iω_0t}{2}$
And $E_3 = \frac {7}{2} \hbar ω_0$ which would give $e^ \frac {- 7iω_0t}{2}$

Am I on the right track?

Yes.