Demonstrating [L,H] Commutes with V Dependent on r Only

  • Thread starter mmwave
  • Start date
  • Tags
    Commutator
In summary, the conversation is about demonstrating that angular momentum and Hamiltonian commute when V is a function of radius r only. The discussion involves using L = Lx + Ly + Lz components and the commutator definition. The conversation also mentions the need to change to a coordinate system with r and two angles, and working out what d/dx is in these coordinates. The person asking for help also mentions using spherical coordinates and filling in the coordinates for x, y, and z. Eventually, a suggestion is given to write d/dx as a product of dr/dx and x*(x^2+y^2+z+2)^3/2.
  • #1
mmwave
647
2
I'm trying to demonstrate that angular momentum and the Hamiltonian commute provided that V is a function of radius r only. Using L = Lx + Ly + Lz components, the definition of Lx = yPz - zPy, etc, the commutator definition and the fact that H = p^2/2m + V. I can reduce [Lx,H] to the following:

[Lx,H] = [yPz,V] - [ZPy,V] and by expanding using and the relation [AB,C] = A[B,C] + [A,C]B and the Pz = hbar/i * d/dz, etc. get

[Lx,H] = hbar/i * ( yf dV/dz - zf dV/y) where f is a test function.

I have not used the fact that V is a function of r only and I can't take this any further.

I know that ( yf dV/dz - zf dV/y) must equal zero but have no idea how to prove it. Please help with a suggestion.

(I believe that H commutes with each component Lx, Ly and Lz and not just L so I don't think the terms above will cancel with terms from [Ly,H] and [Lz,H].)
 
Physics news on Phys.org
  • #2
You should change to a coordinate system with r and two angles. That is , to spherical coordinates. You will have to work out what d/dx is in these coordinates and so on. A lot of writing but that is physics. Don't forget to fill in the spherical coordinates for x, y and z as well.
 
  • #3
Originally posted by mmwave

(I believe that H commutes with each component Lx, Ly and Lz and not just L so I don't think the terms above will cancel with terms from [Ly,H] and [Lz,H].)

This I now think is wrong. You need to look at all 3 components of L to get it to commute.

The book sometimes makes problems hard but it would have a whole section done in rectangular coordinates and then give a problem that must be done spherical coordinates. That's in the next section!

Any other suggestions?
 
  • #4
Hmmm...that would be a bit strange indeed. However, if you have a spherical symmetric potential it would be easiest to use spherical coordinates. What book are you using? In my QM classes we used Griffiths and I can remember that sometimes he gave a sort of sneak preview...You can also write d/dx=d/dr*dr/dx and same for d/dy and d/dz. You can write out dr/dx=x*(x^2+y^2+z+2)^3/2...I think that will work (just gave it a quick glance).

Good luck
 
  • #5
Thanks for the suggestion! Worked great.

dr/dx = x *(x^2+y^2+z^2)^(-1/2), etc.
 

What does it mean for [L,H] to commute with V dependent on r only?

Commute refers to the mathematical concept of two operators being able to be applied in any order without changing the outcome. In this case, [L,H] commuting with V dependent on r only means that the operators [L,H] and V can be applied in any order without changing the result as long as V is only dependent on the variable r.

Why is it important to demonstrate that [L,H] commutes with V dependent on r only?

Demonstrating this property is important in quantum mechanics because it allows us to simplify calculations and make accurate predictions about the behavior of a system. It also helps us understand the relationship between different operators and how they affect the state of a system.

How is the commutator [L,H] related to the operators L and H?

The commutator [L,H] is a mathematical operation that is defined as the difference between the product of the operators L and H and the product of H and L in that order. It is often used to study the behavior of a system and determine if two operators commute with each other.

What does it mean for V to be dependent on r only?

V being dependent on r only means that the operator V only depends on the variable r and is independent of other variables. This allows us to simplify calculations and analyze the behavior of a system more easily.

How can we demonstrate that [L,H] commutes with V dependent on r only?

To demonstrate this property, we can use the properties of the commutator and the operators L and H to show that the commutator [L,H] times V is equal to V times [L,H]. This will prove that the operators commute with each other and that V is dependent on r only.

Similar threads

Replies
8
Views
986
Replies
19
Views
2K
  • Advanced Physics Homework Help
Replies
12
Views
25K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Special and General Relativity
Replies
1
Views
1K
  • Classical Physics
Replies
5
Views
5K
Replies
2
Views
3K
  • Quantum Physics
Replies
19
Views
3K
  • Advanced Physics Homework Help
Replies
3
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Back
Top