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Coherence
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The thread title is probably confusing, but I couldn't really think of a better one. There is a basic feature of quantum mechanics that I have been puzzled by for a long time. My guess is that my issue stems from some fundamental misunderstanding of the theory, so I would appreciate any efforts to help clear things up!
Let's prepare a simple quantum system, like a particle in a one-dimensional infinite square well. If we now measure the energy of the particle at time t = 0, we get some value E_n and its state "collapses" into an energy eigenstate |E_n>, which can be represented in the position basis as a standing wave in the well. Energy eigenstates (equivalently, eigenstates of the Hamiltonian) are stationary states, meaning that we can measure the energy of the particle again at a later time t = t1 and obtain the same energy value E_n. We can then repeat the energy measurement at an even later time t = t2 and should still obtain the same energy value E_n.
Now, let's imagine that instead of measuring the energy again at time t = t1, we now measure the position of the particle at t = t1. We get some value x, and the state "collapses" from the energy eigenstate |E_n> into a position eigenstate |x>, which can be represented in the position basis as a delta function. |x> is obviously not an eigenstate of the Hamiltonian and therefore not stationary. However, we can express |x> as a super-position of energy eigenstates, each one of which has unitary evolution. At time t = t2, the state of the particle is no longer a position eigenstate, but can still be expressed as a super-position of energy eigenstates. If we now measure the energy of the particle at t = t2, there is some probability that we will once again measure value E_n and "collapse" the state back to |E_n>. However, there is also now a probability that we will measure a different energy E_m and "collapse" into a different state |E_m>.
Nothing that I've described so far is mysterious to me from the mathematical formalism of quantum mechanics (i.e., non-commuting operators representing observables that do not have simultaneous eigenstates). My problem is this: how does the wavefunction "know" which observable was measured? In the first example that I described, the energy was measured three times (at t = 0, t = t1, and t = t2), always yielding the result E_n. In the second example that I described, the energy was measured at t = 0, yielding E_n. Then the position was measured at t = t1. And then the energy was measured again at t = t2, generally yielding a different energy E_m. The position measurement at t = t1 clearly has an effect on the result of the energy measurement at t = t2. But how is the wavefunction "aware" of the quantity that is being measured? Certainly we can devise various experimental apparatuses to measure position, and various experimental apparatuses to measure energy. How does the wavefunction "learn" that the arbitrary apparatus we were using was designed to measure position and not energy, and vice versa, causing it to "adjust" its behavior appropriately in the future?
I realize that this whole analysis might be flawed, because in any real experiment, the measurement of the particle energy or position would destroy or significantly perturb the particle (i.e., observer effect), so unitary evolution with the Hamiltonian corresponding to the infinite square well is no longer valid. But it still seems like the example I described is consistent with the kinds of problems you might see in an undergraduate textbook, so there has to be some practical reality to it... or no?
Let's prepare a simple quantum system, like a particle in a one-dimensional infinite square well. If we now measure the energy of the particle at time t = 0, we get some value E_n and its state "collapses" into an energy eigenstate |E_n>, which can be represented in the position basis as a standing wave in the well. Energy eigenstates (equivalently, eigenstates of the Hamiltonian) are stationary states, meaning that we can measure the energy of the particle again at a later time t = t1 and obtain the same energy value E_n. We can then repeat the energy measurement at an even later time t = t2 and should still obtain the same energy value E_n.
Now, let's imagine that instead of measuring the energy again at time t = t1, we now measure the position of the particle at t = t1. We get some value x, and the state "collapses" from the energy eigenstate |E_n> into a position eigenstate |x>, which can be represented in the position basis as a delta function. |x> is obviously not an eigenstate of the Hamiltonian and therefore not stationary. However, we can express |x> as a super-position of energy eigenstates, each one of which has unitary evolution. At time t = t2, the state of the particle is no longer a position eigenstate, but can still be expressed as a super-position of energy eigenstates. If we now measure the energy of the particle at t = t2, there is some probability that we will once again measure value E_n and "collapse" the state back to |E_n>. However, there is also now a probability that we will measure a different energy E_m and "collapse" into a different state |E_m>.
Nothing that I've described so far is mysterious to me from the mathematical formalism of quantum mechanics (i.e., non-commuting operators representing observables that do not have simultaneous eigenstates). My problem is this: how does the wavefunction "know" which observable was measured? In the first example that I described, the energy was measured three times (at t = 0, t = t1, and t = t2), always yielding the result E_n. In the second example that I described, the energy was measured at t = 0, yielding E_n. Then the position was measured at t = t1. And then the energy was measured again at t = t2, generally yielding a different energy E_m. The position measurement at t = t1 clearly has an effect on the result of the energy measurement at t = t2. But how is the wavefunction "aware" of the quantity that is being measured? Certainly we can devise various experimental apparatuses to measure position, and various experimental apparatuses to measure energy. How does the wavefunction "learn" that the arbitrary apparatus we were using was designed to measure position and not energy, and vice versa, causing it to "adjust" its behavior appropriately in the future?
I realize that this whole analysis might be flawed, because in any real experiment, the measurement of the particle energy or position would destroy or significantly perturb the particle (i.e., observer effect), so unitary evolution with the Hamiltonian corresponding to the infinite square well is no longer valid. But it still seems like the example I described is consistent with the kinds of problems you might see in an undergraduate textbook, so there has to be some practical reality to it... or no?