Unitarity And Tome Evolution

Question 1

T-dependent wave function in Hilbert space

Answer 1

Question 2

Time dependent Schrödinger equation in Hilbert space

Answer 2

Question 3

Define exponential of operator

Answer 3

Question 4

Evolution operator

Answer 4

Question 5

Evolution by time t_1 followed by time t_2

Answer 5

Question 6

Adjoint of Evolutionary operator

Answer 6

Question 7

Standard solution to Schrödinger equation in Hilbert space

Answer 7

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Show unitarity of evolution operator

Answer 12

Important point is

e⁰ = 1

Question 13

Given 2 states |Ψ> and |Φ> € H

Answer 13

Question 14

Unitary time evolution preserves

Answer 14

Inner product

Question 15

Given H^^ and some initial state |Ψ(0)> how to find state |Ψ(t)> at some t€R

Answer 15

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

Question 16

If |Ψ> is an Eigenstate of O^ with eigenvalue λ, then measuring O^ in state |Ψ> ?

Answer 16

Will always give eigenvalue λ