Schrödinger Equation in One Dimension: Wave Functions, Probabilities, and Energy Eigenstates | MIT OpenCourseWare

This MIT OpenCourseWare lecture provides a clear introduction to the non-relativistic Schrödinger equation in one dimension. It explains why the wave function is complex, how to interpret |ψ(x,t)|^2 as a probability density, and how probability current relates to the continuity equation. The session covers normalization, unitary time evolution, and the linear superposition principle. It then introduces stationary states as energy eigenstates for time-independent potentials, derives the time-independent Schrödinger equation, and discusses spectra and degeneracy. The speaker explains how a general state evolves by expanding in the eigenbasis with coefficients b_n, and concludes with expectation values, notably the energy, and remarks on non-normalizable states and idealized potentials.

Overview of the Schrödinger Equation

The lecture presents the non-relativistic Schrödinger equation in one dimension, iħ ∂ψ/∂t = Ĥ ψ, where Ĥ = -ħ^2/(2m) ∂^2/∂x^2 + V(x,t). The wave function ψ(x,t) is complex-valued, reflecting quantum amplitudes, with the potential possibly depending on x and time. This sets up the fundamental dynamical framework of quantum mechanics.

Wave Function and Complex Numbers

The instructor emphasizes that unlike real classical fields, ψ must be complex. This is essential for describing interference and time evolution through phase factors. Complex numbers are introduced as a natural language for quantum amplitudes, with complex conjugation and modulus playing central roles in probabilities.

Probability Interpretation and Normalization

The probability density is ρ(x,t) = |ψ(x,t)|^2. The probability of finding the particle in an interval is ρ(x,t) dx, and the total probability over all space must be 1 for all times. This fixes the normalization condition ∫ |ψ|^2 dx = 1 and implies that ψ has units of length^-1/2 in one dimension. The normalization requirement ensures a consistent statistical interpretation of the wave function.

Probability Current and the Continuity Equation

The probability current J(x,t) is J = (ħ/m) Im{ψ* ∂ψ/∂x}. Together with ρ, it satisfies the continuity equation ∂ρ/∂t + ∂J/∂x = 0, expressing conservation of probability. This connects the local flow of probability to the time change of the density, mirroring charge conservation in classical electromagnetism but applied to quantum probability flow.

Stationary States and the Time-Independent Schrödinger Equation

For time-independent potentials V(x), stationary states separate time and space: ψ(x,t) = φ(x) e^{-iEt/ħ}. Substituting into the Schrödinger equation yields the time-independent equation Ĥ φ = E φ, with Ĥ = -ħ^2/(2m) ∂^2/∂x^2 + V(x). The spatial function φ(x) satisfies a second-order differential equation, and the eigenvalues E form the energy spectrum of the system. Stationary states are called stationary because expectation values of time-independent observables remain constant in these states.

Energy Eigenstates, Spectrum, and Completeness

The solutions to the time-independent equation form energy eigenstates φ_n with energies E_n. The spectrum may be discrete (bound states) or continuous (scattering states), and degeneracies can occur when multiple independent eigenfunctions share the same energy. The eigenfunctions can be chosen orthonormal, and the set is complete: any reasonable ψ(x,0) can be expanded as ψ(x,0) = ∑_n b_n φ_n(x). This completeness is the cornerstone of solving dynamics by spectral decomposition.

Time Evolution and Spectral Decomposition

Once the coefficients b_n are known from the initial state, the full time evolution becomes ψ(x,t) = ∑_n b_n e^{-iE_n t/ħ} φ_n(x). Each component acquires its own phase factor over time, so observables evolve according to this superposition. The dual notions of orthonormality and completeness underlie this powerful representation, linking spatial structure to energy labels.

Observables, Expectation Values, and Energy Conservation

For a time-independent observable Â, the expectation value ⟨A⟩(t) = ∫ ψ*(x,t) Â ψ(x,t) dx is generally time-dependent if the state is not stationary. In the energy context, ⟨H⟩ = ∑_n |b_n|^2 E_n, a time-independent weighted average reflecting energy conservation under a time-independent Hamiltonian. These results illustrate how spectral decomposition encodes measurable predictions and conservation laws.

Normalization, Non-Normalizable States, and Potentials

The discussion covers non-normalizable states, such as momentum eigenstates, which are necessary for describing continuous spectra and scattering. Potentials can be piecewise continuous and may include delta functions; the formalism remains valid provided ψ and its derivative behave suitably at infinity. Normalization can be performed after the fact for non-normalized states, enabling practical calculations with idealized states.

Summary and Outlook

The lecture closes by outlining next topics, including deeper analysis of the spectrum in one dimension and the variational principle. The core ideas—linearity, superposition, unitary time evolution, and the expansion in energy eigenstates—form the foundation for understanding quantum dynamics in more complex systems.