Observables and eigenvalues
An observable is something that can be measured. In quantum mechanics, Susskind represents observables by Hermitian operators.
Plain-language picture
An observable is not a passive label attached to the system. It is a question you can ask experimentally: spin along z, energy, momentum, position, and so on. The operator encodes the structure of that question.
Eigenvalue rule
If M is an observable, its possible measurement results are eigenvalues:
M|m⟩ = m|m⟩The vector |m⟩ is an eigenstate of M. If the system is already in |m⟩, then measuring M gives m with certainty.
Why Hermitian
Hermitian operators have real eigenvalues and orthogonal eigenvectors for distinct eigenvalues. Real eigenvalues are needed for real measurement readouts; orthogonal eigenvectors are needed for mutually exclusive outcomes.
Expectation value
For a state |Ψ⟩:
⟨M⟩ = ⟨Ψ|M|Ψ⟩This is an average over many identically prepared experiments, not necessarily an outcome seen in any one run.
Common pitfalls
- Do not assume the expectation value must be one of the eigenvalues.
- Do not confuse “operator” with “matrix in this basis”; the matrix representation can change.
- Do not forget measurement context: the observable determines the relevant eigenbasis.