Principles of Quantum Mechanics · Problem 1.6.4

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · unitary-operators-determinant

Problem

It is assumed that you know (1) what a determinant is, (2) that det⁡ΩT=det⁡Ω\det \Omega^T = \det \Omega (TT denotes transpose), (3) that the determinant of a product of matrices is the product of the determinants. [If you do not, verify these properties for a two-dimensional case Ω=(αβγδ)\Omega = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} with det⁡Ω=(αδ−βγ)\det \Omega = (\alpha\delta - \beta\gamma).] Prove that the determinant of a unitary matrix is a complex number of unit modulus.

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