Lukas' Notes

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Exercise Sheet 1

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Exercise 1

Instruction

Show that the Pauli matrices

are Hermitian, unitary, and that they square to the identity.

Solution

Given that , all three Pauli matrices are hermitian.

Given that all matrices are Hermitian and , all three Pauli matrices are unitary.

Exercise 2

Instruction

Show:

  1. The eigenvalues of a self-adjoined operator are always real.
  2. Projection operators have only and as eigenvalues.

Solution

Let be a self-adjoined operator, i.e. . Assume there exists an eigenvalue of such that is not equal to its complex conjugate :

By definition, there is a corresponding eigenvector for each such that:

Given that is self-adjoined, the following holds:

Choose and substituting yields:

Rewriting it into Bra-Ket notation:

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