Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Modern Physics

IIT JAM : Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

The document Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev is a part of the IIT JAM Course Modern Physics.
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Q.1. Consider the Gaussian probability distributionTools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev where -∞ < x < ∞ where A,a and λ are positive real constants. 
(a) Determine A such that f(x) is probability density
(b) Find 〈x〉 ,〈x2 and σ = Δx 
(c) Sketch the graph of f(x)

(a) 
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(c) 
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.2. If |ϕ1〉 and |ϕ2〉 be two orthonormal state vectors such that A = |ϕ1〉 〈ϕ2| + |ϕ2〉 〈ϕ1|  then
(a) Prove that A is Hermitian
(b) Find the value of A2.

For A to be a projection operator, A should be Hermitian and A2 should be equal to A. The Hermitian adjoint of |ϕ1〉 〈ϕ2| is |ϕ2〉 〈ϕ1| and that of |ϕ2〉 〈ϕ1| is |ϕ1〉 〈ϕ2|. So
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Hence A is Hermitian.
Now,

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Since |ϕ1〉 and 〈ϕ2| are orthonormal,
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.3. The needle on a broken car speedometer is free to swing, and bounces perfectly of the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π 
(a) What is the probability density, f (θ)? 
(b) Compute 〈θ〉, 〈θ2〉 and σ = Δθ, for this distribution

f(θ) = A ,0 < θ < π
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.4. (a) Find the Eigen State of momentum operator Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev If eigen value is λ by relation Pxϕ = λ/ϕ where λ/ℏ = k.
(b) Expand the wave function ψ(x) = A sin kx sin 2kx in basis of Eigen functions of momentum operator P
x

Pxϕ = λ/ϕ where λ/ℏ = k.
Case 1:  If l is positive
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Case 2: If l is negative 

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(b) Expand the function ψ(x) = A sin kx sin 2kx as a linear combination of eigenfunctions of the momentum operator Px.
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev 


Q.5. Consider the function ψ(x, t) = Ae-λ|x|e-ωt  -∞ < x < ∞, where λ and ω are are constant. Find the value of A such that ψ (x, t) is normalized.

(a) ψ(x, t) = Ae-λ|x|e-ωt

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.6. Prove that operator Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev is Hermitian but Dx =  d/dx is not Hermitian.

The given operator Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev is the same as Px . Let ψ1 (x) and ψ2 (x) be two arbitrary 

functions.
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Integrating by parts, this is equal to
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
For the wave function to be square integrable, it must go to zero as x goes to -∞ or +∞.

Thus, the first term in the square bracket is zero. So,
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

From (i) and (ii), 〈ψ1|Pxψ2〉 = 〈Pxψ12
Hence Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev is Hermitian. Similar calculation with A = d/dx will give,

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Hence, 〈ψ1|Aψ2〉 ≠ 〈Aψ12〉 and so, A = d/dx is not Hermitian.


Q.7.

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(a) Find the value of A such that |ϕn〉 is normalized 
(b) 〈ϕmn〉 = δm,n 
(c) If H operator is defined as Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev then prove that the matrix element

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.8. Show that operator O = (1 + i) AB + (1 - i) BA is Hermitian if A and B is Hermitian.

[(1 + i) AB + (1 - i) BA] = [(1 + i)AB] + [(1 - i)BA]
= (1+ i)* (AB) + (1 - i)* (BA)
= (1 - i)* BA + (1 + i)* AB if A and B is hermitian.
= (1 - i) BA + (1 + i)AB
(1 + i) AB + (1 - i)BA
Thus the given operator is Hermitian


Q.9. (a) If ϕ1(θ, ϕ) = A find the value of A such that ϕ1 (θ, ϕ) is normalized.
(b) Prove that Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev is orthogonal to ϕ1 

The wave function ϕ1 (θ, ϕ) = A is defined in spherical symmetry variable is solid angle
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.10. Using Bohr-Somerfield theory, find the energy for a particle of mass m is confined in potential V(x) = k|x|.

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.11.
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(a) Find normalization constant A, B, C for ket |ϕ1〉 |ϕ2〉 |ϕ3

(b)  Prove that |ϕ1〉, |ϕ2〉 and |ϕ3〉 are orthogonal
(c)  Check whether |ϕ1〉, |ϕ2〉 and |ϕ3〉are linearly independent or not.
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(e) Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev then find value of C such that |ψ〉 is normalized
(f) If operator A is defined as Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev where n = 1, 2, 3... then find value of A|ψ〉
(g) If operator A is defined as Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev where n = 1, 2, 3... then find value of Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

(h) If operator A is defined as Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev where n = 1, 2, 3... then find value of 〈ψ|A|ψ〉

