Postulates of Quantum Mechanics | Modern Physics PDF Download

Introduction 

• The quantum mechanical postulates enable us to understand. 
• How a quantum state is described mathematically at a given time t. 
• How to calculate the various physical quantities from this quantum state. 
• Knowing the system’s state at a time t , how to find the state at any later time t .
  i.e., how to describe the time evolution of a system. 

There are following set of postulates:
Postulate 1: The state of any physical system is specified, at each time t , by a state vector |ψ(t)〉 in the Hilbert space. |ψ(t)〉 contains all the needed information about the system. Any superposition of state vectors  is also a state vector.
Postulate 2: To every measurable quantity A to be called as an observable or dynamical variable, there corresponds a linear Hermitian operator Aˆ whose eigen vectors form a complete basis  A|ϕ_n〉 = a_n |ϕ_n〉
Postulate 3: The measurement of an observable A may be represented formally by an action of  on a state vector |ψ(t)〉.
The state of the system immediately after the measurement is the normalized projection  on to the eigen subspace associated with a_n.
Postulate 4 (a): When the physical quantity A is measured on a system in the state |ψ〉, the probability P(a_n) of obtaining the non-degenerate eigen value a_n of the corresponding observable 

Postulate 4 (b): When the physical quantity A is measured on a system in the state |ψ〉.
The probability P(a_n) of obtaining the eigen value a_n of the corresponding observable A is,

Where g_n is the degree of degeneracy of a_n and(i = 1, 2, 3, &, g_n) is orthonormal set of vector which forms a basis in the eigen subspace and associated with eigenvalue a_n of A.
Postulate 5: The time evolution of the state vector |ψ(t)〉 is governed by schrodinger equation given by:

where H is Hamiltonion of the system.
The solution of schodinger equation must be
(a) Single valued and  the value must be finite
(b) Continuous
(c) Differentiable
(d) Square integrable.

Expectation Value

The expectation value of operator A is given  

For continuous variable-  

• 
• Error in measurement of A is

Fourier transformation

Change in basis from one representation to another representation |p〉 is defined as,

The expansion of Ψ(x) in terms of |p〉 can be written as,

where a (p) can be found as,

In 3D:
 where a(p) being expansion coefficient of |p〉.
If any function Ψ(x) can be expressed as a linear combination of state function ϕ_n 

which is popularly derived from fourier trick. 

Parity operator: The parity operator
defined by its action on the basis.

If ψ(-r) = ψ(r), then state has even parity and
If ψ(-r) = -ψ(r) , then state has odd parity.
So, parity operator have +1 and -1 eigen value.
Representation of postulate (4) in continuous basis. 

Example 14: A state function is given by  It is given that  〈ϕ_i | ϕ_j〉, δ_ij , then
(a) check whether
is normalized or not
(b) write down normalized wavefunction.
(c) it is given H |ϕ_n〉 = (n + 1)ℏω| ϕ_n〉 where n = 0,1, 2, 3, 4, .... .If H is measured on |ψ〉, then what will be measurement and with what probability?  
(d) Find the expectation value of H i.e., 〈H〉
(e) Find the error in the measurement of H.

(a) To check normalization, one should verify- 

The value of 〈ψ|ψ〉 = 3/2 , so |ψ〉 is not normalized.
(b) Now we need to find normalized |ψ〉 let A be normalization constant.

So,

(c) It is given that  
H|ϕ_n〉 = (n + 1) ℏω, where n = 0,1, 2, 3, 4, ....
H |ϕ₁〉 2ℏω and H |ϕ₂〉 = 3ℏω
when H will be measured on |ψ〉, it will measured either 2ℏω or 3ℏω
The probability of measurement 2ℏω is P(2ℏω) is given by

So, when H is measured on state |ψ〉, the following outcome will come:
Measurement of H on state : |ϕ₁〉 |ϕ₂〉
Measurement  : 2ℏω 3ℏω
Probability  : 2/3  1/3



(e) The error in measurement in H is given as

Example 15: The wave function of a particle is given by  where ϕ₀ and ϕ₁ are the normalised eigenfunctions with energy E₀ and E₁ corresponding  to ground state and first excited state.
(a) Find the value of B such that Ψ is normalised.
(b) What are the measurements
(c) What is the probability of getting energy E1
(d) What is 〈E〉

For normalized |ψ〉,


(b) Measurement are E₀ ,E₁
(c) Probability of getting 

Example 16: (a) Plot Ψ_I (x) = A₁e^-|x| ; -∞ < x < ∞
(b)
(c) Discuss why ψ_I is not the solution of Schrödinger wave function rather ψ_II is solution of Schrödinger wave function.

(a) ψ_I (x) = A₁e^+x ; x < 0
ψ_II (x) = A₁e^-x ; x > 0
The plot is given by
(b)

(c) Both the function ψ_I and ψ_II are single valued, continuous, square integrable but ψ_I is not differentiable at x = 0 , rather ψ_II is differentiable at x = 0
So, ψ_II can be solution of Schrödinger wave function but ψ_I is not the solution of Schrödinger wave function. 

Example 17: At time t = 0 , the state vector |ψ(0)〉 is given as, 

It is given that, Hamiltonian is defined as H |ϕ_n〉 = n²∈₀|ϕ_n〉
(a) What is wave function |ψ(t)〉 at later time t.
(b) Write down expression of evolution of |ψ(x, t)|²
(c) Find Δ H
(d) Find the value of Δ HΔt

(b) Evolution of shape of the wave packet 

(c)  Δ H = (〈H²〉 - 〈H〉²)^1/2 


(d)

Example 18: Consider a one-dimensional particle which is confined within the region 0 ≤ x ≤ a and whose wave function is  Find the potential V(x).

From the fifth postulate:

Example 19: Eigenvalues of operator A are 0, 2a₀ , 2a₀ and corresponding normalized eigen vector are respectively,  then if the system is in state  then
(a) When A is measured on system in state  then what is the probability of getting value 0, 2a₀, respectively?  
(b) What is the expectation value of A ?

λ₂ = λ₃ = 2a₀ i.e., λ = 2a₀ is doubly degenerate.

Example 20: A free particle which is initially localized in the range -a < x < a is released at time t = 0.

(a) Find A such that ψ (x) is normalized.
(b) Find ϕ(x) i.e., wave function in momentum space.
(c) Find ψ (x, t) i.e., wave function after time t.