In an experiment, do we measure the eigenvalue or expectation value?

[touqra]

In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?

 

[CompuChip]

When we make a measurement, we will always measure an eigenvalue. If we start with a system which wasn't in an eigenstate to begin with, however, it's not sure which eigenvalue we measure. The expectation value is sort of an average; it doesn't even have to be an eigenvalue.

Favorite example: consider an ordinary die with six sides numbered one through six. If we throw it, we can get 1, 2, 3, 4, 5 or 6; these correspond to the eigenvalues of the observable we're measuring in QM. If the chance to get each outcome is the same (that is, 1/6), the expectation value would be
1/6(1 + 2 + 3 + 4 + 5 + 6) = 3,5.
Note that this is not a possible outcome. Instead, if we would make many many throws (theoretically, infinitely many) then the average of the outcomes would be 3.5 (and if not, the die would be flawed )

 

[touqra]

When we make a measurement, we will always measure an eigenvalue. If we start with a system which wasn't in an eigenstate to begin with, however, it's not sure which eigenvalue we measure. The expectation value is sort of an average; it doesn't even have to be an eigenvalue.

Favorite example: consider an ordinary die with six sides numbered one through six. If we throw it, we can get 1, 2, 3, 4, 5 or 6; these correspond to the eigenvalues of the observable we're measuring in QM. If the chance to get each outcome is the same (that is, 1/6), the expectation value would be
1/6(1 + 2 + 3 + 4 + 5 + 6) = 3,5.
Note that this is not a possible outcome. Instead, if we would make many many throws (theoretically, infinitely many) then the average of the outcomes would be 3.5 (and if not, the die would be flawed )

So, how about the wavefunction? Say, I want to measure the Hamiltonian outcome (eigenvalue) of a quantum state (wavefunction). From the measured energy, can I infer the quantum state? Or is the state really just some theoretical construct, which can have absolutely no means of being determined its value from experiments?

 

[kdv]

So, how about the wavefunction? Say, I want to measure the Hamiltonian outcome (eigenvalue) of a quantum state (wavefunction). From the measured energy, can I infer the quantum state? Or is the state really just some theoretical construct, which can have absolutely no means of being determined its value from experiments?

You can never infer completely what the quantum state was before your measurement but you know what the quantum state is just after your measurement if there is no degeneracy (at least in the context of the Copenhagen interpretation).

 

[CompuChip]

To continue my earlier analogy (yes, I know continuing analogies is dangerous because the analogy will stop at some point): suppose I have some construct where I can only read of the sum of n throws.
For one throw, the sum is exactly the outcome of my throw, so I can unambiguously say how many eyes I had (but nothing about earlier throws). If I throw twice (or two dice at the same time) and the sum is 2 or 12, I also know the values of the separate dice unambiguously, but for example the outcome 6 is degenerate (could be (1, 5), (2, 4), (3, 3), (4, 2) or (5, 1), which is 5 or 3 possibilities depending on whether I can discern the dice).

So depending on the system, the eigenvalue will or will not tell you what the state is/was, but it will usually at least narrow down the possibilities.