Introductory Quantum HW (Angular Momentum)

[stefan10]

Homework Statement

Consider a three-dimensional system with wave function ψ. If ψ is in the l = 0 state, we already know that Lzψ=0. Show that Lxψ=0 and Lyψ=0 as well.

Homework Equations

[Lx,Ly]ψ = i*h-bar*Lzψ

The Attempt at a Solution

I'm having trouble figuring out where to start this. I think it should be clear and straight-forward, but for some reason I'm just not seeing how I can derive this. I tried using the above equation, to get

[Lx,Ly]ψ=0.

I assume from here I would prove that this is only true of Lxψ and Lyψ are 0. That is assuming, I'm on the right track.

 

[Simon Bridge]

Have you tried that out to see what you get?
i.e. expand out the commutator.

Since you have ##\psi:\text{L}_z\psi=0## do you know what happens when you try to do ##\text{L}_x\psi##?

 

[stefan10]

Have you tried that out to see what you get?
i.e. expand out the commutator.

Since you have ##\psi:\text{L}_z\psi=0## do you know what happens when you try to do ##\text{L}_x\psi##?

The question states Lz ψ = 0, I don't have any specific function for ψ. I think the question makes this claim because Lz= m*h-bar, and m=0 if l=0.

Could I possibly conclude that L^2 = 0 since l=0, and therefore L^2 = Lz^2+Lx^2+Ly^2 implies Lx = Ly =0?

L^2 = l(l+1)h-bar

 

[Simon Bridge]

You don't need a specific function - you have to exploit the relationships: using your understanding of QM.
The question is telling you that the system is prepared in an eigenstate of the Lz operator.

Could I possibly conclude that L^2 = 0 since l=0, and therefore L^2 = Lz^2+Lx^2+Ly^2 implies Lx = Ly =0?

... consider: can operators take values by themselves?
(you will need to be careful about this to do the problem)
What does the ##l## quantum number refer to in this case?

Part of the reason these questions get set is to force you to do a long-winded exploration before finding the simple solution. Thus, you are best advised to settle on a direction for your inquiry and see it through rather than take random jumps around the theory.

Until you settle on something, there's not much I can do to help.
Did you try any of the other suggestions?

 

[HalfLostAndSearching]

Firstly, let's clarify the notation. Lx, Ly, and Lz represent the x, y, and z components of the angular momentum operator, respectively. The wave function ψ represents the state of the system, which can be described by a combination of quantum numbers, including the angular momentum quantum number l.

In this problem, we are given that the system is in the l = 0 state, which means that the total angular momentum is 0, and thus Lx, Ly, and Lz must all equal 0. We can use this information to show that Lxψ=0 and Lyψ=0.

Starting with the given equation [Lx, Ly]ψ = i*h-bar*Lzψ, we can solve for Lxψ and Lyψ separately.

[Lx, Ly]ψ = i*h-bar*Lzψ
LxLyψ - LyLxψ = i*h-bar*Lzψ
Since Lzψ = 0 for the l = 0 state, we can substitute this in to get:
LxLyψ - LyLxψ = 0

Using the commutator relation for angular momentum operators, [Lx, Ly] = i*h-bar*Lz, we can rewrite this equation as:
i*h-bar*Lzψ - LyLxψ = 0
Lzψ = LyLxψ

Since Lzψ = 0 for the l = 0 state, we can conclude that LyLxψ = 0.

Similarly, we can use the commutator relation [Ly, Lx] = -i*h-bar*Lz to show that LxLyψ = 0.

Therefore, we have shown that both Lxψ and Lyψ equal 0 for the l = 0 state, as required. This result makes sense intuitively, as the angular momentum operators represent the three components of angular momentum, and if the total angular momentum is 0, then each component must also be 0.