MIT OpenCourseWare: Quantum Harmonic Oscillator and Schrödinger Equation

Overview

The MIT OpenCourseWare lecture introduces the quantum harmonic oscillator by connecting its classical cousin, a mass on a spring, to its quantum counterpart through canonical operators X and P with the commutation relation [X, P] = iħ. It emphasizes defining the Hilbert space of states as complex square-integrable functions on the real line, and it introduces the factorization of the Hamiltonian in terms of creation and annihilation operators to reveal the ground state energy and the ladder of excited states. The talk then shows how unitary time evolution leads to the Schrödinger equation, linking energy eigenstates to the spectrum and to the number operator N.

Throughout, the lecturer highlights the elegance of the algebraic approach, the role of the number operator, and the interplay between algebraic factors and physical content, culminating in a structured picture of the harmonic oscillator that generalizes to other quantum systems.

Intro and Setup

The lecture begins with a motivation from the classical harmonic oscillator energy, E = P^2/(2m) + (1/2) m ω^2 X^2, and moves to the quantum system by promoting X and P to operators with the fundamental commutation relation [X, P] = iħ. The state space is identified as a Hilbert space of complex functions on the real line, with wavefunctions that are square-integrable. The Hamiltonian becomes H = (P^2)/(2m) + (1/2) m ω^2 X^2, and one seeks its spectrum via algebraic methods rather than solving differential equations alone.

Creation and Annihilation Operators

Introducing the ladder operators A and A† (often written as a and a† in the notes), the Hamiltonian is factorized as H = (1/2)ħω + ħω A†A. The operators are defined so that A† creates excitations and A annihilates them, with the commutator [A, A†] = 1. The relations X and P can be rewritten in terms of A and A†, providing a direct route to compute the spectrum. The “number operator” N = A†A emerges as central, with H = ħω (N + 1/2). Eigenstates of N and H coincide; their eigenvalues label the energy levels as E_n = ħω(n + 1/2) for n = 0, 1, 2, ... .

Spectrum, Ground State, and Norms

The notes detail how acting with A† on an eigenstate raises the energy by ħω and increases the number operator eigenvalue by one, while A lowers it. The norms of the newly generated states can be analyzed via commutators to ensure positivity as long as n ≥ 0. A crucial point is that attempting to continue lowering below n = 0 leads to a vanishing state, thereby establishing the ground state and the ladder of excited states. The ground state |0⟩ is characterized by A|0⟩ = 0, and its wavefunction in the position basis is a Gaussian, ψ_0(x) ∝ exp(−(mω x^2)/(2ħ)).

Spectrum and Orthogonality

The eigenstates |n⟩ = (1/√(n!)) (A†)^n |0⟩ form an orthonormal basis for the Hilbert space, with N|n⟩ = n|n⟩ and H|n⟩ = ħω(n + 1/2)|n⟩. The notes emphasize that energy eigenstates are non-degenerate, and that the ladder structure provides a complete description of the spectrum without solving differential equations.

From Unitary Time Evolution to Schrödinger Equation

Time evolution in quantum mechanics is unitary: |ψ(t)⟩ = U(t, t0) |ψ(t0)⟩, with U being unitary and satisfying composition rules. Differentiating this evolution yields a Schrödinger-type equation. The lecturer shows how the generator of time evolution, the Hamiltonian, arises from the time derivative of the evolution operator, leading to iħ ∂ψ/∂t = H ψ. This derivation clarifies that the Schrödinger equation is a consequence of the postulate of unitary time evolution, and that given a Hamiltonian one can determine the unitary evolution, or vice versa the Hamiltonian can be inferred from a known unitary evolution.

Conclusion and Outlook

The talk closes by summarizing the utility of the factorization method and the ladder operator approach, and by foreshadowing similar treatments for angular momentum. The instructor invites questions and hints at applying these ideas to other systems such as the hydrogen atom and more complex Hamiltonians, reinforcing the pedagogical power of the operator method in quantum mechanics.