Three-Dimensional Wave Mechanics

Three-dimensional wave mechanics is not merely the one-dimensional Schrödinger equation repeated three times. Position and momentum become vectors, probability is normalized with a volume measure, fixed free-particle energy defines a sphere of momentum directions, boundaries become surfaces, and coordinate systems must be chosen to match the geometry and symmetry of the problem.

This chapter develops that transition through two complementary routes. Cartesian separation leads to boxes, product states, and additive spectra. Spherical separation leads to angular functions, radial half-line equations, and the central-potential problems of the next chapter. The same framework also explains why degeneracy and density-of-states counting become unavoidable in three dimensions.

Scope and ownership

This chapter owns the coordinate-space formulation and its first exactly solvable three-dimensional models. It establishes the volume measure, Laplacian, probability current, separation method, free-particle momentum shells, spherical-coordinate conventions, and the first radial reduction.

Nearby subjects have more specialized canonical homes:

• Central Potentials and the hydrogenic pages develop radial spectra and bound-state models.
• Symmetry, Angular Momentum, and Spin owns the operator algebra of rotations, orbital angular momentum, and representation-theoretic explanations of multiplets.
• Spherical Harmonics owns the mathematical normalization, completeness, and phase conventions for the angular functions.
• Advanced partial-wave scattering belongs in Approximation and Semiclassical Methods.
• Band, many-body, and thermodynamic densities of states belong in later application volumes; this chapter derives only the one-particle free-space result.

The three-dimensional equation

For one nonrelativistic particle of mass $m$ in a scalar potential $V(\mathbf r,t)$,

$$i\hbar \frac{\partial\psi(\mathbf r,t)}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 +V(\mathbf r,t) \right] \psi(\mathbf r,t).$$

The wavefunction is normalized over its spatial domain $\Omega$:

$$\int_\Omega \lvert\psi(\mathbf r,t)\rvert^2,d^3r =1.$$

Consequently, $\lvert\psi\rvert^2$ is probability per unit volume and a normalized position-space wavefunction has dimensions of length$^{-3/2}$. For a subregion $A\subseteq\Omega$,

$$P(A,t) = \int_A \lvert\psi(\mathbf r,t)\rvert^2,d^3r.$$

The probability density and current satisfy

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf j =0,$$

with

$$\rho=\lvert\psi\rvert^2, \qquad \mathbf j = \frac{\hbar}{2mi} \left( \psi^*\nabla\psi - \psi\nabla\psi^* \right).$$

Integrating the continuity equation over a region and applying the divergence theorem gives

$$\frac{d}{dt} \int_A\rho,d^3r = -\int_{\partial A} \mathbf j\cdot d\mathbf S.$$

Probability changes inside $A$ only through flux across its boundary. Schrödinger Equation in Three Dimensions develops these definitions together with stationary states and interface conditions.

Geometry, coordinates, and boundary data

A differential equation does not define a quantum model by itself. One must also state the domain, coordinate measure, and boundary conditions. Hard walls impose conditions on surfaces; finite interfaces require matching across surfaces; bound states on all of $\mathbb R^3$ require square integrability; scattering states require asymptotic incoming and outgoing behavior.

The coordinate system should follow the potential and boundary geometry:

Geometry or symmetry	Natural coordinates	Typical separated labels
Rectangular region or additive potential	Cartesian $(x,y,z)$	$(n_x,n_y,n_z)$ or $(k_x,k_y,k_z)$
Rotationally invariant potential	Spherical $(r,\theta,\phi)$	radial label, $\ell$, $m$
Axial symmetry	Cylindrical $(\rho,\phi,z)$	radial label, $m$, $k_z$ or axial mode
No compatible coordinate symmetry	Problem dependent	usually numerical or approximate labels

Coordinate changes alter the volume element and the differential expression for the Laplacian. They do not change the underlying physics. A correct calculation must transform the measure, operators, and boundary surfaces together.

Separation as an eigenvalue method

For a time-independent potential, first write

$$\psi(\mathbf r,t) = \varphi(\mathbf r)e^{-iEt/\hbar}.$$

The spatial function obeys

$$\left[ -\frac{\hbar^2}{2m}\nabla^2 +V(\mathbf r) \right] \varphi =E\varphi.$$

If the operator, domain, and boundary conditions have compatible product structure, one can factor $\varphi$ into one-coordinate functions. For example, when

$$V(x,y,z) = V_x(x)+V_y(y)+V_z(z),$$

the ansatz $\varphi=XYZ$ yields

$$E=E_x+E_y+E_z.$$

The separation constants become physical quantum labels only after regularity, boundary conditions, and normalization restrict their values. A separated product is one basis mode, not generally the most general state; arbitrary states require sums or integrals over a complete set of modes.

Separation of Variables gives the workflow in Cartesian, spherical, and cylindrical settings and explains when the method fails.

