Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Time development of Expectation values

  1. Oct 21, 2004 #1
    Here is a question that I have found in my quantum text which I have been thinking about for a few days and am unable to make sense of.

    If there is an operator A whose commutator with the Hamiltonian H is the constant c.

    [H,A]=c

    Find <A> at t>0, given that the system is in a normalized eigenstate of A at t=0 corresponding to the eigenvalue of a.


    -So here is what I am thinking. Please tell me what I am doing wrong.

    I know that where p is momentum operator [H,p]=ihdV/dx, and V is some potential in the Hamiltonian

    and d<p>/dt=-<dV/dx>

    So where [H,A]=c, wouldn't d<A>/dt=0? And then there would be no change in the eigenstate of A with time, and <A>=a which is its initial value at 0, which would be a(Psi) where Psi is my wave function or eigenvector or whatever I need it to be.

    Am I understanding the question wrong?
     
  2. jcsd
  3. Oct 21, 2004 #2

    Tom Mattson

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Nope.

    Check out Heisenberg's equation of motion:

    (-i*hbar)(dA/dt)=[H,A]

    So if [H,A]=c, then the operator A is evolving in time at a constant rate. The only way you will have dA/dt=0 is if c=0.

    More here.
     
  4. Oct 21, 2004 #3
    Ok, I follow your argument as to why dA/dt=0, but what I am unclear about is what this does to <A>. And what is the significance of eigenvalue a of the eigenstate A in describing <A> where t>0? We know at t=0,A=a I believe.
     
  5. Oct 22, 2004 #4
    Solve the equation -i hbar dA/dt = c for A.
    From this calculate the time evolution for <A> with the given initial value.

    I did not try it. The fact that the initial average value is an eigenvalue will play a role in the second step: you know the initial state.
     
    Last edited: Oct 22, 2004
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Time development of Expectation values
  1. Expectation value (Replies: 2)

Loading...