Unitary Operator

An operator is unitary if and only if

U^+U^=U^U^+=I\begin{gather*} \hat{U}^+\hat{U}=\hat{U}\hat{U}^+=I \end{gather*}
  • it preserves inner product
∣α′⟩=U^∣α⟩∣β′⟩=U^∣β⟩⟨α′∣β′⟩=⟨α∣U^+U^∣β⟩=⟨α∣I∣β⟩=⟨α∣β⟩\begin{gather*} \left\lvert \alpha ' \right\rangle=\hat{U}\left\lvert \alpha \right\rangle\\ \left\lvert \beta' \right\rangle=\hat{U}\left\lvert \beta \right\rangle\\ \left\langle \alpha'|\beta' \right\rangle=\left\langle \alpha \right\rvert\hat{U}^+\hat{U}\left\lvert \beta \right\rangle\\ =\left\langle \alpha \right\rvert I\left\lvert \beta \right\rangle\\ =\left\langle \alpha|\beta \right\rangle \end{gather*}

  • U is basis chagne between orthonormal basis because you can write U^\hat{U} as

    • as chatgpt puts it: unitary Matrix/operators represent a rotation or change of coordinates in Hilbert space, without changing lengths or angles.
U^=U^I\begin{gather*} \hat{U}=\hat{U}I \end{gather*}

I can use Completeness relation so that if

U^∣αn⟩=∣βn⟩\begin{gather*} \hat{U}\left\lvert \alpha_n \right\rangle=\left\lvert \beta_n \right\rangle \end{gather*}

then we can change basis

U^∑n∣αn⟩⟨αn∣=∑nU∣αn⟩⟨αn∣=∑n∣βn⟩⟨αn∣\begin{gather*} \hat{U}\sum_n\left\lvert \alpha_n \right\rangle\left\langle \alpha_n \right\rvert\\ =\sum_nU\left\lvert \alpha_n \right\rangle\left\langle \alpha_n \right\rvert\\ =\sum_n\left\lvert \beta_n \right\rangle\left\langle \alpha_n \right\rvert \end{gather*}

Both {∣αn⟩}\{\left\lvert \alpha_n \right\rangle\} and {∣βn⟩}\{\left\lvert \beta_n \right\rangle\} are orthonormal basis