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touqra
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In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?
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CompuChip said: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 said: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?
An eigenvalue is a number that represents the possible outcomes of a measurement in quantum mechanics. It is associated with an eigenvector, which describes the state of the system. An expectation value is the average value of a measurement that is expected to be obtained in multiple trials of an experiment.
Both eigenvalues and expectation values are important in an experiment. The eigenvalue represents the possible outcomes of a measurement, while the expectation value gives an indication of the most likely outcome and provides a measure of the overall behavior of the system.
Yes, it is possible to measure both the eigenvalue and expectation value in an experiment. The eigenvalue can be directly measured through a single measurement, while the expectation value can be obtained through multiple measurements and calculating the average.
The choice of which value to measure in an experiment depends on the specific question being investigated. If the goal is to determine the possible outcomes of a measurement, then the eigenvalue should be measured. If the goal is to understand the overall behavior of the system, then the expectation value should be measured.
The eigenvalue and expectation value can both be affected by experimental errors. However, the impact of errors may be more significant on the expectation value, as it is a calculated average. It is important to minimize and account for errors in experimental measurements to obtain accurate results for both values.