Is my procedure for finding the S_z matrix correct?

In summary, the conversation is discussing how to find the matrices representing the operators S^2 and S_z for the case of two spin half particles. The speaker suggests using the basis |++>, |+->, |-+>, |--> and adding the two spin operators together. They also discuss the importance of understanding the properties of S_z and S^2, and the use of the tensor product symbol (\otimes) in this context. However, it is noted that the speaker is not familiar with this symbol and asks if their alternate method of adding 4x4 matrices would yield the correct result. The other speaker explains the formal notation for the vectors and how to properly use the tensor product in finding the matrices. They also clarify that the
  • #1
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Homework Statement
the matrices representing the operators S^2 and S_z
Relevant Equations
the matrices representing the operators S^2 and S_z
I have this homework: consider the case of two spin half particles. Use the basis: |++>, |+->, |-+>, |--> to find the matrices representing the operators S^2 and S_z.
My idea for the solution for S_z is: S_z=S_z(1)+S_z(2) where S_z(1) is the operator for the first particle ... etc
So I will first find the S_z(1) matrix. The first element in the matrix will be: <++|S_z(1)|++>=(hbar/2) the second element will be <++|S_z(1)|+->=0 ... etc Where I finally get a diagonal matrix. Is this procedure correct?
 
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  • #2
Viona said:
Homework Statement:: the matrices representing the operators S^2 and S_z
Relevant Equations:: the matrices representing the operators S^2 and S_z

I have this homework: consider the case of two spin half particles. Use the basis: |++>, |+->, |-+>, |--> to find the matrices representing the operators S^2 and S_z.
My idea for the solution for S_z is: S_z=S_z(1)+S_z(2) where S_z(1) is the operator for the first particle ... etc
So I will first find the S_z(1) matrix. The first element in the matrix will be: <++|S_z(1)|++>=(hbar/2) the second element will be <++|S_z(1)|+->=0 ... etc Where I finally get a diagonal matrix. Is this procedure correct?
What do you mean by ##S_z = S_z(1) + S_z(2)##?

What dimension of matrices are you looking for?
 
  • #3
I am looking for 4x4 matrices.
 
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  • #4
PeroK said:
What do you mean by ##S_z = S_z(1) + S_z(2)##?
I think he means ##S_z=s_{z1}+s_{z2}##. Then $$(s_{z1}+s_{z2})|+-\rangle=\frac{\hbar}{2}|+-\rangle+\left(-\frac{\hbar}{2}\right)|+-\rangle.$$
 
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  • #5
kuruman said:
I think he means ##S_z=s_{z1}+s_{z2}##. Then $$(s_{z1}+s_{z2})|+-\rangle=\frac{\hbar}{2}|+-\rangle+\left(-\frac{\hbar}{2}\right)|+-\rangle.$$
Yes this what i mean.
 
  • #6
Viona said:
I am looking for 4x4 matrices.
What do you know about the properties of ##S_z## and ##S^2## on the space of two-particle spin states? Do you know how they operate on the spin states you are given? Or, do you have to work that out from first principles?
 
  • #7
Viona said:
Yes this what i mean.
Technically that means that $$S'_z = S_z \otimes I + I \otimes S_z$$Where ##S'_z## is the z-spin operator on the two-particle space and ##I## is the identity operator. I think it helps sometimes to know this so that you don't go wrong oversimplifying the product of operators.
 
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  • #8
PeroK said:
Technically that means that $$S'_z = S_z \otimes I + I \otimes S_z$$Where ##S'_z## is the z-spin operator on the two-particle space and ##I## is the identity operator. I think it helps sometimes to know this so that you don't wrong oversimplifying the product of operators.
The problem is I am not familiar with this type of product using this symbol: \otimes ! So will it be wrong if I used the way I described above a got the correct matrices?
 
  • #9
Viona said:
The problem is I am not familiar with this type of product using this symbol: \otimes ! So will it be wrong if I used the way I described above a got the correct matrices?
You haven't said what you're going to do about ##S^2## - that may be the tricky one. What do you get for ##S_z##?

PS I don't understand why, in your original post, you appear to be using only ##S_z(1)##.
 
  • #10
For Sz: I added the two 4x4 matrices together: Sz1 + Sz2 and I got 4x4 diagonal matrix with diagonal elements: hbar, 0, 0, hbar.
 
