2.7: The Schrödinger Equation in Higher Dimensions

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Analyzing quantum wells and bulk materials requires that we solve the Schrödinger Equation in 2-d and 3-d. The equation in 1-d

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

is extended to higher dimensions as follows:

The Kinetic Energy operator

In 1-d

$\hat{T}=\frac{\hat{p}_{x}^{2}}{2m} \nonumber$

Now, the magnitude of the momentum in 3-d can be written

$|\textbf{p}|^{2} = p_{x}^{2}+p_{y}^{2}+p_{z}^{2} \nonumber$

Where $$p_{x}$$, $$p_{y}$$ and $$p_{z}$$ are the components of momentum on the x, y and z axes, respectively. It follows that in 3-d

$\hat{T}=\frac{\hat{p}_{x}^{2}}{2m}+\frac{\hat{p}_{y}^{2}}{2m}+\dfrac{\hat{p}_{z}^{2}}{2m} = -\frac{\hbar^{2}}{2m}\left(\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}}+\frac{d^{2}}{dz^{2}}\right) \nonumber$

Separable Potential – Quantum Well

A quantum well is shown in Figure 2.6.2(a). We will assume that the potential can be separated into x, y, and z dependent terms

$V(x,y,z)=V_{x}(x)+V_{y}(y)+V_{z}(z) \nonumber$

For example, a quantum well potential is given by

$V_{x}(x) = 0 \nonumber$

$$V_{y}(y) = 0$$

$$V_{z}(z) = 0$$

where in the infinite square well approximation $$V_{0} \rightarrow \infty$$, and u is the unit step function.

For potentials of this form the Schrödinger Equation can be separated:

${\left[-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+V_{x}(x)\right] \psi(x, y, z)+\left[-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d y^{2}}+V_{y}(y)\right] \psi(x, y, z)} +\left[-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d z^{2}}+V_{z}(z)\right] \psi(x, y, z)=\left(E_{x}+E_{y}+E_{z}\right) \psi(x, y, z) \nonumber$

The wavefunction can also be separated

$\psi(x,y,z) = \psi_{x}(x)\psi_{y}(y)\psi_{z}(z) \nonumber$

From Equations 1.28.1 and 1.28.2, the solutions to the infinite quantum well potential are

$\psi(x, y, z)=\psi_{x}(x) \psi_{y}(y) \psi_{z}(z)=\sqrt{\frac{2}{L}} \sin \left(n \frac{\pi z}{L}\right) \cdot \exp \left[i k_{x} x\right] \cdot \exp \left[i k_{y} y\right] \nonumber$

with

$E =E_{x}+E_{y}+E_{z} = \frac{\hbar^{2}k_{x}^{2}}{2m}+\frac{\hbar^{2}k_{y}^{2}}{2m}+\frac{n^{2}\hbar^{2}\pi^{2}}{2mL^{2}} \nonumber$

This dispersion relation is shown in Figure 2.7.1 for the lowest three modes of the quantum well.

Separable Potential – Quantum Wire

A quantum wire with rectangular cross-section is shown in Figure 2.6.2(b). Again, we will assume that the potential is infinite at the boundaries of the wire:

$V(x,y,z) = V_{0}u(-x)+V_{0}u(x-L_{x})+V_{0}u(-y)+V_{0}u(x-L_{y}) \nonumber$

where $$V_{0} \rightarrow \infty$$. The associated wavefunction is confined in the x-y plane and composed of plane waves in the z direction, thus we chose the trial wavefunction

$\psi(x,y,z) = \psi(x,y)e^{ik_{z}z} \nonumber$

Inserting Equation 2.7.12 into the Schrödinger equation gives (for $$0 \leq x \leq L_{x}$$ and $$0 \leq y \leq L_{y}$$):

$-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi(x, y)-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d y^{2}} \psi(x, y)+\frac{\hbar^{2} k_{z}^{2}}{2 m} \psi(x, y)=E \psi(x, y) \nonumber$

Since the potential goes to infinity at the edges of the wire, $$\psi(x=0)=\psi(x=L_{x})=\psi(y=0)=\psi(y=L_{y})$$. Thus, the solution is

$\psi(x,y)=\psi_{0}\sin(k_{x}x)\sin(k_{y}y), \nonumber$

where

$k_{x}=\frac{n_{x}\pi}{L_{x}},\ n_{x}=1,2,…,\ k_{y}=1,2,… \nonumber$

Thus, the constraint in the x- and y-directions defines the discrete energy levels

$E_{n_{x}, n_{y}}=\frac{\hbar^{2} \pi^{2}}{2 m}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right), \quad n_{x}, n_{y}=1,2, \ldots \nonumber$

The total energy is

$E_{n_{x}, n_{y}}=\frac{\hbar^{2} \pi^{2}}{2 m}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right) + \frac{\hbar^{2}k_{z}^{2}}{2m}, \quad n_{x}, n_{y}=1,2, \ldots \nonumber$

This dispersion relation is plotted in Figure 2.7.3 for the lowest three modes.

This page titled 2.7: The Schrödinger Equation in Higher Dimensions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.