Help with commutator question please

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In summary, the conversation discusses finding the commutator [H,P] for a general one-dimensional potential v(x), and how to ensure that it is not equal to zero. The participants suggest using the commutation relation [x,p]=i \hbar and expanding the potential as a Taylor series. They also discuss the linearity of the commutator and the role of the h bar term. Finally, they provide a specific example of how to calculate the commutator using differential operators.
  • #1
jus8727
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consider a general one dimensional potential v(x) drive an expression for the commutator [H,P] where h is hamiltonian operator and momentum operator. i keep getting zero and i don't think i should. since next part of homework question sais what condition must v(x) satisfy so that momentum will be constant of the motion recall that this is true if the momentum communtes with the hamiltonian. so i don't think it should zero. any help would be great trtying to do all homework problems to prepare for final!
 
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  • #2
[tex] [P^{2}/2m + V(x) , P] = [ V , P] = -i \frac{dV}{dx}[/tex]
 
  • #3
thats how i started but didnt arrive at that can u show a little bit more detail if u don't mind.
thanks a lot though
 
  • #4
Consider the action of the commutator on a wavefunction

[H, P] psi(x)

(so, we work in the position representation). Write out the commutator explicitely:

H P psi - P H psi
 
  • #5
yes that's what I am doing but I am still getting zero
 
  • #6
Expand the potential as a Taylor series and use the commutation relation [tex][x,p]=i \hbar[/tex].
 
  • #7
samalkhaiat said:
[tex] [P^{2}/2m + V(x) , P] = [ V , P] = -i \frac{dV}{dx}[/tex]

[tex][P^{2}/2m, P] + [V , P] = [V,P][/tex]

because [P,P] = 0.

then

[tex]
[H,P] \phi(x) = [V(x) , id/dx] \phi(x) = iV \frac{d \phi}{dx} -i\frac{d}{dx} \{V(x)\phi(x)\} = \{-i dV/dx\} \phi(x)
[/tex]
 
  • #8
um is there an easier way? like the first respnse seems right but idk how to get there its an elementary problem that i have I am sure
 
  • #9
samal where is the h bar term? is that a property of commutator i have never seen that before
 
  • #10
i think the problem is i don't know what to do when we have an additon in the commutator i don't know how to treat it
 
  • #11
The commutator is linear [A+B,C] = [A,C] + [B,C]. Just write it out and you'll see that.
 
  • #12
jus8727 said:
samal where is the h bar term? is that a property of commutator i have never seen that before

I always set my h bar equal to 1. you can put it back in the equation. The first point you need to know is

[A + B , C] = [A,C] + [B,C]

the second point, an (differential) operator in the commutator acts on all the stuff on the right, so

[tex][A , i\hbar d/dx] f(x) = A(i\hbar df(x)/dx) - i\hbar (d/dx)(Af(x))[/tex]
 
  • #13
oh got it thanks a lot guys!
 

Related to Help with commutator question please

1. What is a commutator and why is it important?

A commutator is a mechanical component that helps reverse the direction of current flow in an electric motor. It is important because it allows the motor to continue spinning in the same direction by switching the direction of current flow in the motor's coils.

2. How does a commutator work?

A commutator is made up of a series of metal segments that are connected to the motor's coils. As the motor spins, the segments pass under the brushes, which are connected to the power source. This causes the current to flow through the coils in a specific direction, allowing the motor to continue spinning in the same direction.

3. What is the purpose of the brushes in a commutator?

The brushes in a commutator play a crucial role in keeping the motor running smoothly. They maintain constant contact with the commutator segments and ensure that the current flows through the coils in the correct direction.

4. How do I troubleshoot issues with a commutator?

If you are experiencing problems with your motor, it is important to check the commutator for any signs of damage or wear. You can also test the continuity of the commutator segments with a multimeter to ensure they are making proper contact with the brushes.

5. Can a commutator be replaced or repaired?

If the commutator is damaged or worn, it may need to be replaced. In some cases, it may be possible to repair a damaged commutator by cleaning it and replacing any worn segments. However, it is typically more cost-effective to replace the commutator entirely.

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