Eigenvalues and Eigenfunctions

Introduction: Stationary Waves and Quantization

In classical physics, a stretched string fixed at both ends can only vibrate with certain discrete frequencies (harmonics). The amplitude function \( f(x) \) for such a standing wave must satisfy two conditions: it must be zero at both ends (fixed boundaries) and be single‑valued and finite. Only for specific wavelengths (or frequencies) do such solutions exist. This is a classic example of an eigenvalue problem: the wave equation allows solutions only for discrete values of the parameter (frequency).

In quantum mechanics, the Schrödinger equation plays a similar role. The wavefunction \( \psi \) must be well‑behaved (finite, single‑valued, continuous) and must vanish at infinity for bound states. These physical boundary conditions restrict the possible values of the total energy \( E \) to a set of discrete eigenvalues, and the corresponding wavefunctions are called eigenfunctions. This quantization of energy is the essence of quantum mechanics.

The Eigenvalue Equation in Quantum Mechanics

In quantum mechanics, every observable physical quantity (energy, momentum, angular momentum) is represented by a linear operator. The eigenvalues of the operator are the possible measured values of that observable. For an operator \( \hat{A} \), the eigenvalue equation is:

Here, \( \psi \) is an eigenfunction (or eigenstate) of the operator, and \( a \) is the corresponding eigenvalue. The most important operator is the Hamiltonian \( \hat{H} \), which represents the total energy. The time‑independent Schrödinger equation is precisely the eigenvalue equation for the Hamiltonian:

Solving this equation yields the allowed energy levels \( E_n \) (eigenvalues) and the corresponding wavefunctions \( \psi_n \) (eigenfunctions).

Conditions for Acceptable Eigenfunctions

For a wavefunction to represent a physical quantum state, it must satisfy three key conditions. These conditions ensure that the probabilistic interpretation of quantum mechanics is consistent and that the wavefunction is square‑integrable (normalizable).

Only solutions that satisfy these conditions are physically admissible. The requirement that \( \psi \) vanish at infinity for bound states leads to quantization of energy eigenvalues.

The Time‑Independent Schrödinger Equation

For a particle of mass \( m \) moving in a potential \( V(\mathbf{r}) \), the Hamiltonian operator is:

The time‑independent Schrödinger equation therefore becomes:

This is an eigenvalue equation. For a given potential \( V \), we seek the values of \( E \) (eigenvalues) for which a well‑behaved solution \( \psi \) exists. Typically, these eigenvalues are discrete for bound states and continuous for free particles.

Interactive Visualization: Eigenfunctions of the 1D Box

The eigenfunctions (wavefunctions) and probability densities for the first three states of a particle in a one‑dimensional box are shown below. Note how the number of nodes increases with the quantum number \( n \).

Properties of Eigenfunctions

Eigenfunctions of a Hermitian operator (such as the Hamiltonian) have two crucial properties:

• Orthogonality: Eigenfunctions corresponding to distinct eigenvalues are orthogonal. For the particle in a box:
\[ \int_{0}^{a} \psi_m^{*}(x) \psi_n(x) dx = 0 \quad (m \neq n) \]
• Completeness: Any well‑behaved wavefunction can be expanded as a linear combination of the eigenfunctions:
\[ \Psi(x) = \sum_{n=1}^{\infty} c_n \psi_n(x) \]

The coefficients \( c_n \) give the probability amplitude to measure the energy \( E_n \).

These properties make eigenfunctions an ideal basis for representing quantum states.

Connection to Bohr’s Atomic Model

The eigenvalues of the Hamiltonian for the hydrogen atom give the same energy levels as Bohr’s model: \( E_n = -13.6\ \text{eV} / n^{2} \). However, the wave mechanical approach also provides the electron probability distribution (orbitals), which Bohr’s model could not. Thus, Bohr’s quantization condition emerges as a consequence of the Schrödinger equation and the boundary conditions on the wavefunction.

Eigenvalue Problems Beyond Quantum Mechanics

The concept of eigenvalues and eigenfunctions is not limited to quantum mechanics. In mathematics, it appears in:

• Vibrating strings and membranes: The normal modes are eigenfunctions of the wave operator.
• Heat conduction: The temperature distribution satisfies an eigenvalue equation related to the Laplacian.
• Sturm‑Liouville theory: A broad class of differential equations with boundary conditions.

In each case, the boundary conditions restrict the possible eigenvalues to a discrete set, and the eigenfunctions form a complete basis.

Video Lecture: Eigenvalues and Eigenfunctions (Urdu/Hindi)

Summary and Key Takeaways

• An eigenvalue equation \( \hat{A}\psi = a\psi \) associates an eigenvalue \( a \) (measured value) with an eigenfunction \( \psi \) (quantum state).
• The time‑independent Schrödinger equation \( \hat{H}\psi = E\psi \) is the eigenvalue equation for the Hamiltonian (energy).
• Boundary conditions (wavefunction zero at infinity for bound states, continuity, single‑valuedness, finiteness) force the energy eigenvalues to be discrete.
• Eigenfunctions of Hermitian operators are orthogonal and form a complete basis.
• The particle in a box is the simplest example, yielding sine wave eigenfunctions and quadratic energy levels.
• The eigenvalues of the hydrogen atom Schrödinger equation reproduce Bohr’s energy levels but also give correct electron probability distributions.

Understanding eigenvalues and eigenfunctions is essential for solving any quantum mechanical problem, from atomic physics to quantum chemistry.