Help Finding eigenvalues of angular momentum operators

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Homework Help Overview

The discussion revolves around finding the eigenvalues of a linear combination of angular momentum operators, specifically the expression 3/5 Lx - 4/5 Ly. Participants are exploring connections to the L^2 and Lz operators in the context of angular momentum in quantum mechanics.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to connect the given expression to known operators and their properties, questioning how to apply the relationship L^2 = Lx^2 + Ly^2 + Lz^2. There are mentions of diagonalization and the use of rotation operators, as well as considerations of symmetry along the z-axis.

Discussion Status

The discussion is active, with various approaches being suggested, including the use of rotation operators and ladder operators. Some participants express confusion about how to proceed without matrix forms, while others are attempting to clarify the requirements of the problem, specifically whether eigenvectors are also needed.

Contextual Notes

Participants note that the problem is derived from an old homework assignment, indicating potential constraints on the methods they can use. There is also uncertainty regarding the completeness of the problem statement, as indicated by questions about the ellipsis in the original post.

thebigstar25
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urgent help!.. Finding eigenvalues of angular momentum operators

the question is asking to find the eigenvalues of:

3/5 Lx - 4/5 Ly ...


I feel that i have to connect it with the L^2 and Lz operators but i just have no idea how to start .. any suggestions will be greatly appreciated ..
 
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Why is there a ..., is there more to the problem?

If you know the matrix forms of the operators (are you in 2D, 3D, what?), then all you have to do is subtract and diagonalize.
 


... R just dots , and I am supposed to do it without considering the matrix form :(
 


Ah, okay, take a general ket and work with it then.

You should know L^2|l,m> and Lz|l,m> and that L^2=Lx^2+Ly^2+Lz^2
 


that what I was confused about .. how would I apply L^2=Lx^2+Ly^2+Lz^2 in order to find 3/5 Lx - 4/5 Ly , since the expression I have without "square" .. I am not sure how to connect them together..
 


Once you know the answer, it seems like it should have been obvious. One suggestion is to think about rotations.

A different approach would be to write Lx and Ly in terms of the ladder operators L+ and L-, but I'm not sure how this way works out.
 


i tried ur second suggestion but things got messy there .. How would i make use of the symmetry along z axis..
 


Does it only ask for eigenvalues, or eigenvectors as well? Because eigenvalues are very simple to get using rotations as vela noted.
 
Last edited:


it is just the eigenvalues .. My problem is that I am not sure how to apply that to my question .. Anyway, this question from an old homework and i just wanted to know how to solve it ..
 
  • #10


First, do you know what the rotation operator for angular momentum is? Since that will help you solve this problem.
 
  • #11


no i don't know, or maybe i do but i don't know that it is called rotation operator?.. I am not sure :/
 

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