Postulates of quantum mechanics, useful for BSc and MSc chemistry students.

1. Dr. Pradeep Samantaroy
Department of Chemistry
Rayagada Autonomous College, Rayagada
pksroy82@gmail.com; pksroy82@yahoo.in
9444078968

2. The physical state of a system at time t is described by the
wave function Ψ (x, t).
Postulate 1
Prepared by Dr. Pradeep Samantaroy

3. The wave function Ψ (x, t) and its first and second derivative
must satisfy the following conditions:
Finite
Continuous
Single valued
Must satisfy the ortho-normal condition
Postulate 2
Prepared by Dr. Pradeep Samantaroy

4. Ortho-normal Condition
Considering one dimension:
Considering three dimension:
Normalized condition:
Orthogonal condition:
Orthonormal condition:
Prepared by Dr. Pradeep Samantaroy

5. Prepared by Dr. Pradeep Samantaroy

6. Which if the following are acceptable functions:
a) ψ = x
b) ψ = x2
c) ψ = sin x
d) ψ = exp(−x)
e) ψ = exp(−x2)
Prepared by Dr. Pradeep Samantaroy

7. A physically observable quantity can be represented by a
Hermitian operator.
An operator  is said to be Hermitian if it satisfies the following
condition.
Where Ψi and Ψj are wave functions representing the physical
states of quantum systems.
Postulate 3
Prepared by Dr. Pradeep Samantaroy

8. Postulate 4
The allowed values of an observable A are the eigenvalues, ai, in
the operator equation
ÂΨi = aiΨi
Where  is the operator
Ψ i is the eigen function
ai is the eigen value
Prepared by Dr. Pradeep Samantaroy

9. An eigen function is for an operator d2/dx2 is Ψ = e2x .
Find the corresponding eigen value.
d2/dx2 (e2x) = d/dx {d/dx (e2x)}
= d/dx (2. e2x)
= 2. 2. e2x
= 4 e2x
Prepared by Dr. Pradeep Samantaroy

10. The average value/expectation value of A i.e. <A>, corresponding to Â,
is obtained from the relation
Postulate 5
Prepared by Dr. Pradeep Samantaroy

11. The quantum mechanical operators corresponding to the observables
are constructed by writing the classical expression in terms of the
variable and converting the expressions to the operators.
Postulate 6
Prepared by Dr. Pradeep Samantaroy