Hermitian Operators
Quantum Mechanical operators are hermitian. If a quantum mechanical operator is represented by a matrix
then
so thatis equal to the complex conjugate transpose of
For example, the spin Pauli operators may be represented by the matrices
Every quantum mechanical operatoris associated with an eigenvalue equation
where
is an eigenvalue
If the quantity corresponding to the operatoris measured, the only possible values of the quantity
that 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:
Ifthen
since
Hence
Now take the complex transpose of both sides:
But
so
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.
