How can one write the electron wavefunction in a periodic lattice as $\psi(x)=\sum_GC_{k-G}e^{i(k-G)x}?$

Question

The following is taken from Kittel's solid state physics.

An electron moving in a one-dimensional lattice of lattice constant $a$, suffers a periodic potential $U(x)$ where $U(x+a)=U(x)$. The periodicity of $U(x)$ implies $$U(x)=\sum_G U_Ge^{iGx}.$$ where $G=G_n=2\pi m/a$, the reciprocal lattice vectors. Due to periodic boundary condition, the wavefunction satisfies the periodicity condition $\psi(x+L)=\psi(x)$ so that $$\psi(x)=\sum_k C_ke^{ikx}$$ where $k=k_n=2\pi n/L$. Substituting these expansions into time-independent Schrodinger equation for the electron in the lattice, we get $$\Big(\frac{\hbar^2 k^2}{2m}-E_k\Big)C_k+\sum_GU_GC_{k-G}=0.$$

How does the equation above mean that $\psi(x)$ given above can be rewritten as $$\psi(x)=\sum_GC_{k-G}e^{i(k-G)x}?$$

Answer 1

Let $\sim$ be an equivalence relation on the set of possible $k$'s, where $k\sim k' \iff k-k' = G$ for some reciprocal lattice vector $G$. That is, $\{k_0,k_{\pm N},k_{\pm 2N},\ldots\}$ is one equivalence class, $\{k_1,k_{1\pm N},k_{1\pm 2N},\ldots\}$ is another, and $\{k_{-1},k_{-1\pm N},k_{-1\pm 2N},\ldots\}$ is a third. We can label each equivalence class by the smallest $k$ (in absolute value) contained within it; then $k_0=0$ is the representative of the first class, $k_1 = \frac{2\pi}{L}$ is the representative of the second, and $k_{-1} = -\frac{2\pi}{L}$ is the representative of the third.

By convention, we choose the set of $N$ distinct representatives to be $$FBZ := \left\{-\frac{\pi N}{L}, -\frac{\pi (N-2)}{L},\ldots,\frac{\pi (N-2)}{L}\right\}$$ Noting that $L=Na$, this can also be written $$FBZ:= \left\{-\frac{\pi}{a},-\frac{\pi}{a}+\frac{2\pi}{L}, \ldots ,-\frac{\pi}{a} + (N-1)\frac{2\pi}{L}\right\}$$ which spans the range $\big[-\frac{\pi}{a},\frac{\pi}{a}\big)$. $FBZ$ is called the first Brillouin zone. In particular, observe that every possible value of $k$ is related to exactly one $k\in FBZ$ via translation by some reciprocal lattice vector.

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This being the case, a generic wavefunction $\Psi$ can be decomposed as

$$\Psi(x) = \sum_{\text{all }k}C_k e^{ikx} = \sum_{k\in FBZ}\left( \sum_{G} C_{k-G} e^{i(k-G)x}\right) = \sum_{k\in FBZ} \psi_k(x)$$ where $$\psi_k(x) \equiv \sum_{G} C_{k-G} e^{i(k-G)x} = e^{ikx}\sum_G C_{k-G}e^{-iGx} \equiv e^{ikx} u_k(x)$$ with $u_k(x)\equiv \sum_G C_{k-G}e^{-iGx}$ a manifestly periodic function with period $a$. Furthermore, if we look carefully at the equation

$$\left(\frac{\hbar^2 k^2}{2m}-E\right)C_k = -\sum_G U_G C_{k-G}$$

we see that it only couples $C_k$'s from within the same equivalence class. That is, the Hamltonian operator is "block diagonal" and does not couple these $\psi_k$'s together:

$$H \Psi(x) = \sum_{k\in FBZ} H_k \psi_k$$

As a result, we can solve the Schrödinger equation for one $k\in FBZ$ at a time. The functions $\psi_k$ are called Bloch waves.