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Homework Help: 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


    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


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    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


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    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
    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
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