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According to the Heisenberg Uncertainty Principle, I can not know the position and momentum (energy) of an electron at the same time. This appears to be the result of the wave-particle duality of the electron. My question is: if I measure the electron\'s position, collapsing its wave function, even though I will not know the momentum (energy) of the electron, will it still behave the exact same as a system? Or will the collapse of the wave function potentially increase, or decrease, the total amount of energy available within the system for the duration of the measurement?
Question Date: 2013-10-28
Answer 1:

Sorry, I'm not sure I understand what you mean by "will it still behave the exact same as a system?" However, the answer to the second part of your question is: yes, measuring the position changes the energy of the system.

Before we take a measurement, we say that the state of the electron is some superposition of eigenstates. Once we measure the position of the electron exactly, the system will be in a particular position eigenstate. So what happened to the momentum? Well, now the momentum of the particle is still some superposition of the momentum eigenstates, but now with different weightings for the eigenstates. Furthermore, because the position is measured more exactly, such that what was once a gaussian-looking distribution of the states looks more like a delta function, the standard deviation of the position --> 0 in the limit of extremely precise measurement. Heisenberg Uncertainty essentially says:

x σp) 1/2 ≥ h/2

Where σx is the standard deviation of the position, and σp is the standard deviation of the momentum. From this, we see that as σx becomes very small, as it does with a more precise measurement, σp must become very large. We also know that σp = sqrt(2>) ~ sqrt(E)

That is, the standard deviation is the square root of the expectation value of the momentum, and this is proportional to the square root of the energy. So what we can gather is that as σx --> 0, σp --> infinity, which implies that the expectation value of the system we measured position for is infinity!

Answer 2:

In measuring the position of the electron, you will alter its momentum; for instance, if you bounce a photon off of the electron to see it, then that photon imparted energy on the electron, thus changing its momentum. Since the photon is also subject to the uncertainty principle, in having collapsed the wave function of the electron's position you have expanded the wave function for its momentum, since you do not know how much energy the photon imparted on it.

Answer 3:

That is a very advanced question, and one that has required a fair bit of thought on my part. So the first part of the question relates to collapsing the wave function of an electron. So, the short answer is this, if you collapse the wave function of an electron, it no longer acts the way it would have before. You have fundamentally changed its state. The behavior of a particle is dictated exactly by its wave function. This is what sets the probability of finding it in any particular location and of measuring any particular momentum. Once the wave function has collapsed, then any following measurements will be investigating this new, collapsed wave function. The collapse of the wave function, on its own, should not cause a change in the energy of the system. However, what causes the wave function collapse (the measurement) very much has the ability to change the energy of the system. Measurements, at the quantum level, are often done with photons. The absorption or emission of photons can change the energy of a system. Similarly, even if measurements are not taken using photons, the measurement is still some sort of transfer of energy to or from the system. And so yes, the act of measurement can change the system you're measuring.

I hope that helps a little bit. There are many resources on the web where you can learn much more about quantum physics. My favorites are any book or lecture (many available online) by Richard Feynman. He has intuition about quantum mechanics that few other scientists or educators ever have.

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