Can the Commutation of Spin Operator and Magnetic Field Yield a Cross Product?

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Homework Help Overview

The discussion revolves around the commutation of the spin operator with a uniform magnetic field, specifically exploring whether this commutation can yield a result analogous to the cross product of the two entities. The subject area is quantum mechanics, focusing on spin operators and their interactions with magnetic fields.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss representing the spin operator and magnetic field using Pauli spin matrices. There is uncertainty about the correct representation of the spin operator and whether the commutation results in zero. Some participants question the necessity of using specific commutation relations and explore the implications of the Hamiltonian's structure.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the use of Pauli matrices and commutation relations, but there is no explicit consensus on the approach or resolution of the questions raised.

Contextual Notes

There is a noted ambiguity in the original problem statement, with participants suggesting that the question may need to be reframed to clarify the relationship between the spin operator and the Hamiltonian in the context of a uniform magnetic field.

n0_3sc
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Homework Statement



I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them.
Does this make sense? I don't see how it can be possible.

Homework Equations



[s,B]

(The s should also have a hat on it)

The Attempt at a Solution



I have sB - Bs but do i represent s as (sx,sy,sz)? x,y,z are subscripts...
Even if I do that wouldn't the commutation = 0?
 
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Represent both spin and the magnetic field in terms of Pauli spin matrices.
 
n0_3sc said:

Homework Statement



I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them.
Does this make sense? I don't see how it can be possible.

Homework Equations



[s,B]

(The s should also have a hat on it)

The Attempt at a Solution



I have sB - Bs but do i represent s as (sx,sy,sz)? x,y,z are subscripts...
Even if I do that wouldn't the commutation = 0?


As stated, the question does not quite make sense. I think you mean the commutator of the spin with the hamiltonian of a particle in a uniform B field, H = \vec{s} \cdot \vec{B} . Then you simply have to use the commutation relation of the Pauli matrices [S_i,S_j] = i \epsilon_{ijk} S_k and the result follows trivially (except that it seems to me that one gets "i" times the cross product)

Patrick
 
nrqed:
So I evaluate [H,s]? In doing that, why would I need the commutation relation [S_i,S_j] = i \epsilon_{ijk} S_k ? It shouldn't be needed if the product terms are only between terms of H and s_x, s_y, s_z.
 
n0_3sc said:
nrqed:
So I evaluate [H,s]? In doing that, why would I need the commutation relation [S_i,S_j] = i \epsilon_{ijk} S_k ? It shouldn't be needed if the product terms are only between terms of H and s_x, s_y, s_z.

But H contains the spin! See my post.
 

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