
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 waveparticle 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: 20131028   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 gaussianlooking
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:
(σ_{x} σ_{p}) ^{1/2} ≥ h/2
Where σ_{x} is the standard deviation
of the position, and σ_{p} is the
standard deviation of the momentum. From this, we
see that as σ_{x} becomes very small,
as it does with a more precise measurement,
σ_{p} must become very large. We also
know that
σ_{p} = 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 σ_{x} >
0, σ_{p}
> 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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