Understanding the Spin Deviation Operator in Holstein-Primakoff Process

[jackychenp]

Hi,

In Holstein-Primakoff process, n_l = a_l^*a_l and N_l Psi_nl = n_l Psi_nl
Why (1-N_l/2S)^1/2 Psi_nl = (1-n_l/2S)^1/2 Psi_nl ? Since the operator (1-N_l/2S)^1/2 is not linear, I think it should not work as a linear operator. Please let me know if I misunderstood something.

 

[kanato]

You can expand the operator $$(1-n_l/2S)^{1/2}$$ in a Taylor series. Each term will have a power of $$n_l$$. These evaluate directly to $$N_l$$ when operating on a wavefunction if it is an eigenstate of the $$n_l$$ operator. In that case, you wind up with a Taylor series which is the same as your original Taylor series, but with the operator replaced by its eigenvalue. The linearity of the operator will not matter, although you will have significant complications if you have multiple non-commuting operators within your series.

 

[jackychenp]

Kanato, thanks a lot. That can explain.