Schrödinger Equation in Three Dimensions

The three-dimensional Schrödinger equation is the coordinate-space equation of motion for a nonrelativistic particle moving in ordinary space. It replaces the one-dimensional second derivative with the Laplacian and interprets $\lvert\psi(\mathbf r,t)\rvert^2$ as a probability density per unit volume.

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

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

Here $\mathbf r=(x,y,z)$ and $\nabla^2$ is the spatial Laplacian.

Normalization In Volume

For a normalized wavefunction on a region $\Omega\subseteq\mathbb R^3$,

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

The probability of finding the particle in a subregion $A\subseteq\Omega$ is

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

Thus $\lvert\psi\rvert^2$ has units of inverse volume, and $\psi$ has units of length$^{-3/2}$ in ordinary three-dimensional normalization.

The volume element depends on coordinates. In Cartesian coordinates,

$$d^3r=dx\,dy\,dz.$$

In spherical coordinates,

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

Forgetting the coordinate volume element changes the probability distribution.

The Laplacian

In Cartesian coordinates,

$$\nabla^2 =\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}.$$

The time-dependent equation becomes

$$i\hbar\frac{\partial\psi}{\partial t} =-\frac{\hbar^2}{2m} \left( \frac{\partial^2\psi}{\partial x^2} +\frac{\partial^2\psi}{\partial y^2} +\frac{\partial^2\psi}{\partial z^2} \right) +V\psi.$$

In curvilinear coordinates the Laplacian contains metric and coordinate-volume factors. For example, the spherical-coordinate Laplacian is not obtained by replacing $x$ with $r$ in the one-dimensional formula. This is one reason central potentials require their own radial equation.

Stationary States

If the potential is time independent, separated stationary solutions have the form

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

Substitution gives the three-dimensional time-independent Schrödinger equation:

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

This is an eigenvalue problem for a differential operator plus boundary conditions. Bound systems usually impose square integrability and surface conditions; scattering systems impose incoming and outgoing behavior at large distances.

Probability Current

The probability density is

$$\rho(\mathbf r,t)=\lvert\psi(\mathbf r,t)\rvert^2.$$

For a real scalar potential and suitable boundary conditions, it obeys the continuity equation

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

where the probability-current density is

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

This is the three-dimensional version of the current used in one-dimensional scattering. Probability is conserved by flowing through space or across boundaries, not by being arbitrarily created or destroyed.

Boundary Conditions In Regions

A three-dimensional wave-mechanics problem is not specified by the differential equation alone. One must also specify the region and boundary behavior.

For a hard-wall box, the wavefunction vanishes on the boundary:

$$\psi=0 \quad\text{on the wall}.$$

At a finite potential interface, $\psi$ is continuous. For equal mass and no singular surface interaction, the normal derivative is also continuous:

$$\frac{\partial\psi}{\partial n}\bigg\rvert_{\mathrm{left}} = \frac{\partial\psi}{\partial n}\bigg\rvert_{\mathrm{right}}.$$

For bound states on all of space, square integrability requires decay at infinity:

$$\int_{\mathbb R^3} \lvert\psi(\mathbf r)\rvert^2\,d^3r \lt \infty.$$

For scattering states, one instead uses asymptotic incoming and outgoing wave conditions. The boundary conditions encode the physics just as much as the Hamiltonian does.

Separability

Some three-dimensional problems reduce to simpler ordinary differential equations by separation of variables. The method works when the potential, coordinate system, and boundary conditions are compatible.

Examples:

• a rectangular box separates in Cartesian coordinates;
• a central potential separates into radial and angular equations in spherical coordinates;
• cylindrical symmetry can separate axial, radial, and angular variables.

Separation is a powerful method, not a universal guarantee. The first wave-mechanics treatment is Separation of Variables, with mathematical background in the Mathematical Toolkit.

Common Mistakes

• Forgetting that $\lvert\psi\rvert^2$ is a probability density per unit volume.
• Omitting the volume element in spherical or cylindrical coordinates.
• Treating $\nabla^2$ as a one-dimensional second derivative in radial problems.
• Assuming every three-dimensional potential separates.
• Specifying a Hamiltonian without specifying the spatial region and boundary conditions.
• Confusing evanescent decay in one coordinate with full square integrability in three dimensions.

Where This Is Used

• Wavefunctions and Probability Density introduces coordinate-space probabilities.
• Time-Dependent Schrödinger Equation gives the general wave-mechanics equation of motion.
• Time-Independent Schrödinger Equation explains stationary-state eigenvalue problems.
• Boundary Conditions explains endpoint, interface, and singular matching conditions.
• Separation of Variables explains how solvable 3D models are reduced to lower-dimensional equations.
• Free Particle in Three Dimensions applies the equation to plane waves, momentum vectors, and energy shells.
• Spherical Coordinates records the measure, Laplacian, and angular-coordinate convention used in central potentials.
• Angular and Radial Separation shows how central potentials reduce the equation to angular and radial parts.
• Degeneracy in Separable Systems explains repeated energies after three-dimensional quantum numbers are introduced.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.

Exercises

1. What are the units of a normalized three-dimensional wavefunction?

Solution

Normalization requires

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

Since $d^3r$ has units of length$^3$, $\lvert\psi\rvert^2$ has units of length$^{-3}$. Therefore $\psi$ has units of length$^{-3/2}$.

2. Show that if $\psi(x,y,z)=X(x)Y(y)Z(z)$ is normalized in a rectangular box and each factor is normalized on its interval, then the full wavefunction is normalized.

Solution

The norm factors:

$$\int\lvert X(x)Y(y)Z(z)\rvert^2\,dx\,dy\,dz = \left(\int\lvert X(x)\rvert^2\,dx\right) \left(\int\lvert Y(y)\rvert^2\,dy\right) \left(\int\lvert Z(z)\rvert^2\,dz\right).$$

If each factor is normalized, each parenthesis equals $1$, so the product equals $1$.

3. Why is the spherical volume element essential for radial probabilities?

Solution

In spherical coordinates,

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

The factor $r^2\sin\theta$ accounts for how much physical volume corresponds to a coordinate interval. Omitting it would give the wrong probability assigned to spherical shells and angular regions.