Quantum mechanics - Find S_x and S_y

AI Thread Summary
The discussion focuses on finding the operators S_x and S_y in quantum mechanics, specifically addressing confusion around the origin of the factor 1/2 in the calculations. Participants suggest using the relationship between the raising and lowering operators, S_+ and S_-, to derive S_x and S_y. The algebraic manipulation involves solving a system of equations derived from these operators. Clarification is provided on how to express the operators in terms of matrices to finalize their representations. The conversation emphasizes the importance of understanding the algebra involved in quantum mechanics calculations.
Graham87
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Homework Statement
Lecture slide (see photo)
Relevant Equations
See second last row on picture
I have a lecture slide that shows how to find S_x and S_y. I get all the steps except the last row.
Where did 1/2 come from? I think my linear algebra needs polishing.

Thanks!

3B929476-C8AE-400E-8B4E-053387D6B2CF.jpeg
 
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Have you tried to use ##\hat S_{\pm} = \hat S_x \pm i\hat S_y## to solve for ##\hat S_x## and ##\hat S_y##?

I suggest that if you do the algebra yourself, you'll see where the ##\frac 1 2## arises.
 
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PeroK said:
Have you tried to use ##\hat S_{\pm} = \hat S_x \pm i\hat S_y## to solve for ##\hat S_x## and ##\hat S_y##?

I suggest that if you do the algebra yourself, you'll see where the ##\frac 1 2## arises.
Thanks. I have, and I get something like this:
I think what confuses me is the algebra.
7AD68287-B620-4D29-8B18-915F7F5FDFE4.jpeg
 
$$S_+ + S_- = (S_x + iS_y) + (S_x - iS_y) = ?$$
 
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It is basically just a system of equations
##A = x + \mathrm{i}y##
##B = x -\mathrm{i}y##
solve for ##x## and ##y## in terms of ##A## and ##B##
 
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Aha thanks alot! Got it!
 
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and then when you are done with that, insert the matrices, and find the final matrix representation of those operators.
 
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