View source for A solved problem for spin waves

A chosen problem with solution by Hossein.

'''Problem:'''

(a) Making use of the spin commutation relation,<math>[s_{i}^{\alpha },s_{j}^{\beta }]=i\delta _{ij}\varepsilon ^{\alpha \beta \gamma }s_{i }^{\gamma }</math> , apply the identity

<math> i\dot{s}_{i}=[\hat{s_{i}},\hat{H}]</math>

To express the equation of motion of a spin in a nearest neighbor spin S one dimensional Heisenberg ferromagnet.

(b)Interpreting the spins as classical vectors, and taking the continuum limit, show that the equation of motion of the hydrodynamics modes takes the form 

<math>\dot{s}=Ja^{2}S\times {\partial }^{2}S,</math> 

Where a denotes the lattice spacing.

(c) Confirm that the equation of motion is solved by the Ansatz,

<math>S=(cCos(\omega t-kx),cSin(kx-\omega t)),\sqrt{S^{2}-c^{2}})</math> 

and determine the dispersion, sketch a snapshot configuration of the spins in the chain.

'''Solution''':

(a) Making the use of equation of motion <math> i\dot{s}_{i}=[\hat{s_{i}},\hat{H}]</math> and the commutation relation ,<math>[s_{i}^{\alpha },s_{j}^{\beta }]=i\delta _{ij}\varepsilon ^{\alpha \beta \gamma }s_{i }^{\gamma }</math>, we obtain 

<math> i\dot{s}_{i}=J_{i}\hat{s}_{i}\times (\hat{s}_{i+1}+\hat{s}_{i-1})</math>

(b) Interpreting the spins as classical vectors, and applying the Taylor expansion 

<math>\hat{S}_{i+1}=S_{i}+a{\partial }S_{i}+(\frac{a^{2}}{2}){\partial }^{2}S_{i}</math>

We obtain the classical equation of motion 


<math>\dot{s}=Ja^{2}S\times {\partial }^{2}S,</math>

Substituting, we find that <math>S=(cCos(\omega t-kx),cSin(kx-\omega t)),\sqrt{S^{2}-c^{2}})</math>,
satisfies the equation of motion with  <math>\omega =J(ka^{2})\sqrt{S^{2}-c^{2}}
</math>.

(c) The corresponding spin wave solution has the precessional from shown in blow figurer.

[[Image:2.JPG|thumb|center|450.px| Spin-wave dispersion.]]