Spherical Coordinates

Spherical coordinates are the natural coordinate system for central potentials, angular wavefunctions, rigid rotors, and partial-wave expansions. They are useful because they separate distance from direction, but they are also a common source of errors: the volume element, unit vectors, and Laplacian all change.

This page fixes the coordinate convention used in this volume:

$$r\ge0, \qquad 0\le\theta\le\pi, \qquad 0\le\phi\lt 2\pi.$$

Here $\theta$ is the polar angle measured down from the positive $z$ axis, and $\phi$ is the azimuthal angle in the $xy$ plane.

The convention used here is the standard physics convention: $r$ is distance from the origin, $\theta$ is the polar angle from the positive $z$ axis, and $\phi$ is the azimuthal angle of the projection into the $xy$ plane.

Coordinate Definitions

The Cartesian coordinates are

$$x=r\sin\theta\cos\phi,$$ $$y=r\sin\theta\sin\phi,$$

and

$$z=r\cos\theta.$$

Conversely,

$$r=\sqrt{x^2+y^2+z^2},$$ $$\cos\theta=\frac{z}{r},$$

and

$$\phi=\operatorname{atan2}(y,x).$$

The function $\operatorname{atan2}$ chooses the correct quadrant for the azimuth. At $r=0$, the angles are undefined because every direction labels the same physical point.

Unit Vectors

The spherical unit vectors are

$$\hat{\mathbf e}_r = (\sin\theta\cos\phi,\, \sin\theta\sin\phi,\, \cos\theta),$$ $$\hat{\mathbf e}_\theta = (\cos\theta\cos\phi,\, \cos\theta\sin\phi,\, -\sin\theta),$$

and

$$\hat{\mathbf e}_\phi = (-\sin\phi,\, \cos\phi,\, 0).$$

The vector $\hat{\mathbf e}_r$ points outward. The vector $\hat{\mathbf e}_\theta$ points in the direction of increasing polar angle, and $\hat{\mathbf e}_\phi$ points in the direction of increasing azimuth.

Unlike Cartesian unit vectors, these unit vectors depend on position. Differentiating a vector field in spherical coordinates therefore requires differentiating both components and basis vectors.

Volume And Solid Angle

The volume element is

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

The solid-angle element is

$$d\Omega = \sin\theta\,d\theta\,d\phi.$$

Thus

$$d^3r=r^2\,dr\,d\Omega.$$

A normalized wavefunction obeys

$$\int_0^\infty \int_0^\pi \int_0^{2\pi} \lvert\psi(r,\theta,\phi)\rvert^2 r^2\sin\theta\,d\phi\,d\theta\,dr =1.$$

This is the same normalization as $\int_{\mathbb R^3}\lvert\psi(\mathbf r)\rvert^2\,d^3r=1$, written in spherical coordinates. The factor $r^2\sin\theta$ is not optional; it is the Jacobian of the coordinate transformation.

Radial And Angular Probability

For a spherically symmetric probability density, the probability of finding the particle between $r$ and $r+dr$ is

$$dP = 4\pi r^2\lvert\psi(r)\rvert^2\,dr.$$

For a separated central-potential wavefunction

$$\psi(r,\theta,\phi) = R(r)Y(\theta,\phi),$$

with

$$\int_{S^2}\lvert Y(\theta,\phi)\rvert^2\,d\Omega=1,$$

the radial normalization is

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

This is why the radial probability density is $\lvert R(r)\rvert^2r^2$, not just $\lvert R(r)\rvert^2$. If $u(r)=rR(r)$, then the same condition becomes

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

Gradient

For a scalar function $f(r,\theta,\phi)$,

$$\nabla f = \hat{\mathbf e}_r\frac{\partial f}{\partial r} + \hat{\mathbf e}_\theta \frac{1}{r} \frac{\partial f}{\partial\theta} + \hat{\mathbf e}_\phi \frac{1}{r\sin\theta} \frac{\partial f}{\partial\phi}.$$

The factors $1/r$ and $1/(r\sin\theta)$ appear because a change in angle corresponds to an arc length, not a Cartesian distance. Near the poles, $\sin\theta=0$, so the coordinate expression is singular even though the underlying space is not.

