Given a transformation ##U## such that ##|\psi'>=U|\psi>##, the invariance ##<\psi'|\psi'>=<\psi|\psi>## of the scalar product under the transformation ##U## means that ##U## is either linear and unitary, or antilinear and antiunitary.(adsbygoogle = window.adsbygoogle || []).push({});

How do I prove this?

##<\psi'|\psi'>##

##= <U\psi|U\psi>##

For ##U## unitary and linear, we have

##<U\psi|U\psi>##

##<U(\alpha\psi_{a}+\beta\psi_{\beta})|U(\alpha\psi_{a}+\beta\psi_{\beta})>##

##<(\alpha U\psi_{a}+\beta U\psi_{\beta})|(\alpha U\psi_{a}+\beta U\psi_{\beta})>##

##=(<\psi_{a}|(U^{\dagger})\alpha^{*}+<\psi_{b}|(U^{\dagger})\beta^{*})(\alpha U|\psi_{a}>+\beta U|\psi_{\beta}>)##

##=|\alpha|^{2}(<\psi_{a}|(U^{\dagger})(U)|\psi_{a}>+|\beta|^{2}(<\psi_{b}|(U^{\dagger})(U)|\beta>##

##=|\alpha|^{2}+|\beta|^{2}##

Is this how the proof should go for ##U## linear and unitary?

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# Unitary and linear operator in quantum mechanics

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