Tools & Postulates of Quantum Mechanics: Assignment | Modern Physics PDF Download

Q.1. Consider the Gaussian probability distribution 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) 





(c) 

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

Hence A is Hermitian.
Now,

Since |ϕ₁〉 and 〈ϕ₂| are orthonormal,

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

Q.4. (a) Find the Eigen State of momentum operator  If eigen value is λ by relation P_xϕ = 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

P_xϕ = λ/ϕ where λ/ℏ = k.
Case 1:  If l is positive

Case 2: If l is negative 

(b) Expand the function ψ(x) = A sin kx sin 2kx as a linear combination of eigenfunctions of the momentum operator Px.

 

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

Q.6. Prove that operator  is Hermitian but D_x =  d/dx is not Hermitian.

The given operator  is the same as P_x . Let ψ₁ (x) and ψ₂ (x) be two arbitrary 

functions.

Integrating by parts, this is equal to

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,

From (i) and (ii), 〈ψ₁|P_xψ₂〉 = 〈P_xψ₁|ψ₂〉
Hence  is Hermitian. Similar calculation with A = d/dx will give,

Hence, 〈ψ₁|Aψ₂〉 ≠ 〈Aψ₁|ψ₂〉 and so, A = d/dx is not Hermitian.

Q.7.

(a) Find the value of A such that |ϕ_n〉 is normalized 
(b) 〈ϕ_m|ϕ_n〉 = δ_m,n 
(c) If H operator is defined as  then prove that the matrix element

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)* B^†A^† + (1 + i)* A^†B^† 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  is orthogonal to ϕ₁ 

The wave function ϕ₁ (θ, ϕ) = A is defined in spherical symmetry variable is solid angle

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

Q.11.

(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.

(e)  then find value of C such that |ψ〉 is normalized
(f) If operator A is defined as  where n = 1, 2, 3... then find value of A|ψ〉
(g) If operator A is defined as  where n = 1, 2, 3... then find value of
(h) If operator A is defined as  where n = 1, 2, 3... then find value of 〈ψ|A|ψ〉

then 〈ϕ₁| = A^* (1  0  0), 〈ϕ₂| = B^* (0  -i  -i), 〈ϕ₂| = C^* (0  -i  -i)

c₁ = 0 c₂ + c₃ = 0 and c₂ - c₃ = 0 ⇒ c₁ = 0, c₂ = 0, c₃ = 0
So |ϕ₁〉, |ϕ₂〉 and |ϕ₃〉 are linearly independent

Q.12. If Hamiltonian of system is  then Find commutation [H, x] and [[H, x], x]

As, H = p² / 2m + V(x) 

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 is 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

 H|θ₁〉 = E₀|θ₁〉 and H|θ₂〉 = 4E₀ |θ₂〉
H is measured on state ψ at t = 0 what is measurement is eigen value E₀ and 4E₀

(b) A|ψ〉 = a₀|ψ〉
Probability of getting a₀ on state ψ at t = 0

Q.14. Consider the operator  acting on smooth function of x. find the commutation [α , cos x]

[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) =  where α is a real constant. The expectation value of the momentum operator in the state ψ₁ is

Integrate by parts

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 Δp_x, the value of the momentum itself should have at least a value comparable to Δp_x. You cannot have an uncertainty of 5 units if the value never exceeds 2 units. So we assume that p ≈ Δp_x. Similarly x ≈ Δ_x. The expression for energy is

From uncertainty principle,  As we are looking for the lowest energy, let us write Δp_x = ℏ/2Δx , 

For E to be minimum, dE/d(Δx) should be zero. This gives,

(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,

de broglie wavelength is

For an electron, h²/2m = 1.5eV nm².

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,

So, α = 12.25.

Q.17. Consider a system whose initial state  and Hamiltonian is defined as
(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

The probabilities of finding the values E₁ = -5 , E₂ = 3 , E₃ = 5 are given by

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

Hence,

(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, 〈ϕ_n)|ϕ_m〉 = δ_nm and since  we have

The second method uses matrix algebra:

The energy at time t is 

As expected E (t) = E (0) since

Q.18. A Particle of mass m is subjected to the potential energy  At a particular time it has wave function  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 If 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

If Hψ(x) has to have the same functional form as ψ(x), one should not have the term. So, 

or