1. Postulates of Quantum Mechanics
(from “quantum mechanics” by Claude Cohen-Tannoudji)
6th postulate: The time evolution of the state vector
is governed by the Schroedinger equation
where H(t) is the observable associated with the total energy of the system.
1st postulate: At a fixed time t0, the state of a physical system is defined by specifying a ket

2. Postulates of Quantum Mechanics
(from “quantum mechanics” by Claude Cohen-Tannoudji)
2nd postulate: Every measurable physical quantity
is described by an operator
This operator is an observable.
3rd postulate: The only possible result of the measurement of a physical quantity
is one of the eigenvalues
of the corresponding observable
4th postulate (non-degenerate): When the physical quantity
is measured on a system in the normalized state
the probability of
obtaining the eigenvalue
of the corresponding observable is
where
is the normalized eigenvector of
associated with the eigenvalue

3. Physical interpretation of
is a probability density. The probability of
finding the particle in the volume element
at time is
General solution for
Try separation of variables:
Plug into TDSE to arrive at the pair of linked equations:
and

4. Orthogonality:
For
which are different eigenvectors of
we have orthogonality:
Let us prove this to introduce the bra/ket notation used in the textbook