Hermitian Operators

Quantum Mechanical operators are hermitian. If a quantum mechanical operator is represented by a matrixthenso thatis equal to the complex conjugate transpose ofFor example, the spin Pauli operators may be represented by the matrices

Every quantum mechanical operatoris associated with an eigenvalue equationwhereis an eigenvalueIf the quantity corresponding to the operatoris measured, the only possible values of the quantitythat may be observed are the eigenvalues of the operator

The eigenvalues of the matrix represent actual observable quantities and must be real numbers ie not complex numbers. This is a feature of Hermitian operators – that the eigenvalues are real.

Proof:

Ifthensince

Hence

Now take the complex transpose of both sides:

Butso

For the spin Pauli matrices above, the eigenvalues are +-1 , but in general eigenvalues may take any one of a possibly infinite range of values. This is the case for the energy of a harmonic oscillator, the position of momentum of a free particle, or the energy of an electron in an atom.

Notice that the spin is quantized, since the eigenvalues are +- 1. This falls quite naturally out of the eigenvalue equations. If a physical quantity is quantized then the set of eigenvalues will form a (possibly infinite) discrete set.