1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ehrenfest theorem problem

  1. Oct 3, 2009 #1
    1. The problem statement, all variables and given/known data
    Use ehrenfest theorem ([tex]i*\hbar*d<Q>/dt=(\varphi(t),[Q,H],\varphi(t))[/tex] to show that the expectation value of the position of a particale that moves in 3 dimensions with the Hamiltonian [tex] H=p^2/2m+V(r)[/tex] satisfies [tex] d<r>/dt=<p>/m[/tex]


    2. Relevant equations


    ([tex]i*\hbar*d<Q>/dt=(\varphi(t),[Q,H],\varphi(t))[/tex]

    or [tex] d<Q>/dt=<-i[Q,H]/(\hbar)[/tex]
    3. The attempt at a solution

    [tex] [Q,H]=QH-HQ=Q((-i*\hbar*d/dx)^2/2m+V(r))-((-i*\hbar*d/dx)^2/2m+V(r))(Q)=Q*(\hbar)^2 d^2/dx^2*1/2m +QV(r)-(\hbar)^2 d^2Q/dx^2*1/2m+V(r)Q=QV(r)-(\hbar)^2 d^2Q/dx^2*1/2m+V(r)Q[/tex] not sure how to continue this problem

    Perhaps i should say: [tex] i*\hbar*d<r>/dt=[\varphi, [r,H]\varphi][/tex]
     
    Last edited: Oct 3, 2009
  2. jcsd
  3. Oct 4, 2009 #2
    anybody find a hard time reading the latex code or all of the country
     
  4. Oct 5, 2009 #3

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    That looks like the best starting point to me...what do you get when you do that?
     
  5. Oct 5, 2009 #4
    [tex]i*\hbar*d/dt=[\varphi, [r,H]\varphi]=[\varphi, [r,p^2/2m+V(x,t)]\varphi]=[\varphi, (r*p^2/2m+V(x,t)-p^2/2m+V(x,t)*r)\varphi]=1/(i*\hbar*2*m)*<[x,p]*d(p^2)/dp>=(<i*\hbar*2*p>)/(i*\hbar*2*m)=<p>/m[/tex]? Please take a look at my latex code because I don't think latex displayed all of my solution
     
  6. Oct 5, 2009 #5

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Your [itex]\LaTeX[/itex] is terrible!:yuck:

    Click on the image below to see how to generate something more legible:

    [tex]i\hbar\frac{d\langle r\rangle}{dt}=(\varphi, [r,H]\varphi)=(\varphi, [r,p^2/2m+V(r)]\varphi)[/tex]

    You need to be careful to only use square brackets to represent commutators,and round brackets otherwise. There is also no need to use the * symbol to represent multiplication, it just makes things look messy. And you should use \frac when appropriate. Also, your potential is given to you as a function of [itex]r[/itex]...why would you write it as a function of [itex]x[/itex] and [itex]t[/itex]?

    continue from here...
     
  7. Oct 5, 2009 #6
    [tex]
    i\hbar\frac{d\langle r\rangle}{dt}=(\varphi, [r,H]\varphi)=(\varphi, [r,p^2/2m+V(r)]\varphi)=1/(2*\hbar*m*i)*(<[x,p]d(p^2)/dp>)=1/(2*\hbar*m*i)*(<[i*\hbar*2p>)=<p>/m
    [/tex] hope this is better
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Ehrenfest theorem problem
  1. Ehrenfest's theorem (Replies: 2)

  2. Ehrenfest's Theorem (Replies: 4)

Loading...