Transition Moments and Selection Rules

[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>=∫Ψ_fμΨ_idτ 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?

 

[DrClaude]

My instructor was talking about allowed transitions for IR spectroscopy and how you can predict them relatively well using PIB wavefunctions.

The harmonic oscillator is generally speaking a better model.

So for fun I plugged in μ=x²

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

I'm mostly wondering what a negative transition moment means, my gut tells me emission.

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 $|\langle\mu\rangle|^2$.

And can there be any transitions which are forbidden if the integral value is never zero?

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.

 

[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.

 

[DrClaude]

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.

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

 

[DrDu]

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.

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.