Isolated System

This builds on Schrödinger’s equation. State changes from t1→t2t_1\rightarrow t_2 according to some linear operator U(t2,t1)U(t_2, t_1).

∣ψ(t2)⟩=U(t2,t1)∣ψ(t1)⟩\left\lvert \psi(t_2) \right\rangle=U(t_2, t_1)\left\lvert \psi(t_1) \right\rangle

Properties

  1. t2=t1t_2=t_1 gives no change
∣ψ(t1)⟩=U(t1,t1)∣ψ(t1)⟩ \left\lvert \psi(t_1) \right\rangle=U(t_1, t_1)\left\lvert \psi(t_1) \right\rangle U(t1,t1)=I U(t_1, t_1)=I
  1. composition
∣ψ(t3)⟩=U(t3,t2)∣ψ(t2)⟩ \left\lvert \psi(t_3) \right\rangle=U(t_3, t_2)\left\lvert \psi(t_2) \right\rangle =U(t3,t2)U(t2,t1)∣ψ(t1)⟩ =U(t_3, t_2)U(t_2, t_1)\left\lvert \psi(t_1) \right\rangle =U(t3,t1)∣ψ(t1)⟩ =U(t_3, t_1)\left\lvert \psi(t_1) \right\rangle 
hence
U(t3,t1)=U(t3,t2)U(t2,t1) U(t_3, t_1)=U(t_3, t_2)U(t_2, t_1)
  1. inverse
I=U(t1,t2)U(t2,t1) I=U(t_1,t_2)U(t_2,t_1) U(t1,t2)=U−1(t2,t1)=U†(t2,t1) U(t_1,t_2)=U^{-1}(t_2,t_1)=U^\dagger(t_2, t_1)
  1. diagonalizable
U†U=UU† U^\dagger U = U U^\dagger 
this means [spectral theorem](/spectral-decomposition) applies.
For every [orthonormal basis](/orthonormality) $\{\left\lvert k \right\rangle\}$,
U=∑kλk∣k⟩⟨k∣ U=\sum_k\lambda_k\left\lvert k \right\rangle\left\langle k \right\rvert I=⟨k∣U†U∣k⟩ I=\left\langle k \right\rvert U^\dagger U\left\lvert k \right\rangle λk=eiθk \lambda_k=e^{i\theta_k}