Separation of Variables

Separation of variables is a method for solving suitable partial differential equations by looking for product solutions. Instead of solving for one function of many variables at once, one tries

$$u(q_1,q_2,\ldots,q_n) = Q_1(q_1)Q_2(q_2)\cdots Q_n(q_n).$$

If the equation and boundary conditions cooperate, substitution reduces the original PDE to several ordinary differential equations tied together by separation constants. In quantum mechanics, those constants often become energy, angular momentum, momentum, or other spectral labels.

Separation is a method, not a guarantee. It works when the operator, coordinate system, and boundary data have compatible structure.

Core Idea

Suppose a PDE for $u(x,y)$ admits a product trial function

$$u(x,y)=X(x)Y(y).$$

After substitution, a separable equation can often be rearranged into the form

$$F(x)+G(y)=0.$$

Since $x$ and $y$ can be varied independently, this can hold for all allowed $x$ and $y$ only if each part is constant:

$$F(x)=\alpha, \qquad G(y)=-\alpha.$$

The constant $\alpha$ is a separation constant. The original PDE has become two ODEs.

The sign convention for a separation constant is chosen for convenience. In box problems, negative constants often lead to sine and cosine equations; in angular problems, constants are chosen to match standard angular-momentum notation.

Why Quantum Mechanics Uses It

The time-dependent Schrödinger equation for a time-independent Hamiltonian admits separated stationary solutions:

$$\psi(\mathbf r,t)=\varphi(\mathbf r)T(t).$$

Substitution into

$$i\hbar\partial_t\psi=H\psi$$

gives

$$i\hbar\frac{1}{T}\frac{dT}{dt} = \frac{1}{\varphi}H\varphi.$$

The left side depends only on $t$, while the right side depends only on spatial coordinates. Both equal a constant $E$:

$$i\hbar\frac{dT}{dt}=ET, \qquad H\varphi=E\varphi.$$

Thus

$$T(t)=e^{-iEt/\hbar}.$$

This is the usual bridge from the time-dependent Schrödinger equation to the time-independent Schrödinger equation. The spatial part is an eigenvalue problem.

Product Solutions and Superposition

For a homogeneous linear equation, product solutions can be combined:

$$u=\sum_n c_nX_n(x)Y_n(y)$$

or, when a continuous label is present,

$$u=\int c(\lambda)X_\lambda(x)Y_\lambda(y)\,d\lambda.$$

An individual separated solution is usually only one mode. General solutions are built by superposing separated modes, subject to convergence and boundary-condition requirements. This is why Sequences, Series, and Convergence matters even when each separated ODE is elementary.

Rectangular Box Example

For a particle in a three-dimensional rectangular infinite well, the stationary equation inside the box is

$$-\frac{\hbar^2}{2m} \left( \frac{\partial^2\varphi}{\partial x^2} +\frac{\partial^2\varphi}{\partial y^2} +\frac{\partial^2\varphi}{\partial z^2} \right) =E\varphi.$$

The boundary conditions are imposed on coordinate-aligned planes:

$$\varphi=0 \quad \text{at} \quad x=0,L_x;\quad y=0,L_y;\quad z=0,L_z.$$

Try

$$\varphi(x,y,z)=X(x)Y(y)Z(z).$$

After division by $XYZ$, one obtains

$$\frac{X''}{X} +\frac{Y''}{Y} +\frac{Z''}{Z} = -\frac{2mE}{\hbar^2}.$$

Each term depends on only one coordinate, so write

$$\frac{X''}{X}=-k_x^2, \qquad \frac{Y''}{Y}=-k_y^2, \qquad \frac{Z''}{Z}=-k_z^2.$$

The one-dimensional equations are

$$X''+k_x^2X=0, \qquad Y''+k_y^2Y=0, \qquad Z''+k_z^2Z=0.$$

The boundary conditions give

$$k_x=\frac{n_x\pi}{L_x}, \qquad k_y=\frac{n_y\pi}{L_y}, \qquad k_z=\frac{n_z\pi}{L_z},$$

and

$$E = \frac{\hbar^2}{2m} \left( k_x^2+k_y^2+k_z^2 \right).$$

The success of the method here depends on both the Hamiltonian and the walls separating in Cartesian coordinates. A rotated, curved, or irregular boundary would generally spoil this simple product form.

Central Potentials

For a central potential $V(r)$, spherical coordinates are adapted to the symmetry. The stationary Schrödinger equation separates with

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

The angular equation is the eigenvalue problem for angular momentum:

$$L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad L_zY_\ell^m = \hbar mY_\ell^m.$$

The functions $Y_\ell^m$ are spherical harmonics. The radial equation can be written as

$$-\frac{\hbar^2}{2m} \frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{dR}{dr} \right) + \left[ V(r) +\frac{\hbar^2\ell(\ell+1)}{2mr^2} \right]R = ER.$$

The term

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}$$

acts like an angular-momentum barrier in the radial equation. The angular separation constants are not arbitrary; single-valuedness, regularity, and angular boundary conditions restrict $\ell$ and $m$.