Laplacian

For scalar wavefunctions, the spherical-coordinate Laplacian is

$$\nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2\frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta} \frac{\partial}{\partial\theta} \left( \sin\theta\frac{\partial f}{\partial\theta} \right) + \frac{1}{r^2\sin^2\theta} \frac{\partial^2 f}{\partial\phi^2}.$$

It is often useful to separate the radial and angular parts:

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

where the unit-sphere angular Laplacian is

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

In quantum mechanics,

$$\hat L^2=-\hbar^2\Delta_{S^2},$$

so the Laplacian may also be written as

$$\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2\frac{\partial}{\partial r} \right) - \frac{\hat L^2}{\hbar^2r^2}.$$

This identity is the coordinate bridge to Angular and Radial Separation, angular momentum, and the radial Schrödinger equation.

Separation Ansatz

For a central potential,

$$V(\mathbf r)=V(r),$$

try a product form

$$\psi(r,\theta,\phi) = R(r)Y(\theta,\phi).$$

The angular equation is the eigenvalue problem on the unit sphere:

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

The angular functions are spherical harmonics,

$$Y_\ell^m(\theta,\phi),$$

with

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

The radial function then obeys a one-dimensional equation on the half-line, with the angular eigenvalue producing the centrifugal term. The separation step is derived in Angular and Radial Separation, and the deeper radial boundary conventions belong to Radial Schrödinger Equation.

Coordinate Singularities

Spherical coordinates are singular at the origin and at the polar axis:

• at $r=0$, every direction labels the same point;
• at $\theta=0$ and $\theta=\pi$, the azimuth $\phi$ is undefined;
• the factors $1/\sin\theta$ in derivative formulas reflect a coordinate singularity, not a physical singularity.

Physical scalar wavefunctions must be single-valued and regular where the space itself is regular. In particular,

$$\psi(r,\theta,\phi+2\pi) = \psi(r,\theta,\phi)$$

for ordinary scalar wavefunctions. Spinor sign changes under rotations are a different issue and belong to the spin-volume discussion of $SU(2)$.

Common Mistakes

• Swapping the physics convention for $\theta$ and $\phi$ without saying so.
• Forgetting the Jacobian $r^2\sin\theta$ in normalization and expectation values.
• Replacing the spherical Laplacian by only $\partial^2/\partial r^2$ for radial functions.
• Treating the poles as physical singularities rather than coordinate singularities.
• Forgetting that spherical unit vectors depend on angle.
• Using $\arctan(y/x)$ instead of $\operatorname{atan2}(y,x)$ and losing the quadrant of $\phi$.

Where This Is Used

• Schrödinger Equation in Three Dimensions introduces the three-dimensional equation and volume normalization.
• Separation of Variables explains why central potentials separate in spherical coordinates.
• Angular and Radial Separation derives the central-potential split into $R(r)$ and $Y_\ell^m(\theta,\phi)$.
• Central Potentials explains the rotationally invariant systems that use these coordinates.
• Radial Schrödinger Equation uses the spherical Laplacian to derive the radial equation.
• Hydrogen Atom uses $R_{n\ell}(r)Y_\ell^m(\theta,\phi)$ and radial normalization.
• Particle on a Sphere freezes $r=R$ and keeps only the angular kinetic-energy problem on $S^2$.
• Position-Space Representation writes orbital angular momentum as differential operators in these coordinates.
• Spherical Coordinates for Angular Momentum focuses on the symmetry-side split into $L^2$, $L_z$, and spherical harmonics.
• Spherical Harmonics gives the angular eigenfunctions on $S^2$.
• Rigid Rotor uses the angular part of the Laplacian as the whole kinetic-energy operator.

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.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2012.
• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2006.

Exercises

1. Compute the volume of a ball of radius $a$ using the spherical volume element.

Solution

Use

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

Then

$$V = \int_0^a r^2\,dr \int_0^\pi\sin\theta\,d\theta \int_0^{2\pi}d\phi.$$

The three factors are

$$\frac{a^3}{3}, \qquad 2, \qquad 2\pi.$$

Thus

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

2. For a radial function $f(r)$, simplify the spherical Laplacian.

Solution

If $f$ depends only on $r$, all angular derivatives vanish. Therefore

$$\nabla^2 f(r) = \frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{df}{dr} \right).$$

Equivalently,

$$\nabla^2 f(r) = \frac{d^2f}{dr^2} + \frac{2}{r}\frac{df}{dr}.$$

The second term is the part often missed by treating $r$ as if it were a Cartesian coordinate.

3. Show that the unit-sphere area is $4\pi$.

Solution

The area element on the unit sphere is

$$d\Omega=\sin\theta\,d\theta\,d\phi.$$

Thus

$$\int_{S^2}d\Omega = \int_0^\pi\sin\theta\,d\theta \int_0^{2\pi}d\phi = 2\cdot2\pi = 4\pi.$$