Three-Dimensional Wave Mechanics

In three dimensions, a wavefunction assigns an amplitude to every point in space. The main new choices are geometric: which coordinates fit the potential, which boundary conditions describe the container, and which measure converts a density into probability? This chapter develops these choices through boxes, free particles, and spherical separation.

Geometry, separation, and probability

For a spinless particle with a scalar potential,

$$H=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r), \qquad \int |\psi(\mathbf r)|^2\,d^3r=1.$$

Changing coordinates does not change the state, but it changes how derivatives and volume elements are written. In spherical coordinates,

$$d^3r=r^2\sin\theta\,dr\,d\theta\,d\phi.$$

Consequently, equal intervals in $r$, $\theta$, and $\phi$ do not describe equal volumes. The Jacobian is part of the probability calculation.

A rectangular hard box separates into three one-dimensional wells. If its side lengths are $L_x,L_y,L_z$, then

$$\psi=X_{n_x}(x)Y_{n_y}(y)Z_{n_z}(z), \qquad E_{n_xn_yn_z}=\frac{\pi^2\hbar^2}{2m} \left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right),$$

where each quantum number is a positive integer. Multiplying the three normalized factors gives a normalized spatial state. Adding their energies works because the Hamiltonian is a sum of independent coordinate Hamiltonians on a product domain.

Not every state in this box is a single product. Superpositions of product eigenfunctions are also allowed, and some cannot be factored. Nor does every three-dimensional potential separate: a coupling such as $xy$ generally defeats separation in the original Cartesian coordinates. A useful coordinate change must simplify both the equation and its boundaries.

For a central potential, spherical coordinates separate angular motion from radial motion. Start with Spherical Coordinates if the volume measure or Laplacian is unfamiliar, then follow Angular and Radial Separation.

Reading sequence

Page	What you will learn
Schrödinger Equation in Three Dimensions	Equation of motion, volume normalization, current, and boundary data
Separation of Variables	Product modes, separation constants, coordinate choice, and failure conditions
Three-Dimensional Box	Hard-wall products, additive spectrum, labels, and cubic degeneracy
Free Particle in Three Dimensions	Momentum vectors, energy shells, delta normalization, and spherical waves
Spherical Coordinates	Coordinate convention, Jacobian, unit vectors, gradient, and Laplacian
Angular and Radial Separation	Spherical harmonics, radial equation, reduced wavefunction, and centrifugal term
Degeneracy in Separable Systems	Eigenspace dimension, sources of repeated energies, and splitting
Density of States: First Encounter	Large-box shell counting and the spinless free-particle $D(E)$

How to use this chapter

For a first calculation, study the box before moving to spherical coordinates. Use free particles and the density of states when a question asks how many momentum states fit within an energy interval. The density of states is a smoothed count in a large system; an individual finite box still has discrete levels.

Exercises

In a cubic box of side $L$, find the ground-state energy and the degeneracy of the first excited energy, excluding spin. What happens if only $L_x$ is slightly increased?

Solution

Write $E_0=\pi^2\hbar^2/(2mL^2)$. The ground state $(1,1,1)$ has energy $3E_0$. The states $(2,1,1)$, $(1,2,1)$, and $(1,1,2)$ all have energy $6E_0$, so the first excited level has degeneracy three. Increasing $L_x$ lowers the $x$ contribution most for $(2,1,1)$; that state separates from the other two. Equality of the remaining $y$ and $z$ lengths keeps their pair degenerate in this rectangular model.

A normalized isotropic state is written $\psi(\mathbf r)=R(r)/\sqrt{4\pi}$. What is the probability in a thin spherical shell?

Solution

Integrating the angular factor over $d\Omega$ gives one. Thus the probability between $r$ and $r+dr$ is $r^2|R(r)|^2\,dr$, to first order in the shell thickness. The factor $r^2$ counts the growing shell volume; $|R|^2\,dr$ alone is not the radial probability.

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2012.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.