## A problem on spherical harmonics.

Posted by peeterjoot on January 10, 2011

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# Motivation.

One of the PHY356 exam questions from the final I recall screwing up on, and figuring it out after the fact on the drive home. The question actually clarified a difficulty I’d had, but unfortunately I hadn’t had the good luck to perform such a question, to help figure this out before the exam.

From what I recall the question provided an initial state, with some degeneracy in , perhaps of the following form

and a Hamiltonian of the form

From what I recall of the problem, I am going to reattempt it here now.

## Evolved state.

One part of the question was to calculate the evolved state. Application of the time evolution operator gives us

Now we note that , and , so the exponentials reduce this nicely to just

## Probabilities for measurement outcomes.

I believe we were also asked what the probabilities for the outcomes of a measurement of at this time would be. Here is one place that I think that I messed up, and it is really a translation error, attempting to get from the english description of the problem to the math description of the same. I’d had trouble with this process a few times in the problems, and managed to blunder through use of language like “measure”, and “outcome”, but don’t think I really understood how these were used properly.

What are the outcomes that we measure? We measure operators, but the result of a measurement is the eigenvalue associated with the operator. What are the eigenvalues of the operator? These are the values, from the operation . So, given this initial state, there are really two outcomes that are possible, since we have two distinct eigenvalues. These are and for , and respectively.

A measurement of the “outcome” , will be the probability associated with the amplitude (ie: the absolute square of this value). That is

Now, the only other outcome for a measurement of for this state is a measurement of , and the probability of this is then just . On the exam, I think I listed probabilities for three outcomes, with values respectively, but in retrospect that seems blatently wrong.

## Probabilities for measurement outcomes.

What are the probabilities for the outcomes for a measurement of after this? The first question is really what are the outcomes. That’s really a question of what are the possible eigenvalues of that can be measured at this point. Recall that we have

So for a state that has only contributions before the measurement, the eigenvalues that can be observed for the operator are respectively and respectively.

For the case, our probability is , leaving as the probability for measurement of the () eigenvalue. We can compute this two ways, and it seems worthwhile to consider both. This first method makes use of the fact that the operator leaves the state vector intact, but it also seems like a bit of a cheat. Consider instead two possible results of measurement after the observation. When an measurement of is performed our state will be left with only the kets. That is

whereas, when a measurement of is performed our state would then only have the contribution, and would be

We have two possible ways of measuring the eigenvalue for . One is when our state was (, and the resulting state has a component, and the other is after the measurement, where our state is left with a component.

The resulting probability is then a conditional probability result

The result is the same, as expected, but this is likely a more convicing argument.

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