Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Modern Physics

IIT JAM : Postulates of Quantum Mechanics IIT JAM Notes | EduRev

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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|ϕn〉 = ann
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 Postulates of Quantum Mechanics IIT JAM Notes | EduRev on to the eigen subspace associated with an.
Postulate 4 (a): When the physical quantity A is measured on a system in the state |ψ〉, the probability P(an) of obtaining the non-degenerate eigen value an of the corresponding observable 

Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Postulate 4 (b): When the physical quantity A is measured on a system in the state |ψ〉.
The probability P(an) of obtaining the eigen value an of the corresponding observable A is,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Where gn is the degree of degeneracy of an andPostulates of Quantum Mechanics IIT JAM Notes | EduRev(i = 1, 2, 3, &, gn) is orthonormal set of vector which forms a basis in the eigen subspace and associated with eigenvalue an of A.
Postulate 5: The time evolution of the state vector |ψ(t)〉 is governed by schrodinger equation given by:
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
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  

  • Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
For continuous variable-  

  • Postulates of Quantum Mechanics IIT JAM Notes | EduRev
  • Error in measurement of A is Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Fourier transformation

Change in basis from one representation to another representation |p〉 is defined as,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
The expansion of Ψ(x) in terms of |p〉 can be written as,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
where a (p) can be found as,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
In 3D:
Postulates of Quantum Mechanics IIT JAM Notes | EduRev where a(p) being expansion coefficient of |p〉.
If any function Ψ(x) can be expressed as a linear combination of state function ϕn 
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
which is popularly derived from fourier trick. 

Parity operator: The parity operator  defined by its action on the basis.
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

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 Postulates of Quantum Mechanics IIT JAM Notes | EduRev It is given that  〈ϕi | ϕj〉, δij , then
(a) check whether  is normalized or not
(b) write down normalized wavefunction.
(c) it is given H |ϕn〉 = (n + 1)ℏω| ϕn〉 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- 

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
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.
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
So,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(c) It is given that  
H|ϕn〉 = (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
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
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
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(e) The error in measurement in H is given as
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Example 15: The wave function of a particle is given by Postulates of Quantum Mechanics IIT JAM Notes | EduRev 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〉

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
For normalized |ψ〉,
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(b) Measurement are E0 ,E1
(c) Probability of getting 

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Example 16: (a) Plot ΨI (x) = A1e-|x| ; -∞ < x < ∞
(b) Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(c) Discuss why ψI is not the solution of Schrödinger wave function rather ψII is solution of Schrödinger wave function.

(a) ψI (x) = A1e+x ; x < 0
ψII (x) = A1e-x ; x > 0
The plot is given by
Postulates of Quantum Mechanics IIT JAM Notes | EduRev(b)
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

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

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

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
It is given that, Hamiltonian is defined as H |ϕn〉 = n20n
(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

Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(b) Evolution of shape of the wave packet 

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(c)  Δ H = (〈H2〉 - 〈H〉2)1/2 
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(d)
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Example 18: Consider a one-dimensional particle which is confined within the region 0 ≤ x ≤ a and whose wave function is Postulates of Quantum Mechanics IIT JAM Notes | EduRev Find the potential V(x).

From the fifth postulate:
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

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

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
λ2 = λ3 = 2a0 i.e., λ = 2a0 is doubly degenerate.
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

Example 20: A free particle which is initially localized in the range -a < x < a is released at time t = 0.
Postulates of Quantum Mechanics IIT JAM Notes | EduRev
(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.

Postulates of Quantum Mechanics IIT JAM Notes | EduRev
Postulates of Quantum Mechanics IIT JAM Notes | EduRev

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