References: Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Exercises 1.6.3 – 1.6.6.

Here are a few more results about unitary operators.

Shankar defines a unitary operator as one where

From this we can derive the other condition by which they can be defined, namely that a unitary operator preserves the norm of a vector:

This follows, for if we define the effect of by

then

Thus .

Theorem 1The product of two unitary operators and is unitary.

*Proof:* Using Shankar’s definition 1, we have

Theorem 2The determinant of a unitary matrix is a complex number with unit modulus.

*Proof:* The determinant of a hermitian conjugate is the complex conjugate of the determinant of the original matrix, since (where the superscript denotes the transpose) for any matrix, and the hermitian conjugate is the complex conjugate transpose. Therefore

Therefore as required.

Example 1The rotation matrix is unitary. We have

By direct calculation

Example 2Consider the matrixBy calculating

Thus is unitary, but because it is not hermitian. Its determinant is

This is of the required form with .

Example 3Consider the matrix

Thus is unitary, but because it is not hermitian. Its determinant is

This is of the required form with .

Pingback: Spectral theorem for normal operators | Physics pages

Pingback: Time-dependent propagators | Physics pages