# Transition Moments and Selection Rules

1. Dec 10, 2014

### Rambro323

My instructor was talking about allowed transitions for IR spectroscopy and how you can predict them relatively well using PIB wavefunctions. It was stated that you could solve for the probability of a transition moment to determine if a transition is forbidden or not.

<μ>=∫ΨfμΨidτ where μ=x

So for fun I plugged in μ=x2 and started getting alternating negative and positive, whose absolute values decreased as the difference between i and f increased. The integral value never reached zero.

I'm mostly wondering what a negative transition moment means, my gut tells me emission. And can there be any transitions which are forbidden if the integral value is never zero?

2. Dec 11, 2014

### Staff: Mentor

The harmonic oscillator is generally speaking a better model.

Note that this not represent a dipole moment, so you have to be careful not to carry any interpretation too far.

The sign depends on a purely arbitray choice of phase or orientation of the coordinate system. It has nothing to do with emission. The transition rate is proportional to $|\langle\mu\rangle|^2$.

The definition of an electric-dipole forbidden transition is one for which $|\langle\mu\rangle|^2 = 0$, otherwise it is allowed. There can be other processes (electric quadrupole, magnetic dipole, etc.) that allow for transitions, but the corresponding transition rates are generally orders of magnitude smaller, and can be neglected when doing ordinary IR spectroscopy.

3. Dec 11, 2014

### Rambro323

Thank you!

I didn't use harmonic oscillator mostly because I didn't want to deal with its wavefunctions cause that would have taken me more time. So then that means that all transitions are allowed for particle in a box when μ=x2 until |μ|2 gets arbitrarily close to zero. Kind of weird.

4. Dec 12, 2014

### Staff: Mentor

As I said previously, μ=x2 does not correspond to a dipole moment, therefore you cannot conclude anything about allowed or forbidden transitions.

5. Dec 12, 2014

### DrDu

At least, now I know what you meant with PIB wavefunctions.
For a particle in a box, even if you take mu=x, you at least get transitions between all even and odd eigenfunctions. However, the size of the matrix elements decays rapidly with the difference of quantum numbers.