Exploring the Commutation of Spin Operator and Magnetic Field

[n0_3sc]

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?

 

[George Jones]

Represent both spin and the magnetic field in terms of Pauli spin matrices.

 

[nrqed]

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

 

[n0_3sc]

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 $$s_x, s_y, s_z$$.

 

[nrqed]

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.