Schroedinger Equation II

The postulates of Quantum Mechanics:

I. Dynamical variables or observables ( measurable quantities ) are associated with actions of Linear Operators. ( Note that some variables like spin are intrinisically quantum mechanical without any classical analogs).

** We will adopt the covention of representing operators by boldface

Assuming ψ(x,t)= A e^i(kx-ωt) for a free particle it is easy to see
by direct operation that the momentum operator p=-i ℏ(∂/∂ x)
and that for the energy operator E=-i ℏ(∂/∂ t)

II. The Measurement of a dynamical variable or observable A that yields the value a leaves the system in a quantum state given by the wave function Φ_a
or A Φ_a=a Φ_a is an eigenvalue equation.
For example pψ=pψ.

III) Expectation Value: The average value of an observable for a system in the quantum state given by the wave function ψ is given as
⟨ C ⟩=∫ψ*C&psi dx

The average value is understood in the sense of an ensemble average where
simultaneous measurements of the observable is made at time t on a large number of exact replica of the system with identical initial quantum state specified by ψ(x,0).

IV) The time development of the wave function ψ(x,t) is given by the time dependent Schroedinger's equation.