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
then 〈ϕ1| = A* (1  0  0), 〈ϕ2| = B* (0  -i  -i), 〈ϕ2| = C* (0  -i  -i)
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

c1 = 0 c2 + c= 0 and c2 - c3 = 0 ⇒ c1 = 0, c2 = 0, c3 = 0
So 1〉, |ϕ2〉 and |ϕ3〉 are linearly independent
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.12. If Hamiltonian of system is Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev then Find commutation [H, x] and [[H, x], x]

As, H = p2 / 2m + V(x) 

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.13. Let |θ1〉 and |θ2〉 be two eigenfunction of Hamiltonian operator with eigen value E0 and 4E0 respectively. The wave function of the particle at time t = 0 isTools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev which is also eigen function of operator A with eigen value a0. The operator A is associated with observable a.
(a) If H is measured on state ψ at t = 0 what is measurement with what probability.
(b) If A is measured on state ψ at t = 0 what is probability to get eigen value a0.
(c) If A is measured on state ψ at t = t what is probability to get eigen value a0.
(d) Find error in measurement of energy ie ΔE
(e) Find ΔE.Δt

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev H|θ1〉 = E01〉 and H|θ2〉 = 4E02
H is measured on state ψ at t = 0 what is measurement is eigen value E0 and 4E0

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(b) A|ψ〉 = a0|ψ〉
Probability of getting a0 on state ψ at t = 0
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.14. Consider the operator Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev acting on smooth function of x. find the commutation [α , cos x]

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

[a, cos x]ψ(x) = -sin xψ [a, cos x] = -sin x


Q.15. Consider the two lowest normalized energy eigenfunctions ψ0 (x) and y1 (x) of a one dimensional system. They satisfy  ψ0 (x) = ψ*0 (x) and ψ1(x) = Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev where α is a real constant. The expectation value of the momentum operator in the state ψ1 is

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Integrate by parts
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.16. (a) using Heisenberg uncertainty principle estimate the minimum possible energy of linear Harmonic oscillator of mass m. The potential for such a particle is v(x) = (mω2x2)/2/
(b) when an electron is accelerated through a potential difference V, its de Broglie Wave length is α/√V for non relativistic speed. If λ and V represent numerical values in Angstrom and Volt, find the numerical value of α.

(a) In order to have an uncertainty of Δpx, the value of the momentum itself should have at least a value comparable to Δpx. You cannot have an uncertainty of 5 units if the value never exceeds 2 units. So we assume that p ≈ Δpx. Similarly x ≈ Δx. The expression for energy is
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
From uncertainty principle, Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev As we are looking for the lowest energy, let us write Δpx = ℏ/2Δx , 

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

For E to be minimum, dE/d(Δx) should be zero. This gives,
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(b) As an electron is accelerated through a potential difference V, its potential energy is decreased by eV. The kinetic energy gained is equal to this value. So,

eV = mv2 / 2 = p2 / (2m)
or,

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

de broglie wavelength is

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

For an electron, h2/2m = 1.5eV nm2.
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
If  V is put in volts, the energy eV is the same as V electronvolts. Cancelling the unit eV from the numerator and the denominator,

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

So, α = 12.25.


Q.17. Consider a system whose initial state Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev and Hamiltonian is defined as Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(a) If a measurement of energy is carried out what values would be obtained with what probabilities?
(b) Find the state of system at later time t,
(c) Find the average value of energy at time t = 0.
(d) Find the average value of energy at time t = t.

(a) A measurement of the energy yields the values E1 = -5 , E2 = 3 , E3 = 5 ; the respective (orthonormal) eigenvectors of these values are

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

The probabilities of finding the values E1 = -5 , E2 = 3 , E3 = 5 are given by
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

(b) To find |ψ(t)〉 we need to expand |ψ(0)〉 in terms of the eigenvectors

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
Hence,
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
(c) We can calculate the energy at time t = 0 in three quite different ways. The first method uses the bra-ket notation. Since 〈ψ(0)|ψ(0)〉 = 1, 〈ϕn)|ϕm〉 = δnm and since Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev we have
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

The second method uses matrix algebra:
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
The energy at time t is 

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
As expected E (t) = E (0) since Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev


Q.18. A Particle of mass m is subjected to the potential energy Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev At a particular time it has wave function Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev and identified as energy eigen function with definite total mechanical energy .find the value of a.

The operator corresponding to the total mechanical energy is Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRevIf particle has definite value of the total mechanical energy, its wave function should be an eigenfunction of H, that is,

Hψ(x) = λψ where λ is independent of x
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev
If Hψ(x) has to have the same functional form as ψ(x), one should not have the Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRevterm. So, 

Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

or
Tools & Postulates of Quantum Mechanics: Assignment IIT JAM Notes | EduRev

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