Calculating Total Energy for 1-d Electron Gas

[mblaskovic]

HI!

Also:

I have 1-d electron gas in tight banding model with included interaction between electrons of same spins V[itex]_{\uparrow}[/itex]=-N[itex]_{\uparrow}[/itex]U where U > 0, [itex]\sigma[/itex]=[itex]\pm1[/itex] is spin up or down, and pauli interaction with outside field B is included. I have to calculate the total energy of gas at T=0K as function of of polarization x=[itex]\frac{N_{\uparrow}-N_{\downarrow}}{N}[/itex] expand it to series up to 6th order, minimize it and find the nontrivial solution for U [itex]N=N_{\uparrow}+N_{\downarrow}, N_{\uparrow}=\frac{N}{2}(1+x)[/itex]
and [itex]N_{\downarrow}=\frac{N}{2}(1-x)[/itex]
[itex]N_{\sigma}=\frac{N}{2}(1+{\sigma}x)[/itex]
my major problem is calculating the total energy

 

[mblaskovic]

HI!

My problem is next:

I have 2-d electron gas with included interaction between electrons of opposite spins V[itex]_{\sigma}[/itex]=N[itex]_{\sigma}[/itex]U where U > 0 and [itex]\sigma[/itex]=[itex]\pm1[/itex] is spin up or down, and pauli interaction with outside field B is included. I have to calculate the total energy of gas at T=0K as function of N[itex]_{\sigma}[/itex], then i have to minimize the energy as function of polarization parameter x=[itex]\frac{N_{\uparrow}-N_{\downarrow}}{N}[/itex] and calculate the suscepitibility

[itex]N=N_{\uparrow}+N_{\downarrow}, N_{\uparrow}=\frac{N}{2}(1+x)[/itex], [itex]N_{\downarrow}=\frac{N}{2}(1-x), and N_{\sigma}=\frac{N}{2}(1+{\sigma}x) [/itex]

my major problem is calculating the total energy, the rest is not so tough i am just not sure that my energy is calculated correct...