For hydrogenic problems, this separation produces radial Coulomb equations, Laguerre Polynomials in the radial bound states, and angular spherical harmonics. The model card Hydrogen Atom records the canonical physical problem.

For azimuthally symmetric angular dependence, the polar equation reduces to Legendre Polynomials with $Y_\ell^0\propto P_\ell(\cos\theta)$.

Cylindrical separation produces Bessel Functions in the radial coordinate. Free central radial motion and partial-wave scattering use the spherical Bessel functions from the same family.

Coordinate Systems

A coordinate system is useful for separation when it is aligned with the operator and the boundary data. Common quantum examples include:

• Cartesian coordinates for rectangular boxes and translation-invariant directions;
• cylindrical coordinates for axial symmetry;
• spherical coordinates for central potentials and rotors;
• parabolic coordinates for some Coulomb-field and Stark-effect calculations.

Changing coordinates also changes the measure and differential operators. For example, spherical normalization uses

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

The measure is part of the problem, especially when normalizing separated wavefunctions. See Normalization Conventions.

Boundary Conditions Must Separate Too

It is not enough for the differential equation to separate. The boundary conditions must also be compatible with the separated coordinates.

For a rectangular box, each wall fixes one coordinate, so the boundary conditions become separate conditions on $X$, $Y$, and $Z$. For a sphere, the boundary $r=a$ fixes the radial coordinate, while angular regularity belongs to $Y(\theta,\phi)$. For an irregular boundary, a product ansatz in a simple coordinate system may fail even if the local differential operator looks separable.

This is why separation of variables is often a symmetry-and-geometry method as much as an algebraic trick.

Relation to Commuting Operators

In quantum mechanics, successful separation often reflects a set of commuting operators. For a central potential, one may choose simultaneous eigenfunctions of

$$H,\qquad L^2,\qquad L_z.$$

The labels

$$E,\qquad \ell,\qquad m$$

are separation constants interpreted as spectral values of commuting observables. This viewpoint helps explain degeneracy: if the Hamiltonian does not depend on a label, several separated modes can share the same energy.

Common Mistakes

• Assuming every PDE can be separated after a clever guess.
• Checking the differential equation but not the boundary conditions.
• Forgetting coordinate measures when normalizing separated solutions.
• Treating one product solution as the most general solution.
• Choosing separation constants with signs that obscure the boundary conditions.
• Reusing Cartesian intuition for spherical or cylindrical Laplacians.
• Assuming separation constants are mere algebraic artifacts rather than spectral labels.

Cross-Links

• Partial Differential Equations
• Eigenvalue Problems
• Boundary Conditions
• Sturm–Liouville Theory
• Time-Independent Schrödinger Equation
• Spherical Harmonics
• Bessel Functions
• Normalization Conventions

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Suppose $F(x)+G(y)=0$ for all $x$ and $y$ in a rectangle. Show that $F$ and $G$ must be constant.

Solution

Fix two values $y_1$ and $y_2$. Then

$$F(x)+G(y_1)=0, \qquad F(x)+G(y_2)=0$$

for every $x$. Subtracting gives $G(y_1)=G(y_2)$. Since the two $y$ values were arbitrary, $G$ is constant. Then $F=-G$ is constant too.

2. For a time-independent Hamiltonian, derive the time factor in a separated solution $\psi(\mathbf r,t)=\varphi(\mathbf r)T(t)$.

Solution

Substitution into $i\hbar\partial_t\psi=H\psi$ gives

$$i\hbar\varphi\frac{dT}{dt}=T H\varphi.$$

After dividing by $\varphi T$, both sides equal a constant $E$:

$$i\hbar\frac{1}{T}\frac{dT}{dt}=E.$$

Thus

$$T(t)=e^{-iEt/\hbar}$$

up to an overall constant.

3. In a two-dimensional rectangular box, use $\varphi(x,y)=X(x)Y(y)$ to show that the energy has the form

$$E = \frac{\hbar^2}{2m} \left(k_x^2+k_y^2\right).$$

Solution

Inside the box,

$$-\frac{\hbar^2}{2m} \left( \frac{\partial^2\varphi}{\partial x^2} +\frac{\partial^2\varphi}{\partial y^2} \right) =E\varphi.$$

With $\varphi=XY$, division by $XY$ gives

$$-\frac{\hbar^2}{2m} \left( \frac{X''}{X} +\frac{Y''}{Y} \right) =E.$$

Let $X''/X=-k_x^2$ and $Y''/Y=-k_y^2$. Then

$$E = \frac{\hbar^2}{2m} \left(k_x^2+k_y^2\right).$$

4. Why is spherical separation natural for a central potential $V(r)$ but not for a rectangular box?

Solution

A central potential depends only on $r$, so the Hamiltonian is adapted to spherical symmetry and commutes with angular momentum. A rectangular box has coordinate-aligned planar walls, so its boundary conditions separate naturally in Cartesian coordinates. The useful coordinate system must fit both the operator and the boundary data.