# Angular Momentum and ladder operators

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

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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
Staff Emeritus
Gold Member
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

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
Staff Emeritus
Gold Member
Gamma said:
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

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.