Introduction to Quantum Mechanics · Problem 5.39

Identical Particles — David J. Griffiths

Physics — Quantum Mechanics · periodic_boundary_conditions

Problem

In Section 5.3.1 we put the electrons in a box with impenetrable walls. The same results can be obtained using periodic boundary conditions. We still imagine the electrons to be confined to a box with sides of length lxl_x, lyl_y, and lzl_z but instead of requiring the wave function to vanish on each wall, we require it to take the same value on opposite walls: ψ(x,y,z)=ψ(x+lx,y,z)=ψ(x,y+ly,z)=ψ(x,y,z+lz).\psi(x, y, z) = \psi(x + l_x, y, z) = \psi(x, y + l_y, z) = \psi(x, y, z + l_z). In this case we can represent the wave functions as traveling waves, ψ=1lxlylz eik⋅r=1lxlylz ei(kxx+kyy+kzz),\psi = \frac{1}{\sqrt{l_x l_y l_z}}\, e^{i \mathbf{k} \cdot \mathbf{r}} = \frac{1}{\sqrt{l_x l_y l_z}}\, e^{i(k_x x + k_y y + k_z z)}, rather than as standing waves (Equation 5.49). Periodic boundary conditions—while certainly not physical—are often easier to work with (to describe something like electrical current a basis of traveling waves is more natural than a basis of standing waves) and if you are computing bulk properties of a material it shouldn't matter which you use. (a) Show that with periodic boundary conditions the wave vector satisfies kxlx=2nxπ,kyly=2nyπ,kzlz=2nzπ,k_x l_x = 2 n_x \pi, \quad k_y l_y = 2 n_y \pi, \quad k_z l_z = 2 n_z \pi, where each nn is an integer (not necessarily positive). What is the kk-space volume occupied by each block on the grid (corresponding to Equation 5.51)? (b) Compute kFk_F, EFE_F, and EtotE_{\text{tot}} for the free electron gas with periodic boundary conditions. What compensates for the larger volume occupied by each kk-space block (part (a)) to make these all come out the same as in Section 5.3.1?

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