Angular Momentum and ladder operators

[Gamma]

Hi,

I have done most of the problem in this word document (attached). I have some trouble though. In my QM class, we assumed that the z component of angular momentum Lz satisfies, Lz Ylm = m hbar Ylm and the ladder operator L+ and L- were defined as L+_ = Lx +- iLy. We were able to find the eigen values of Lx and Ly using the ladder operators.

In this problem initially they define Lx to satisfy Lx Ylm = m hbar Ylm and the continue to say tha Lx = 1/2 (L+ + L-). How is this possible? Further, in part C, they are asking to find the eigen values of Lz. I am not sure how to find this. I would like to know what is Lz in terms of L+ and L-. Please help me out if possible. Thank You,

Gamma

 

[Gamma]

My question is: can we have

Lx |l,m> = m hbar |l,m> and

Lx = 1/2 (L+ + L-)

The familier thing in QM is to have Lz |l,m> = m hbar |l,m> and define
L+- as L+_ = Lx +_ i Ly.

Any help and ideas would be greatly appreciated. Thannk You.

Gamma

 

[George Jones]

The question looks fine. Note that in the original statement of the problem, the spherical harmonics are not defined as eigenfunction of L_x. However, since x, y, and z are arbitrary labels, if simultaneous eigenfunction of L^2 and L_z can be found, so can simultaneous eigenfunctions of L^2 and L_x. The question denotes these latter eigenfunctions by \Phi.

The answer in blue for (a) is incorrect. The correct answer is, roughly, that both l and l' label definite values of L^2, and if a state has a definite value of L^2, this value is unique, so l = l'.

I haven't had a chance to look at the other answers, and I have to go do work now, but, if no one else steps in, I'll be back in 2 or 3 hours to give more help.

Regards,
George

 

[Gamma]

The answer in blue for (a) is incorrect. The correct answer is, roughly, that both l and l' label definite values of L^2, and if a state has a definite value of L^2, this value is unique, so l = l'.

In part (a), what I was saying is since Ylm' are eigen functions of L2 and Lx, then superposition of Ylm' should also be an eigen function fo L2 and Lx where m' (= -l, ...,+l ).

if simultaneous eigenfunction of L^2 and L_z can be found, so can simultaneous eigenfunctions of L^2 and L_x. The question denotes these latter eigenfunctions by \Phi.

I agree. But we are expected to do part (b) of this prolem using Lx=1/2 (L+ + L-). L+ and L- are ladder operators. So it looks like L+- has been defined as L+_ = Lx +_ iLy or L+_ = Lx +_ iLz.

I would expect L+_ = Ly +_ i Lz. This is where I am confused. Please clarify if possible. Thank You.

Gamma.

 

[George Jones]

In part (a), what I was saying is since Ylm' are eigen functions of L2 and Lx

But the Y_{lm'} are not eigenfunctions of L_x, the \Phi_{lm'} are.

In blue, you write

L_x Y_{lm} = mћ * Y_{lm}.

This is not true.

However, as you wrote in your last post, that an eigenfunction of L_x is a superposition of the Y_{lm'} for m = -l, ... , +l.

Regards,
George

 

[vroom]

okay, let see the problem like this: as George mentioned x,y,z are arbit labels. We generally use the covariant notations 1,2,3 in place of x,y,z.
So this way, there is no priviage for a,y,or, z.
if a relation is true for L_z in a co-ordinate frame. It'll be true for L_x and L_y too, in a suitable frame. Physics is invariant under such choices.
So u can use the same algebra.