Cartesian canonical systems

The hard-wall box

For a rectangular box with side lengths $L_x,L_y,L_z$, the normalized stationary modes are

$$\begin{aligned} \varphi_{n_xn_yn_z}(x,y,z) &= \sqrt{\frac{8}{L_xL_yL_z}} \sin\frac{n_x\pi x}{L_x} \sin\frac{n_y\pi y}{L_y} \sin\frac{n_z\pi z}{L_z},\\ n_x,n_y,n_z &=1,2,3,\ldots . \end{aligned}$$

Their energies are additive:

$$E_{n_xn_yn_z} = \frac{\pi^2\hbar^2}{2m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right).$$

The box is the cleanest first example of a complete three-label basis. In a cube, axis permutations can have equal energy, so the energy eigenvalue alone no longer identifies a unique state. Three-Dimensional Box develops the walls, normalization, spectrum, and cubic degeneracies.

The free particle

On all of $\mathbb R^3$,

$$H = \frac{\mathbf p^2}{2m} = -\frac{\hbar^2}{2m}\nabla^2.$$

A plane wave labeled by $\mathbf k$ has

$$\mathbf p=\hbar\mathbf k, \qquad E(\mathbf k) = \frac{\hbar^2\lvert\mathbf k\rvert^2}{2m}.$$

With momentum normalization,

$$\langle\mathbf r\vert\mathbf p\rangle = \frac{1}{(2\pi\hbar)^{3/2}} e^{i\mathbf p\cdot\mathbf r/\hbar},$$

and

$$\langle\mathbf p\vert\mathbf p'\rangle = \delta^{(3)}(\mathbf p-\mathbf p').$$

At fixed positive energy, $\mathbf p$ can point in any direction on the sphere $\lvert\mathbf p\rvert=\sqrt{2mE}$. This continuous angular degeneracy is the geometric origin of the three-dimensional shell factor used in scattering and state counting.

Free Particle in Three Dimensions treats plane waves, wave packets, current, periodic-box regulation, and the first outgoing spherical waves.

Spherical coordinates

For central potentials, distance and direction should be separated. In the standard physics convention,

$$r\geq0, \qquad 0\leq\theta\leq\pi, \qquad 0\leq\phi\lt2\pi,$$

with volume and solid-angle elements

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

The scalar Laplacian is

$$\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2\frac{\partial}{\partial r} \right) + \frac{1}{r^2}\Delta_{S^2},$$

where

$$\Delta_{S^2} = \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \left( \sin\theta \frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2}.$$

The factors $r^2$ and $\sin\theta$ encode geometry. Omitting them changes normalization, expectation values, and the operator itself. The apparent singularities at $r=0$ and the polar axis are coordinate singularities; physical scalar wavefunctions must remain regular and single-valued where space is regular.

Spherical Coordinates fixes the coordinate convention and collects the unit vectors, gradient, Laplacian, Jacobian, and radial-probability rules used throughout the later central-potential and rotor chapters.

Angular and radial reduction

For a central potential $V(\mathbf r)=V(r)$, use

$$\psi(r,\theta,\phi) = R_{E\ell}(r)Y_\ell^m(\theta,\phi).$$

The angular eigenproblem is

$$\Delta_{S^2}Y_\ell^m = -\ell(\ell+1)Y_\ell^m,$$

with

$$\ell=0,1,2,\ldots, \qquad m=-\ell,-\ell+1,\ldots,\ell.$$

Defining $u_\ell(r)=rR_\ell(r)$ converts the radial equation to

$$-\frac{\hbar^2}{2m} \frac{d^2u_\ell}{dr^2} + \left[ V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2} \right]u_\ell = Eu_\ell, \qquad r\gt0.$$

This resembles a one-dimensional equation, but it lives on the half-line and carries boundary information at the origin. The centrifugal term is angular kinetic energy expressed in radial form, not an additional interaction.

The normalizations are

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2r^2,dr =1,$$

or equivalently

$$\int_0^\infty \lvert u_{n\ell}(r)\rvert^2,dr =1.$$

Angular and Radial Separation derives this split. The full domain analysis and model-specific radial solutions begin with Radial Schrödinger Equation.

Degeneracy and state labels

An energy $E$ is degenerate when its eigenspace has dimension greater than one:

$$g_E = \dim\ker(H-EI).$$

Three-dimensional problems provide several distinct mechanisms:

Source	Example	Robustness
Visible symmetry	The $2\ell+1$ values of $m$ in a central potential	Persists under symmetry-preserving changes
Equal separable parameters	Cubic box or isotropic oscillator	Splits when lengths or frequencies become unequal
Arithmetic coincidence	Different integer triples with equal energy sums	Usually split by generic deformation
Hidden symmetry	Extra Coulomb degeneracy	Persists only while the special structure remains
Continuum energy shell	Free-particle momentum directions	Modified by boundaries or directional fields

Energy alone labels an eigenspace, not a unique vector. A Complete Set of Commuting Observables supplies additional compatible labels. Degeneracy in Separable Systems compares the box, isotropic oscillator, central potentials, and degeneracy lifting.

