Susskind QM Lecture 3 - Principles of Quantum Mechanics
One-line takeaway
States are vectors; observables are Hermitian operators; measurement outcomes are eigenvalues.
Plain-language map
Lecture 3 formalizes the rules that were implicit in the spin examples. A linear operator is a machine that takes a ket and returns another ket. Once a basis is chosen, the operator can be represented by a matrix whose entries are matrix elements.
Physical observables are represented by Hermitian operators. Hermitian operators are special because their eigenvalues are real and their eigenvectors can be used as orthonormal measurement bases. That is exactly what a measurement theory needs: real possible answers and mutually exclusive outcome states.
Core principles
- A system has a vector space of possible states. See Hilbert space.
- A state is represented by a ket |Ψ⟩. See Quantum state.
- A measurable quantity is represented by a Hermitian operator. See Observables and eigenvalues.
- The possible results of measuring an observable are the operator’s eigenvalues.
- After a measurement with a definite outcome, the state is in the corresponding eigenstate or eigenspace. See Measurement and state preparation.
Key equations
M|m⟩ = m|m⟩
⟨M⟩ = ⟨Ψ|M|Ψ⟩For spin, the Pauli matrices provide concrete operators for σx, σy, and σz. They are the recurring test case for the abstract rules.
Common pitfalls
- Do not identify an operator with one particular matrix too strongly; the matrix changes when the basis changes.
- Do not assume the expectation value must be one of the possible measurement outcomes.
- Do not forget degeneracy: one eigenvalue can correspond to more than one independent eigenvector.