Understanding Coupled Spin Operators

In summary, the conversation discusses the derivation of the total spin operator in a hydrogen atom and the issue of applying non-commutative spin operators to obtain eigenvalues and eigenvectors. The importance of understanding the difference between spin magnitude and spin projection is also emphasized.
  • #1
cooev769
114
0
Trying to get my head around this one. Given that you can have a proton and an electron in a hydrogen atom for example, and they can create a singlet or triplet configuration, with spin 1 and spin 0 respectively. The total spin operator can be derived as:

S^2 = (Se + Sp)^2 = Se^2 + Sp^2 + 2Se.Sp

Where the dot is the dot product, but then what you get is a bunch of Sx,Sz,Sy acting on the tensor products to provide eigenvalues and the eigenvectors out. The math is all well and good, but the problem I have is that the spin operators Sz,Sy and Sx don't commute, so how come you can just apply the operators to the respective spins to get out an answer, seems odd to me. Any help would be appreciated.

Cheers.
 
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  • #2
What, are you saying that the components of the spin of the proton don't commute with those of the electron?
 
  • #3
No. I'm saying that the component of the electron don't commute with each other yet they're evaluated in the dot production order to solve for the S^2 operator. But this seems daft to me given they do not commute.
 
  • #4
cooev769 said:
No. I'm saying that the component of the electron don't commute with each other yet they're evaluated in the dot production order to solve for the S^2 operator. But this seems daft to me given they do not commute.
Write out the dot product. Do you get any terms that are affected by noncommutativity?
 
  • #5
I understand what you're saying strange rep, but by that logic wouldn't that mean I can just write out the total spin operator as:

S = Sz + Sy + Sx

And apply this to any state to get the total spin. It's effectively the same thing.
 
  • #6
cooev769 said:
[...] wouldn't that mean I can just write out the total spin operator as:

S = Sz + Sy + Sx
If each of your terms on the rhs are vectors, then S is also a vector. If that's what you meant, then let's write them properly, in bold, i.e.,

S = Sz + Sy + Sx.

Now draw a diagram with 3 orthogonal axes in 3D space and draw a vector from the origin to the point (1,1,1), i.e., x=1, y=1, z=1.

The vector S corresponds to spin-projection along that direction. This is quite different from the spin magnitude, i.e., the length of the vector.

The direction of a vector is not invariant under rotations, but the length is.

[...] And apply this to any state to get the total spin. It's effectively the same thing.
Hopefully you can now see that it's not the same thing?
 
  • #7
Okay I'll keep contemplating your answer. Thanks.
 
  • #8
Ah I get it now, thank you.
 

FAQ: Understanding Coupled Spin Operators

1. What are coupled spin operators?

Coupled spin operators are mathematical operators that are used to describe the behavior of coupled spin systems in quantum mechanics. These operators are used to calculate the spin states and spin interactions of particles with intrinsic angular momentum, such as electrons.

2. How are coupled spin operators related to spin states?

Coupled spin operators are related to spin states through the concept of superposition. Spin states can be represented as linear combinations of eigenstates of coupled spin operators, which describe the possible orientations of the particle's spin.

3. What is the significance of coupled spin operators in quantum mechanics?

Coupled spin operators are crucial in understanding the behavior of particles with spin in quantum mechanics. They allow us to calculate the probabilities of different spin states and the effects of spin interactions, which are fundamental to many physical phenomena.

4. How are coupled spin operators used in experimental research?

Coupled spin operators are used in experimental research to design and interpret experiments that involve particles with spin. By using these operators, scientists can predict the outcomes of experiments and verify the theoretical predictions of quantum mechanics.

5. Are there any practical applications of understanding coupled spin operators?

Yes, understanding coupled spin operators has many practical applications in fields such as quantum computing, magnetic resonance imaging (MRI), and spintronics. These applications rely on our understanding of spin states and spin interactions, which are described by coupled spin operators.

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