  • #11
for S^2 I will say: S^2=(S1+S2)^2= (S1)^2 + (S2)^2 +2S1.S2 where S1.S2=Sx1Sx2+Sy1Sy2+Sz1Sz2
and add all these 4x4 matrices together to find S^2
 
  • #12
Viona said:
The problem is I am not familiar with this type of product using this symbol: \otimes ! So will it be wrong if I used the way I described above a got the correct matrices?
The formal notation for the vectors is, for example:$$|+-\rangle = |+\rangle \otimes |-\rangle$$ In this case, then:$$S'_z|+-\rangle = (S_z \otimes I + I \otimes S_z)(|+\rangle \otimes |-\rangle) = (S_z|+\rangle \otimes I |-\rangle) + (I|+\rangle \otimes S_z |-\rangle)$$$$ = (\frac \hbar 2 |+\rangle \otimes |-\rangle) + (|+\rangle \otimes \frac \hbar 2 |-\rangle) = \hbar (|+\rangle \otimes |-\rangle)$$And we see that ##|+-\rangle## is an eigenstate of ##S'_z## with eigenvalue ##\hbar##.

You can drop the prime - I was just using it to make it clear that it's technically a different operator on the two-particle space.
 
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  • #13
Viona said:
for S^2 I will say: S^2=(S1+S2)^2= (S1)^2 + (S2)^2 +2S1.S2 where S1.S2=Sx1Sx2+Sy1Sy2+Sz1Sz2
and add all these 4x4 matrices together to find S^2
That's not going to work: $$S^2 = S_x^2 + S_y^2 + S_z^2 = \vec S \cdot \vec S$$
 
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  • #14
PS Calculating ##S^2## from first principles is more work than you might think! That's why I asked if you already know how the operator ##S^2## acts on the two-particle space.

This is tricky until you get the hang of it, so don't worry if you are confused at this stage.
 
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  • #15
PeroK said:
What do you know about the properties of ##S_z## and ##S^2## on the space of two-particle spin states? Do you know how they operate on the spin states you are given? Or, do you have to work that out from first principles?
I only studied two spin half particles from the book Introduction to Quantum Mechanics by David J. Griffiths. I do not know if this enough For example see the attached picture.
77777.png
 
  • #16
Viona said:
I only studied two spin half particles from the book Introduction to Quantum Mechanics by David J. Griffiths. I do not know if this enough For example see the attached picture.
View attachment 289739
I know he uses that shorthand notation, but that was something I didn't like. I learned QM from that book but I took some time to find out about the tensor product of hilbert spaces ##\otimes##. As, otherwise, I couldn't figure out what was going on with those composite operations.

If you've got that book, then look at the triplet and singlet states on page 185 (2nd edition). That should give a big clue of how ##S^2## works.

Also, Griffiths does some of the calculations for ##S^2## on page 186: note the decomposition of ##S^2## as I indicated above:

PeroK said:
That's not going to work: $$S^2 = S_x^2 + S_y^2 + S_z^2 = \vec S \cdot \vec S$$
 
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  • #17
Viona said:
For Sz: I added the two 4x4 matrices together: Sz1 + Sz2 and I got 4x4 diagonal matrix with diagonal elements: hbar, 0, 0, hbar.
It might just be a typo, but there's a negative sign missing on one of the ##\hbar##s.
 
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  • #18
vela said:
It might just be a typo, but there's a negative sign missing on one of the ##\hbar##s.
Where? In post #15?
 
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  • #19
In the post I quoted, #10.
 
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  • #20
vela said:
In the post I quoted, #10.
Ah yes. That matrix should be traceless. Thanks.
 
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1. How do I know if my procedure for finding the Sz matrix is correct?

The best way to check the correctness of your procedure is to compare your results with a known or published Sz matrix. If your results match, then your procedure is likely correct. You can also double-check your calculations and make sure you have followed all the necessary steps accurately.

2. What are the key steps I should follow when finding the Sz matrix?

The key steps for finding the Sz matrix include identifying the spin quantum numbers, constructing the basis states, finding the raising and lowering operators, and then applying them to the basis states to obtain the matrix elements. It is also important to pay attention to any necessary normalization factors.

3. Can I use any method to find the Sz matrix or are there specific techniques I should follow?

There are specific techniques and formulas that should be followed when finding the Sz matrix. These include using the spin angular momentum operators and their commutation relations, as well as applying the ladder operators to the basis states. It is important to use the correct formulas and follow the necessary steps for accurate results.

4. Is it possible to have more than one correct Sz matrix for a given system?

Yes, it is possible to have more than one correct Sz matrix for a given system. This can occur if there are degenerate energy levels, meaning multiple states have the same energy. In this case, there may be multiple valid ways to construct the Sz matrix.

5. What are some common mistakes to avoid when finding the Sz matrix?

Some common mistakes to avoid when finding the Sz matrix include forgetting to include normalization factors, using incorrect formulas or commutation relations, and making errors in the application of the ladder operators. It is important to double-check your calculations and follow the necessary steps accurately to avoid these mistakes.

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