Unitarity And Tome Evolution Flashcards
T-dependent wave function in Hilbert space
Time dependent Schrödinger equation in Hilbert space
Define exponential of operator
Evolution operator
Evolution by time t_1 followed by time t_2
Adjoint of Evolutionary operator
Standard solution to Schrödinger equation in Hilbert space
Show unitarity of evolution operator
Important point is
e0 = 1
Given 2 states |Ψ> and |Φ> € H
Unitary time evolution preserves
Inner product
Given H^^ and some initial state |Ψ(0)> how to find state |Ψ(t)> at some t€R
IMPORTANT ALGORITHM
-find eigenvalues E_n and orthonormal basis of eigenstates of H^^
- find expression for initial state in given basis
-act with evolutionary operator
If |Ψ> is an Eigenstate of O^ with eigenvalue λ, then measuring O^ in state |Ψ> ?
Will always give eigenvalue λ