Observables

Observables in Quantum mechanics are represented by linear Hermetian operators

Operators form an associative but not commutative algebra over called the “Operator Algebra” where the product is the composition.
If there is a unique eigenvector for each eigenvalue of an operator , then we label eigenvectors by their eigenvalues and write . There is no confusion because the vectors are always in a ket.

If all eigenvectors are nondegenerate we can write

so that

Define defines if exists.

Matrix elements of in this basis are .

Also
Also

When eigenvectors of are degenerate, we look for a complete set of commuting observables st for all pairs.

Then they are simultaneously diagonalizable (diagonalizable in the same basis)

Operators in

Let’s mention and ignore some issues in spaces such as

• In finite dim all operators are bounded
• In infinite dimensions that’s not true.
  Position and momentum operators are unbounded.
  We would need to pay attention to them (but we don’t)

Operators for composite systems

Let be basis of and a linear operator.
Similarly and on
We define
by

Note 𝟙𝟙