Separation of variables · Quantum Mechanics

Intuition

When a potential is a sum of pieces, each depending on one coordinate, the three-dimensional problem falls apart into three one-dimensional ones. The wavefunction is a product and the energy a sum. The particle in a cubic box is the example: each direction is an infinite well, and the energy is the sum of three. The same energy can be reached by different triples of quantum numbers, so the levels of a cube are degenerate, for a reason that is plainly symmetry: the three directions of a cube are alike.

A cube of jelly shaken along three edges at once wobbles in each direction independently; its motion is three one-dimensional wobbles added together. Shaken the same way along any edge, it cannot tell which edge was chosen.

The lowest levels of a particle in a cubic box, at heights $n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=3,6,9,11,12,14$ in units of $\pi^{2}\hbar^{2}/2mL^{2}$ , with one short line for each state. $(1,1,1)$ is alone; $(1,1,2)$ has three arrangements; $(2,2,2)$ is alone again; $(1,2,3)$ has six.

Separable potentials

If $V(\mathbf{r})=V_{1}(x)+V_{2}(y)+V_{3}(z)$ , the Hamiltonian is a sum of three commuting one-dimensional Hamiltonians, and its eigenstates are products of theirs.

\begin{aligned}\psi(\mathbf{r})&=X(x)\,Y(y)\,Z(z)\\ E&=E_{x}+E_{y}+E_{z}\end{aligned}

The cubic box

In a cube of side $L$ with infinite walls, $\psi=\bigl(\tfrac{2}{L}\bigr)^{3/2}\sin\frac{n_{x}\pi x}{L}\sin\frac{n_{y}\pi y}{L}\sin\frac{n_{z}\pi z}{L}$ with each .

A separable potential gives products

\begin{aligned}&V=V_{1}(x)+V_{2}(y)+V_{3}(z)\\ &\Longrightarrow\ \psi=X(x)Y(y)Z(z),\ \ E=E_{x}+E_{y}+E_{z}\end{aligned}

Put a product into the equation and divide by it. The left side becomes a sum of three terms each depending on one coordinate only. A sum of such terms can be constant only if each is, and each constant is an energy of a one-dimensional problem.

Proof steps

\hat{H}=\hat{H}_{x}+\hat{H}_{y}+\hat{H}_{z},\quad \hat{H}_{x}=-\frac{\hbar^{2}}{2m}\partial_{x}^{2}+V_{1}(x)

The Laplacian and the potential both split by coordinates.

\begin{aligned}\hat{H}(XYZ)&=(\hat{H}_{x}X)\,YZ+X\,(\hat{H}_{y}Y)\,Z\\ &\quad+XY\,(\hat{H}_{z}Z)\end{aligned}

Each piece acts on its own factor only.

\frac{\hat{H}_{x}X}{X}+\frac{\hat{H}_{y}Y}{Y}+\frac{\hat{H}_{z}Z}{Z}=E

Divide the eigenvalue equation by $XYZ$ .

\frac{\hat{H}_{x}X}{X}=E_{x},\quad \frac{\hat{H}_{y}Y}{Y}=E_{y},\quad \frac{\hat{H}_{z}Z}{Z}=E_{z}

Varying $x$ alone changes only the first term, so it is a constant; likewise the others.

E=E_{x}+E_{y}+E_{z}

Three one-dimensional problems, with energies that add.

Applications

Practice

Energies Add

In a cubic box the energy is the sum of three one-dimensional box energies, one for each direction.

E=\frac{\pi^{2}\hbar^{2}}{2mL^{2}}\bigl(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\bigr)

Try it

An electron in a line of length 1 nm has lowest energy 0.376 eV. What is its lowest energy in a cube of side 1 nm, in eV? Give three decimal places.