One-Dimensional Quantum Potentials, Bound States, and the Variational Principle | MIT OpenCourseWare

MIT OpenCourseWare delivers a focused lecture on quantum mechanics in one dimension, covering bound states, spectral structure, and essential theorems. The talk defines bound states as energy eigenstates whose wavefunctions vanish at infinity, proves the absence of degeneracy for 1D bound states, and shows how complex solutions can be taken as real. It then discusses parity for even potentials, introduces the node theorem for bound states, and centers on the variational principle with a delta function example to bound the ground-state energy. The discussion also touches continuous versus discrete spectra and boundary conditions, concluding with homework pointers on delta potentials.

Overview

In this MIT OpenCourseWare lecture, the instructor examines quantum mechanics in one dimension with a focus on bound states, spectral properties, and the variational principle. The session starts by defining bound states as energy eigenstates with wavefunctions that vanish as |x| → ∞ and proceeds to establish foundational theorems that shape how we understand one-dimensional systems.

Key Theorems in One Dimension

• Theorem 1: There are no degeneracies among bound states in one-dimensional potentials. The strategy involves assuming degeneracy, manipulating the Schrödinger equations for the two degenerate states, and showing that the two solutions must be proportional, contradicting a true degeneracy unless they describe the same state up to a phase.
• Theorem 2: A complex solution can be used to generate two real solutions, which in turn can be chosen as real without loss of generality. This implies that for one-dimensional bound states, any solution can be taken to be real up to an overall phase.
• Theorem 3: If the potential is even, eigenstates can be chosen to be even or odd. The proof constructs symmetric and antisymmetric combinations of a non‑even, non‑odd solution and uses parity to generate two independent solutions with the same energy, leading to the even/odd classification of bound states in symmetric wells.

Spectrum and Boundary Conditions

The lecturer analyzes the spectrum by considering the continuity of the wavefunction and its derivative under different potential profiles. He emphasizes that in many common cases, the wavefunction ψ and its first derivative ψ′ are continuous, except when the potential contains a delta function or a hard wall which can produce finite jumps in ψ′. This discussion helps illuminate how bound-state spectra arise or become continuous depending on the potential landscape.

Node Theorem and Bound States

The node theorem states that in a discrete bound-state spectrum, the nth bound-state wavefunction has exactly n−1 nodes. The lecturer outlines intuitive and more rigorous arguments, including a pedagogical infinite-square-well argument that is not fully rigorous but illustrates how nodes increase with energy and how the infinite-well case generalizes to more complex potentials.

Variational Principle

The variational principle is introduced as a powerful method to estimate ground-state energies. The idea is to use a trial wavefunction ψ that is normalized and compute the expectation value of the Hamiltonian, which provides an upper bound to the true ground-state energy E0. The functional form F[ψ] = ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ is introduced to handle unnormalized trial functions. The minimum of this functional corresponds to the ground state and, in the infinite-dimensional function space, excited states appear as saddle points rather than global minima.

Delta Function Potential Example

A concrete delta function potential V(x) = −αδ(x) is used to illustrate the variational method. A Gaussian-type trial function with a variational parameter β is chosen, and the energy bound is computed as a function of β. The result demonstrates how the variational bound can be optimized to yield a bound on the true ground-state energy, and how the exact known result for the delta potential can be approached with a sensible choice of trial function.

Implications and Homework

The lecture ends by highlighting corollaries for even potentials, revisiting the nodes theorem, and pointing to homework problems that explore delta-function potentials and variational methods. The instructor hints at extending these ideas to more complex potentials and to the full variational treatment of excited states in subsequent lectures.