Motion in a Periodic Potential

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

We will now consider the motion of electrons in periodic potentials. An example of such a potential is given in Figure 1.

Image:periodic potential.jpg

Figure 1: An example of a periodic potential.

A periodic potential is, by definition, translationally symmetric over a certain period (in Figure 1 it is over a period of a\!); i.e.,

 V(x)=V(x + a).\!

In this case, a\! is the period.

Bloch's Theorem

The Hamiltonian of a system with a periodic potential of period a\! commutes with translations by a\!:

\hat{T}_a\psi(x)=\psi(x+a)\!

It is therefore possible to simultaneously diagonalize both the Hamiltonian and the translation operator. We will now show that the eigenfunctions of the Schrödinger equation for this system,

\left [-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right ]\psi(x)=E\psi(x),

have the form,

\psi(x)=e^{ikx}u_k(x)\!

where

u_k(x+a)=u_k(x)\!

has the same period as the potential; this result is known as Bloch's theorem, and the eigenfunctions are called Bloch waves. To show this, we will prove that eigenfunctions of the translation operator, and thus of the Hamiltonian, have the above form. Applying the translation operator to the proposed wave function, we find that


\begin{align}
\psi(x+a)&=\hat{T}_a\psi(x) \\
&=\hat{T}_a\left [e^{ikx}u_k(x)\right ] \\
&=\left [e^{ik(x+a)}u_k(x+a)\right ] \\
&=e^{ika}\left [e^{ikx}u_k(x)\right ] \\
&=e^{ika}\psi(x)
\end{align}

We see that the proposed wave function is indeed an eigenfunction of the translation operator, with eigenvalue e^{ika}.\! The quantity, \hbar k,\! is sometimes referred to as the crystal momentum, since it is a momentum-like quantity that characterizes the eigenstates of a system with discrete, rather than continuous, translational symmetry.

Also note that, if k\! is complex, then \psi(x)\! will diverge for x\rightarrow\pm\infty, with the choice of sign depending on the sign of the imaginary part of k.\! Therefore,  k \! has to be real if \psi(x)\! is to be a normalizable wave function.

Applying Bloch's theorem to the Schrödinger equation for a given periodic potential will reveal interesting and important results, such as a "band" structure to the energy spectrum as a function of k\!. For materials with weak electron-electron interactions, one can then deduce, given the Fermi energy of the system, whether such a system is metallic (overlapping bands), semiconducting (small gap between bands), or insulating (large gap between bands) (see Figure 2).

Image:Insulator-metal.svg.png

Figure 2. Energy band illustration showing the condition for metal, semiconductor, and insulator.

Dirac Comb Potential

As a simple example, let us consider a Dirac comb potential and the resulting Schrödinger equation,

V(x)=V_0\sum_{n=-\infty}^{\infty}\delta(x-na)

and

\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V_0\sum_{n=-\infty}^{\infty}\delta(x-na)\right)\psi(x)=E\psi(x).

Let us focus on the region, 0 < x < a,\! since the wave function for all x\! may be obtained using Bloch's theorem. Within this region, the equation simply reduces to that of a free particle, and thus the solution is


\begin{align}
\psi_k(x) &= Ae^{iqx}+Be^{-iqx} \\
&= e^{ikx}\left(Ae^{i(q-k)x}+Be^{-i(q+k)x}\right) \\
&= e^{ikx}u_k(x)
\end{align}

with energy E=\frac{\hbar^2q^2}{2m}.

Continuity of the wave function and the periodicity of u_k\! requires that

\psi_k(a^{-})=\psi_k(a^{+})=e^{ika}\psi_k(0^{+}).\!

Applying this condition to our wave function, we obtain

Ae^{iqa}+Be^{-iqa}=e^{ika}(A+B).\! (1)

Recall that the derivative of the wave function at a delta function potential is discontinuous, with the discontinuity given by

\psi'_k(a^{+})-\psi'_k(a^{-})=\frac{2mV_0}{\hbar^2}\psi_k(a).

One may easily verify that the derivative of the wave function satisfies

\psi'_k(x+a)=e^{ika}\psi'_k(x),\!

so that

\psi'_k(a^{-})=\psi'_k(a^{+})=e^{ika}\psi'_k(0^{+}).\!

We may now find the derivative of the wave function just to the left and just to the right of the delta function:

\psi'_k(a^{-})=iq(Ae^{iqa}-Be^{-iqa})\!
\psi'_k(a^{+})=iqe^{ika}(A-B)\!

We thus obtain

iq(Ae^{ika}-Be^{ika}-Ae^{iqa}+Be^{-iqa})=\frac{2mV_0}{\hbar^2}e^{ika}(A+B). (2)

By requiring that Equations (1) and (2) have non-trivial solutions, we obtain the following relation between q\! and k:\!

\cos(ka)=\cos(qa)+\frac{mV_0a}{\hbar^2}\frac{\sin(qa)}{qa} (3)

To obtain the energy bands, we now simply need to solve for q\! for a given value of k\! and substitute the result into the energy given above. Since there can be multiple values of q\! that satisfy the above equation, we see that there are multiple "energy bands" in our system. Note that the maximum value of the left-hand side of (3) is less than that of the right-hand side. This implies that there are some values of q\! that cannot be obtained for any k.\! The energies corresponding to these values of q\! are "forbidden bands" (see Fig. 3). We sketch the energy bands of the system as a function of k\! in Fig. 4.

Image:k_q relation graph.jpg

Figure 3. Sketch of the right-hand side of Equation (3). The curve is the right-hand side, while the box represents the range of the left-hand side. The solid red portions of the curve represent the forbidden bands, while the dashed black portions represent the allowed bands.

Image:dispersion relation small.jpg

Figure 4. Sketch of the energy bands for the Dirac comb potential as a function of k.\! The band gap, corresponding to the forbidden energy bands, is labeled.

We see that, in conjunction with Pauli exclusion principle, the single-particle band spectrum of a periodic potential, such as the one we discussed here, gives us a simple description of band insulators and band metals.

Problem

Let us now consider a more general case, namely a square wave potential, given by

 V(x)=\begin{cases}
0, & na < x < na+c \\
V_0, & na+c < x < (n+1)a,
\end{cases}

where n\! runs over all integers. Determine the energy spectrum for this potential and show that it reduces to the result for the Dirac comb when c\rightarrow a\! and V_0\rightarrow\infty\! in such a way that the product, V_0(c-a),\! remains finite.

Solution

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