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When we try to measure the position of a system, wavefunction collapses to form a spike. After a while, the wavefunction spreads again, and you take another measurement, the results will be different" in general.

What would happen if we measure the energy instead? Again, the wavefunction collapses to one eigenstate, after a while, would the wave spread out again? And how could one measure the energy without knowing its momentum and potential?

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The position collapses into a position eigenstate (a spike in wavefunction) because you measured the position. That spike spreads over time because the position eigenstates are not momentum eigenstates, i.e. there is a range of momenta in the spike. Another way of looking at this is that the spike is not an eigenstate of the Hamiltonian (i.e. a well defined energy) and therefore evolves in time beyond simple a phase change.

If you measure the energy, you will collapse into an energy eigenstate, which is not a position eigenstate. For example, energy eigenstates of a particle in a box (classic quantum homework problem) are distributed across the width of the box, not uniformly, but not strongly localized like the spike. If you measure the energy again, you will get the same answer (it is an eigenstate of the Hamiltonian).

As for measurements, we measure energy differences usually. For example, the energy eigenstates of a hydrogen atom (or hydrogen like atom) correspond to various configurations of the electron "orbiting" the atom plus relative orientation of the electron's magnetic moment (spin) to the "orbit" and the nuclear magnetic moment.

I work with calcium ions that effective have a single electron around a nucleus (the other electrons effectively cancel out). If the calcium ion is in the lowest energy configuration (ground state) and I shine a laser on it, sweeping the frequency of the laser, the ion will remain dark until my laser is in a narrow ~20 MHz window around 755 THz (397 nm UV light) at which point the ion will glow ... something we can see with a sensitive CCD camera. That 755 THz corresponds to the energy difference between two different configuration of the ion. The ion glows because it can absorb the light at that specific frequency and will then re-emit the light into a random direction through spontaneous emission. The absorption and emission happen about 20 million times a second. If the ion is not in the ground state, and there are other metastables levels it could be in, the ion won't glow. In this way I can detect whether it was in the ground state. When I turn off the laser, the ion will emit a single final photon and fall back into the ground state (and you can detect this photon).

If the ion was originally in some superposition of the ground state and one of the metastable states, the ion will either collapse into the ground state and glow or collapse into the metastable state and stay dark. The probability of it being bright or dark depends on the original superposition. Through this measurement process, I have collapsed the ion's state into an energy eigenstate. At no point are we measuring the electron's position or momentum, only the energy difference.

The following link about last year's Nobel prize might help.

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Cool, this is very interesting. I didn't know it's in fact possible to measure energy without regarding Position or momentum! One thing I still want to ask is, it seems you can have different result " either dark or bright"? But I was told that repeat measurement will keep giving you the same eigenstate only if you measure energy , not other quantities like position. So why is it the case? –  el psy Congroo Oct 25 '13 at 17:01
    
@elpsyCongroo Yes, there are two different measurement results: bright and dark. The states the measurement leaves the system in are eigenstates of the ion so I will remeasure the same answer if I repeat the measurement. The energy eigenstates are, by Schrodinger's equation, the states that evolve in time with only an overall phase change. There is a fundamental link between time and energy that comes to us via a beautiful result called Noether's theorem: every invariance in the physics rules implies a conserved quantity. Time invariance => energy conservation. –  Jason A Oct 26 '13 at 0:23
    
Does it mean you get the same answer no matter how long you wait, unless you measure other quantity such as momentum? I don't quite get how it is time invariant. The wave function still has a time depedence part. –  el psy Congroo Oct 26 '13 at 8:47
    
@elpsyCongroo Yes to the question. For energy eigenstates, the time dependence is an overall phase change exp(-i E t/h_bar) that doesn't affect the measurement of the energy. –  Jason A Oct 26 '13 at 15:02
    
I see, I'm much more clearer now, so does it mean ie Position, its eigenstate, will be time dependent? " and is it a dirac delta function? –  el psy Congroo Oct 26 '13 at 17:39

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