Can Spin Comutation Relations be Simplified by Using Average Spin Values?

[Petar Mali]

[tex][\hat{S}^+_{n},\hat{S}^-_{m}]=2S_n^z\delta_{n,m}[/tex]

where [tex]n,m[/tex] are position vectors of spins. Why if we have

[tex]\langle\hat{S}^z\rangle\approx S[/tex] we can use

[tex][\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}[/tex]?

where [tex]S[/tex] is value of the spin, for example [tex]\frac{1}{2}[/tex].

 

[DrDu]

Its an approximation. And we know that

[tex]
\langle [\hat{S}^+_{n},\hat{S}^-_{m}]\rangle \approx 2S\delta_{n,m}
[/tex]
So it does not appear implausible.

 

[Petar Mali]

Yes ok. I don't have problem with

[tex] \langle [\hat{S}^+_{n},\hat{S}^-_{m}]\rangle \approx 2S\delta_{n,m}[/tex]

but where [tex]\langle... \rangle[/tex] disappear?

Why I can use this approximation?

[tex]
[\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}
[/tex]

 

[Petar Mali]

If relation [tex]

[\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}

[/tex]

is correct than

we can define

[tex]\hat{S}^-_n \rightarrow \sqrt{2S}\hat{B}_n^+[/tex]

[tex]\hat{S}^+_n\rightarrow \sqrt{2S}\hat{B}_n[/tex]

which is Bloch approximation but I don't see some reasons for

[tex]

[\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}

[/tex]

 

[DrDu]

Why I can use this approximation?

You can always use whatever approximation you like. The question is whether it is good or bad. This depends on the system you use it for. A point about which you didn't tell us anything.

 

[Petar Mali]

Well I have some magnetic ordered system. For example ferromagnet or antiferromagnet and have some spin hamiltonian. I want to replace operators [tex]\hat{S}^{\pm},\hat{S}^z[/tex] with some functions of Bose operators. Temperatures are low.

 

[DrDu]

Well, what you could try is to express the hamiltonian in terms of [tex] \hat{B}_n^+[/tex] and
[tex]
\hat{S}^-_n - \sqrt{2S}\hat{B}_n^+
[/tex]
considering terms containing the difference as a perturbation. Maybe you can show that they only lead to small corrections in the limit you are considering?