Quantum Mechanics - Matrix representations

In summary, the conversation is about finding the operators Jx and Jy using J+ and J- and forming a matrix with the correct entries. The conversation includes suggestions and tips on how to approach the problem and correct errors.
  • #1
Graham87
63
16
Homework Statement
See pic
Relevant Equations
See pic
1FBB25AF-72CA-4AFE-8E80-1634FE430ABA.jpeg

I have found J^2 and Jz, but I am not sure how to find Jx and Jy.
I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.

07FECEF0-D48C-4642-B066-D8B1A231D831.jpeg


Thanks!
 
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  • #2
Graham87 said:
Homework Statement:: See pic
Relevant Equations:: See pic

View attachment 313615
I have found J^2 and Jz, but I am not sure how to find Jx and Jy.
I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.

View attachment 313617

Thanks!
Using ##J_{\pm}## sounds like a good idea. Show us what you get.
 
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  • #3
Graham87 said:
I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.
You have to do it the other way around: Express ##J_x## and ##J_y## in terms of ##J_+## and ##J_-##.
 
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  • #4
68534854-CFAB-47FA-91C5-B3670758D0A8.jpeg

I got this, but it gets messy when I try to find J+- with this expression:
00BEB4E3-F3C6-4034-979E-40D7D12B4490.jpeg

I don’t know how to form the matrix, what goes where.
 
  • #5
Matrix elements are found using ##\braket{j',m' | J_x | j, m}##, so you can start by calculating ##J_+ \ket{j,m}## and ##J_- \ket{j,m}## explicitly for the 4 ##\ket{j,m}## states you need to consider here.
 
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  • #6
Something like this?

D8B422B4-599F-4EFE-BF29-43F27E1374BF.jpeg

05C651B6-5D22-4A12-AD53-07208E8C2354.jpeg
 
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  • #7
Good effort, but I don't agree with some of the entries in the ##J_{\pm}## matrices.
 
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  • #8
Here's a tip. A lot of physics textbooks write one formula with various ##\pm## and ##\mp##. I find it easier to keep the formulas separate with either ##+## or ##-##.
 
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  • #9
Graham87 said:
Where in the matrix did you not agree ?
The non-zero entries!
 
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  • #10
PeroK said:
The non-zero entries!
Thanks! Just noticed my error.
cheers!
 

Related to Quantum Mechanics - Matrix representations

1. What is the concept of matrix representations in quantum mechanics?

The concept of matrix representations in quantum mechanics involves representing physical quantities such as position, momentum, and energy as matrices. This allows for the mathematical description of quantum systems and their behavior.

2. How are matrix representations used in quantum mechanics?

Matrix representations are used in quantum mechanics to describe the state of a quantum system, calculate probabilities of different outcomes, and predict the evolution of the system over time. They also allow for the calculation of observables, such as energy and momentum, from the system's wave function.

3. What are the benefits of using matrix representations in quantum mechanics?

Matrix representations provide a mathematical framework for understanding and predicting the behavior of quantum systems. They allow for the calculation of probabilities and observables, which can be compared to experimental results. They also provide a way to visualize and manipulate complex quantum systems.

4. How do matrix representations relate to other mathematical concepts in quantum mechanics?

Matrix representations are closely related to other mathematical concepts in quantum mechanics, such as wave functions, operators, and eigenvalues. They provide a way to connect these concepts and make predictions about the behavior of quantum systems.

5. Are there any limitations to using matrix representations in quantum mechanics?

While matrix representations are a powerful tool in quantum mechanics, they do have limitations. They can only be applied to systems with a finite number of states and may not accurately describe systems with continuous variables. Additionally, the calculations involved in matrix representations can become complex and time-consuming for larger systems.

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