How Does Ehrenfest Theorem Apply to Particle Position in Quantum Mechanics?

[noblegas]

Homework Statement

Use ehrenfest theorem ($$i*\hbar*d<Q>/dt=(\varphi(t),[Q,H],\varphi(t))$$ to show that the expectation value of the position of a particlee that moves in 3 dimensions with the Hamiltonian $$H=p^2/2m+V(r)$$ satisfies $$d<r>/dt=<p>/m$$

Homework Equations

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

or $$d<Q>/dt=<-i[Q,H]/(\hbar)$$

The Attempt at a Solution

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

Perhaps i should say: $$i*\hbar*d<r>/dt=[\varphi, [r,H]\varphi]$$

 

[noblegas]

anybody find a hard time reading the latex code or all of the country

 

[gabbagabbahey]

Perhaps i should say: $$i*\hbar*d<r>/dt=[\varphi, [r,H]\varphi]$$

That looks like the best starting point to me...what do you get when you do that?

 

[noblegas]

$$i*\hbar*d/dt=[\varphi, [r,H]\varphi]=[\varphi, [r,p^2/2m+V(x,t)]\varphi]=[\varphi, (r*p^2/2m+V(x,t)-p^2/2m+V(x,t)*r)\varphi]=1/(i*\hbar*2*m)*<[x,p]*d(p^2)/dp>=(<i*\hbar*2*p>)/(i*\hbar*2*m)=<p>/m$$? Please take a look at my latex code because I don't think latex displayed all of my solution

 

[gabbagabbahey]

Your $\LaTeX$ is terrible!

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

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

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 $r$...why would you write it as a function of $x$ and $t$?

continue from here...

 

[noblegas]

$$i\hbar\frac{d\langle r\rangle}{dt}=(\varphi, [r,H]\varphi)=(\varphi, [r,p^2/2m+V(r)]\varphi)=1/(2*\hbar*m*i)*(<[x,p]d(p^2)/dp>)=1/(2*\hbar*m*i)*(<[i*\hbar*2p>)=<p>/m$$ hope this is better