From mode counting to density of states

A periodic cubic box of volume $V$ discretizes free-particle wavevectors as

$$\mathbf k = \frac{2\pi}{L} (n_x,n_y,n_z), \qquad n_i\in\mathbb Z.$$

One state occupies $\mathbf k$-space volume $(2\pi)^3/V$, so

$$\sum_{\mathbf k} \longrightarrow \frac{V}{(2\pi)^3} \int d^3k.$$

For one spinless particle, the number of states inside a sphere of radius $k$ is

$$N(k) = \frac{V}{6\pi^2}k^3.$$

Using $E=\hbar^2k^2/(2m)$ gives

$$D(E) = \frac{dN}{dE} = \frac{V}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt E.$$

An unsplit internal degeneracy multiplies this result. The formula is a smooth large-volume approximation; it is not the exact spike spectrum of a small finite box. Density of States: First Encounter derives the count, compares periodic and hard-wall regulators, and distinguishes density of states from exact degeneracy.

Reading route

1. Start with Schrödinger Equation in Three Dimensions for volume normalization, the Laplacian, current, and boundary surfaces.
2. Learn the reusable reduction method in Separation of Variables.
3. Work through Three-Dimensional Box as the discrete Cartesian model.
4. Continue to Free Particle in Three Dimensions for vector momentum, continuum normalization, and energy shells.
5. Fix the measure and operator conventions in Spherical Coordinates.
6. Use Angular and Radial Separation as the bridge to central potentials.
7. Read Degeneracy in Separable Systems before using energy labels as state labels.
8. Finish with Density of States: First Encounter to connect discrete boxes with continuum state counting.

Page map

Page	Canonical role
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 $D(E)$

Common mistakes

• Treating $\lvert\psi\rvert^2$ as probability rather than probability density per unit volume.
• Changing coordinates without changing the measure and Laplacian.
• Assuming a symmetric potential guarantees separability when the boundary geometry breaks the same symmetry.
• Treating one separated product as the most general state instead of one basis mode.
• Confusing hard-wall sine modes with periodic-box plane waves.
• Interpreting a fixed free-particle energy as a unique momentum vector rather than a momentum shell.
• Normalizing $R(r)$ with $dr$ or $u(r)$ with $r^2dr$.
• Treating the reduced radial equation as a full-line one-dimensional problem.
• Assuming every degeneracy has the same symmetry explanation.
• Confusing exact finite-level degeneracy with a smooth density of states.
• Mixing $\mathbf p$ and $\mathbf k$ normalization conventions without the required powers of $\hbar$.

References

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

Exercises

1. A normalized wavefunction is constant inside a ball of radius $a$ and zero outside. Find the constant amplitude and the probability of finding the particle within radius $a/2$.

Solution

Write $\psi=C$ inside the ball. Normalization gives

$$1 = \lvert C\rvert^2 \frac{4\pi a^3}{3},$$

so, up to a global phase,

$$C = \sqrt{\frac{3}{4\pi a^3}}.$$

The probability inside radius $a/2$ is the ratio of the two volumes:

$$P(r\leq a/2) = \frac{(a/2)^3}{a^3} = \frac18.$$

2. In a cubic hard-wall box, compare the ground state $(1,1,1)$ with the first excited shell, consisting of permutations of $(1,1,2)$. Find the energy ratio and the shell degeneracy.

Solution

For a cube,

$$E_{n_xn_yn_z} = E_0 \left( n_x^2+n_y^2+n_z^2 \right), \qquad E_0=\frac{\pi^2\hbar^2}{2mL^2}.$$

The ground-state sum is $3$, while the first excited sum is $6$. Therefore

$$\frac{E_{112}}{E_{111}} =2.$$

The distinct permutations are $(1,1,2)$, $(1,2,1)$, and $(2,1,1)$, so the shell is threefold degenerate.

3. Let $\psi(r,\theta,\phi)=R(r)Y(\theta,\phi)$ with $\int\lvert Y\rvert^2d\Omega=1$. Show that $u=rR$ has ordinary half-line normalization, and explain why this does not make $u$ a wavefunction on the full real line.

Solution

The three-dimensional norm factors as

$$1 = \int_0^\infty \lvert R(r)\rvert^2r^2,dr \int_{S^2} \lvert Y\rvert^2,d\Omega.$$

The angular factor is one, and $u=rR$ gives

$$1 = \int_0^\infty \lvert u(r)\rvert^2,dr.$$

The coordinate $r$ is a distance, so its domain is $[0,\infty)$ rather than $(-\infty,\infty)$. Regularity and the domain of the original three-dimensional Hamiltonian impose boundary behavior at $r=0$; extending $u$ to negative $r$ would introduce points that do not represent new physical positions.

4. In $d$ spatial dimensions, suppose the number of free-particle states below wavenumber $k$ scales as $N(k)\propto k^d$. Derive the energy dependence of the density of states for $E=\hbar^2k^2/(2m)$, and specialize to $d=1,2,3$.

Solution

Because $k\propto E^{1/2}$,

$$N(E) \propto E^{d/2}.$$

Differentiating gives

$$D(E) = \frac{dN}{dE} \propto E^{d/2-1}.$$

Therefore

$$\begin{array}{c|c} d & D(E)\\ \hline 1 & E^{-1/2}\\ 2 & E^0\\ 3 & E^{1/2} \end{array}$$

up to dimension-dependent geometric factors, normalization volume, mass and $\hbar$ factors, and any internal degeneracy.