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$_{\uparrow}$=-N$_{\uparrow}$U where U > 0, $\sigma$=$\pm1$ 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=$\frac{N_{\uparrow}-N_{\downarrow}}{N}$ expand it to series up to 6th order, minimize it and find the nontrivial solution for U $N=N_{\uparrow}+N_{\downarrow}, N_{\uparrow}=\frac{N}{2}(1+x)$
and $N_{\downarrow}=\frac{N}{2}(1-x)$
$N_{\sigma}=\frac{N}{2}(1+{\sigma}x)$
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$_{\sigma}$=N$_{\sigma}$U where U > 0 and $\sigma$=$\pm1$ 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$_{\sigma}$, then i have to minimize the energy as function of polarization parameter x=$\frac{N_{\uparrow}-N_{\downarrow}}{N}$ and calculate the suscepitibility

$N=N_{\uparrow}+N_{\downarrow}, N_{\uparrow}=\frac{N}{2}(1+x)$, $N_{\downarrow}=\frac{N}{2}(1-x), and N_{\sigma}=\frac{N}{2}(1+{\sigma}x)$

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...