Non-Stationary State Wavefunction - Normalized? <L^2>? Uncertainty on L^2?

[mkosmos2]

Homework Statement

Consider the nonstationary state:

$\Psi = \sqrt{\frac{1}{3}}\Psi_{22-1} + \sqrt{\frac{2}{3}}\Psi_{110}$

Where $\Psi_{22-1}$ and $\Psi_{110}$ are normalized, orthogonal and stationary states of some radial potential. Is $\Psi$ properly normalized? Calculate the expectation value of $L^{2}$ and the uncertainty in $L^{2}$ for a particle in this state.

Homework Equations

$|a_{n_{1}l_{1}m_{1}}|^{2} + |a_{n_{2}l_{2}m_{2}}|^{2} = 1$ (1)

$<L^{2}> = \int_{all{}\Omega}\Psi^{*}(-\hbar^{2}\Lambda^{2})\Psi d\tau$ (2)

$\Delta L^{2} = \sqrt{<(L^{2})^{2}> - <L^{2}>^{2}}$ (3)

The Attempt at a Solution

For the first part, I think it's safe in this situation to use equation (1) where the $a_{nlm}$ terms are the two coefficients in $\Psi$, but I'm not sure if it applies when the quantum numbers aren't the same for the two stationary states.

For the second part, still not positive, but I think if I use equation (2), the $L^{2}$ operator results in $\hbar^{2}l(l+1)$ coming outside the integral, and because the two states are orthogonal, the integral and therefore $L^{2}$ goes to 0.

The last part has me tripped up the most because I don't know whether $L^{2}$ being zero automatically means the uncertainty is zero, but something tells me it's not that simple.

I'm just fuzzy on a lot of the concepts surrounding this topic so any clarification/confirmation will be much, much appreciated.

Thank you very much for your time, and if these answers are correct, sorry I wasted it :s

-Mike

 

[sean_mp]

Ok, for the first part, all you need to know is that the states are orthogonal and normalized. This means
$$\big< \Psi_{n'l'm'} \big\lvert \, \Psi_{nlm} \big> =\big<n'l'm' \big\lvert \,nlm\big> = 0~~ \text{if}~~nlm=n'l'm';~1~~\text{if}~~nlm \ne n'l'm'$$
So for your state, we have
$$\big< \Psi \, \big\lvert \, \Psi \big> =\bigg( \sqrt{ \frac{1}{3}} \Psi_{22-1}^*+ \sqrt{ \frac{2}{3}} \Psi_{110}^* \bigg)\bigg( \sqrt{ \frac{1}{3}} \Psi_{22-1}+ \sqrt{ \frac{2}{3}} \Psi_{110} \bigg)= \frac{1}{3}(1)+ \frac{2}{3}(1) + \frac{2}{6}(0)\frac{2}{6}(0)= \frac{3}{3} =1$$
So yes, it is normalized.
For the second part,
$$L^2 \big\lvert \,nlm \big> = \hbar^2 l(l+1) \big\lvert \,nlm \big>$$
or
$$\big<n'l'm' \big\lvert L^2 \big\lvert \,nlm \big> = \big<n'l'm'\big\lvert \hbar^2 l(l+1) \big\lvert \,nlm \big> = \hbar^2 l(l+1)~~\text{if}~ l=l';~~0~~ \text{if} ~~l \ne l'$$
so,
$$\begin{multline}\big<L^2 \big> = \frac{1}{3} \big<22-1 \big\lvert \,\hbar^2 2(2+1) \big\lvert 22-1 \big> +\frac{2}{3} \big<110 \big\lvert \,\hbar^2 1(1+1) \big\lvert 110 \big>\\ = \frac{1}{3} \hbar^2 2(2+1) \big<22-1 \big\lvert \ 22-1 \big> +\frac{2}{3} \hbar^2 1(1+1) \big<110 \big\lvert \, 110 \big> = \frac{1}{3} \hbar^2 2(2+1) +\frac{2}{3} \hbar^2 1(1+1) \end{multline}$$

and etc. You get the idea. By the way, you need to do NO integration on this problem. That's the beauty of orthonormality.