Exploring the Quantum Theory of Magnetism: Callen's Result & the P_S Function

[Petar Mali]

[tex]\langle\hat{S}^z\rangle=\hbar\frac{(S-P_S)(1+P_S)^{2S+1}+(1+S+P_S)P_S^{2S+1}}{(1+P_S)^{2S+1}-P_S^{2S+1}}[/tex]

Can you tell me something more about Callen result? And this function [tex]P_S[/tex]?

In book Nolting "Quantum theory of magnetism" this relation is get from diferential equation

[tex]\frac{d^2\Omega}{d\alpha^2}+\frac{(1+P_S)+P_Se^{-a\hbar}}{(1+P_S)-P_Se^{-a\hbar}}\hbar\frac{d\Omega}{d\alpha}-\hbar^2S(S+1)\Omega=0[/tex]

two boundary conditions are

[tex]\Omega(0)=1[/tex]

and

[tex]\prod^{S}_{m_S=-S}(\frac{d}{d\alpha}-\hbar m_S)\Omega(\alpha)|_{\alpha=0}=0[/tex]

How he get this diferential equation?

 

[BigJDubs]

The differential equation is derived using the standard approach of transforming the Hamiltonian into a form suitable for a variational calculation. The starting point is the mean-field Hamiltonian for a Heisenberg spin system, which can be written asH=-\sum_{i,j}J_{ij}{{\bf S}_i}\cdot{{\bf S}_j}where the sum is over all pairs of spins i and j. This can be written in terms of raising and lowering operators asH=\sum_{i,j}J_{ij}\left(\frac{1}{2}\left(S_i^+S_j^-+S_i^-S_j^+\right)-\frac{1}{2}\left(S_i^zS_j^z+S_i^z+S_j^z+1\right)\right)The next step is to introduce a variational parameter \alpha which is related to the average spin \langle S_i^z\rangle by\langle S_i^z\rangle=S-\alphaThis allows us to write the Hamiltonian asH=\sum_{i,j}J_{ij}\left(\frac{1}{2}\left(e^{-\alpha\hbar}S_i^+S_j^-+e^{\alpha\hbar}S_i^-S_j^+\right)-\frac{1}{2}\left(S_i^zS_j^z+(S-\alpha)(S_j^z+1)+\alpha(S_i^z+1)\right)\right)We can then define a “trial wave function” \Omega(\alpha) which depends on \alpha as\Omega(\alpha)=\langle\alpha|\Psi\ranglewhere \Psi is the ground state wave function. We can then calculate the expectation value of the Hamiltonian with respect to this wave function by\langle H\rangle_\Omega=\langle\alpha|H|\alpha\rangleSubstituting the expression for the Hamiltonian into this equation yields\langle H\rangle_