Overview
MIT OpenCourseWare presents a detailed lecture on quantum mechanics focusing on position and momentum operators. The instructor derives X and P, shows their noncommutativity with [X,P] = iħ, and explains how wave functions are treated in the X representation. The talk traverses momentum space via the Fourier transform, introduces momentum eigenstates, and discusses how the same physics is encoded in both X and P representations. The session culminates with the Stern-Gerlach discussion introducing spin, the magnetic moment, and the two-state spin-1/2 system, setting the stage for the spin formalism and measurement sequences that follow in the course.
Overview
The lecture begins with a review of the variational principle and then delves into position and momentum operators in quantum mechanics. The instructor defines the position operator X̂ and momentum operator P̂ in the X representation and shows their fundamental noncommutativity through the commutator [X̂, P̂] = iħ. Time is treated as a background arena; for the moment, wave functions are considered with no explicit time dependence, as psi(X) in the position representation.
Operators and Representations
The discussion then connects wave functions to vectors and operators to matrices, offering an intuitive picture: a function psi(X) on an interval [0, A] can be discretized into components and viewed as a vector. In this discretized view, X̂ becomes a diagonal matrix with entries corresponding to position values 0, ε, 2ε, ..., Nε, illustrating how the operator acts by simple multiplication in this basis. The expectation value
is defined as the integral of psi*(X) X psi(X) over X, and eigenstates of X̂ are delta functions δ(X − X0), which are not normalizable but serve as useful formal tools.Momentum Operator and Fourier Transform
The momentum operator P̂ can be written as −iħ d/dX in the X representation. Verifying the commutation relation shows consistency between the operator definitions. The spectrum of P̂ is explored via momentum eigenstates e^{iPX/ħ}, and the relationship between position and momentum representations is made precise through Fourier transforms: psi~(P) is the transform of psi(X), and psi(X) can be recovered from psi~(P). This establishes that the same physical state can be expressed either in position space or momentum space, with the corresponding operators acting in the appropriate representation.Momentum Representation and Operator Equivalence
In momentum space, P̂ acts by multiplication, while X̂ becomes iħ d/dP. The dual pictures are connected by the Fourier transform, illustrating the abstract idea that operators have different representations depending on the chosen basis. The lecturer notes that these concepts, though introduced here in a basic form, will be revisited with full notation later in the course.Stern-Gerlach and Spin
The transition to spin is introduced with the Stern-Gerlach experiment, which reveals the spin-1/2 nature of the electron. The magnetic moment and Bohr magneton are discussed as the link between angular momentum and magnetic effects. For elementary particles like the electron, the g-factor is approximately 2, leading to the magnetic moment μ = −g μ_B Ŝ/ħ. The two-state spin-1/2 system is then introduced through Sz eigenstates with eigenvalues ±ħ/2, and the experimental setup is shown to project states onto Z and X bases, leading to distinct measurement outcomes. The lecture emphasizes that a general spin state is a superposition of these basis states and that Stern-Gerlach type experiments crystallize the idea of quantum state vectors in a two-dimensional complex space.Spin Measurements and Quantum Logic
Three representative experiments are discussed: measuring Z then Z, measuring Z then X, and measuring Z then X with a subsequent Z analysis. The results reveal that spin states along Z and X are not orthogonal in the quantum sense, and that after certain sequences the memory of prior measurements can be altered or destroyed. This paves the way for a more precise mathematical treatment of spin and the Pauli matrices in later lectures.
