I in understanding the measurement process.

[QuantumDefect]

Hello everyone,
I am in my Undergraduate Quantum Mechanics class right now and it has been great fun so far. However, I am still hooked up on a topic that just does not click with me, but is of vital importance. I need help in understanding what the wavefunction does and why it does it when we measure the:

a)Position
b)Momentum
c)Energy

I get that when we measure the position, the probability density "collapses" into a delta function at the measured value. The many momenta needed to create that spike, quickly make the probability density spread out again, so if we measure the position immediatly after, we will not necessarily get the same value again.

We recently had a test where the system was a particle in an infinite square well. He asked us that if we measure the particle's location and find it is at some value, will subsequent measurements yield the same result.

His response was that this is an eigenfunction of position, not of energy. Since the position operator does not commute with the energy operator for a square well, this wavefunction is not a stationary state and will evolve in time.

And then we go on to measuring the energy. He asked if we measure the energy, will subsequent measurements yield the same value? He said that it is not an eigenstate of energy. Therefore it won't evolve in time and repeated measurements of energy will give the same result.

I am very confused at the moment, and I would greatly appreciate any help you could give me. Thank you for your time.

 

[physics girl phd]

first: I'm thinking you are a bit confused here... but via

it won't evolve in time and repeated measurements of energy will give the same result.

I'm going to guess you likely have an energy eigenstate.

Then say you know what operator the wavefunction IS an eigenstate of... how do you write that operator (as a function of x and/or its derivatives)?? How dow you write the other operators in term of these? Then do the operators commute?

I'm also thinking there is another way to think about the problem: Can you write out your wavefunction for the state? That one should be easy to write out. So what is the expectation value of the operator (just for practice)? Then - most importantly (you could get away with just doing this)... what is the standard deviation of that (this is the important thing... can you tell us why)?

 

[Rach3]

We recently had a test where the system was a particle in an infinite square well. He asked us that if we measure the particle's location and find it is at some value, will subsequent measurements yield the same result.

His response was that this is an eigenfunction of position, not of energy. Since the position operator does not commute with the energy operator for a square well, this wavefunction is not a stationary state and will evolve in time.

The point of the measurement process is, if you measure again immediately afterwards, the outcome is the same (in the limiting sense; the faster you re-measure, the more similar the outcomes). In general, your state will continue evolving after being measured; after your first measurement you're in an eigenstate of your observable $$\hat{O}\psi(0)=\lambda\psi(0)$$, and then you continue evolving as $$\psi(t)=\hat{U}(t)\psi(0)$$, in general leaving the eigenstate. Since U(t)->1 as t->0, "fast" repeated measurements get you the same outcome with high probability.

So be careful to distinguish (i) "ideal" repeated measurements in the limiting sense and (ii) "real" repeated measurements, in which the states do actually undergo unitary evolution (unless [H,O]=0).

And then we go on to measuring the energy. He asked if we measure the energy, will subsequent measurements yield the same value? He said that it is not an eigenstate of energy. Therefore it won't evolve in time and repeated measurements of energy will give the same result.

When you measure energy, you collapse the state into an energy eigenstate; it is then in an eigenstate of the Hamiltonian! Any measurements after that will give the same result - the system does not evolve any more. (Because $$U(t)=e^{-i H t / \hbar}=e^{-i E t/ \hbar}$$ is just a complex phase factor.)

 

[CarlB]

Half of the difficulty of QM is the mathematics, the other half is the interpretation. If you want to get rid of half the difficulty, take a look at Julian Schwinger's measurement algebra. It is devoted to the measurement problem alone. And since it is restricted to discrete degrees of freedom (think spin), the mathematics is a lot simpler.

I put a collection of links to papers on Schwinger's measurement algebra on the web here:
http://www.d1013329.mydomainwebhost.com/applications/DokuWiki/doku.php?id=smapapers

Carl