My understanding is that an arbitrary quantum-mechanical wavefunction can be wri
ID: 1382367 • Letter: M
Question
My understanding is that an arbitrary quantum-mechanical wavefunction can be written as a linear combination of eigenfunctions of some Hermitian operator, most commonly the Hamiltonian; when a measurement corresponding to that operator is made on this superposition state, the wavefunction collapses and only specific values are observed -- namely, the eigenvalues of the particular eigenstates that comprised the wavefunction. (Moreover, the probability of measuring the eigenvalue Ei is proportional to ?ci?2, the square of the coefficient of that eigenstate in the linear combination, etc.)
And yet, in many situations, it seems to be assumed that the system is already in an eigenstate and that superposition is not possible. For example:
the electron in the hydrogen atom is said to be in, e.g., the 1s 2S1/2 state or the 2s 2S1/2 state, but never a superposition of the two.
the possible angular momentum vectors for a QM rigid rotor with fixed l are sometimes drawn as discrete "cones"... but couldn't the average L point in any direction, since a rigid rotor might be in a superposition of states?
when deriving Boltzmann statistics, we consider how to place Ni particles into the level with energy ?i, but there is no consideration that a particle might occupy two (or more) energy levels simultaneously.
Explanation / Answer
Often, we don't observe single quantum systems. Often, we observe ensembles of identical or nearly identical systems, often in thermal equilibrium.
In some cases, it can be utterly indistinguishable whether each individual system is in an eigenstate, or each individual system is in a superposition state with random relative phase. See this question for one such case: Are these two quantum systems distinguishable?
When a substantial fraction of an ensemble is in a superposition state and this population shares a common relative phase between the superposed states, we can observe the superposition much more easily.
As a concrete example, I've used a femtosecond laser to line up a bunch of nitrogen molecules. This changes the index of refraction of nitrogen gas, and you can watch how the index changes with time due to molecular rotation. It turns out that quantum superposition effects explain observed results beautifully. However, collisions between molecules disrupt the synchronous rotation, and the nitrogens drift out of relative alignment with one another. They might be driven towards rotational eigenstates, or they might stay in superposition states but be driven out of sync. The interesting thing is, you can't tell the difference between these two cases by watching the index of refraction - and I think you actually can't even tell the difference in principle, which is kinda neat.
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