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**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〉 ,〈x ^{2}**

(a)(c)

**Q.2. If |ϕ _{1}〉 and |ϕ_{2}〉 be two orthonormal state vectors such that A = |ϕ_{1}〉 〈ϕ_{2}| + |ϕ_{2}〉 〈ϕ_{1}| then **

(a) Prove that A is Hermitian

(b) Find the value of A^{2}.

For A to be a projection operator, A should be Hermitian and A

^{2}should be equal to A. The Hermitian adjoint of |ϕ_{1}〉 〈ϕ_{2}| is |ϕ_{2}〉 〈ϕ_{1}| and that of |ϕ_{2}〉 〈ϕ_{1}| is |ϕ_{1}〉 〈ϕ_{2}|. So

Hence A is Hermitian.

Now,Since |ϕ

_{1}〉 and 〈ϕ_{2}| 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〉, 〈θ ^{2}〉 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 _{x}ϕ = 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**

P

_{x}ϕ = 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

(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 ψ_{1}(x) and ψ_{2}(x) be two arbitraryfunctions.

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), 〈ψ

_{1}|P_{x}ψ_{2}〉 = 〈P_{x}ψ_{1}|ψ_{2}〉

Hence is Hermitian. Similar calculation with A = d/dx will give,

Hence, 〈ψ_{1}|Aψ_{2}〉 ≠ 〈Aψ_{1}|ψ_{2}〉 and so, A = d/dx is not Hermitian.

**Q.7.**

**(a) Find the value of A such that |ϕ _{n}〉 is normalized **

**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 ϕ _{1}(θ, ϕ) = A find the value of A such that ϕ_{1} (θ, ϕ) is normalized.**

(b) Prove that is orthogonal to ϕ_{1}

The wave function ϕ

_{1}(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 |ϕ**

(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

then 〈ϕ_{1}| = A^{*}(1 0 0), 〈ϕ_{2}| = B^{*}(0 -i -i), 〈ϕ_{2}| = C^{*}(0 -i -i)c

_{1}= 0 c_{2}+ c_{3 }= 0 and c_{2}- c_{3}= 0 ⇒ c_{1}= 0, c_{2}= 0, c_{3}= 0

So|ϕ_{1}〉, |ϕ_{2}〉 and |ϕ_{3}〉 are linearly independent

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

As, H = p

^{2}/ 2m + V(x)

**Q.13. ****Let |θ _{1}〉 and |θ_{2}〉 be two eigenfunction of Hamiltonian operator with eigen value E_{0} and 4E_{0} respectively. The wave function of the particle at time t = 0 is which is also eigen function of operator A with eigen value a_{0}. 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 a_{0}.

(c) If A is measured on state ψ at t = t what is probability to get eigen value a_{0}.

(d) Find error in measurement of energy ie ΔE

(e) Find ΔE.Δt

H|θ

_{1}〉 = E_{0}|θ_{1}〉 and H|θ_{2}〉 = 4E_{0}|θ_{2}〉

H is measured on state ψ at t = 0 what is measurement is eigen value E_{0}and 4E_{0}

(b)A|ψ〉 = a_{0}|ψ〉

Probability of getting a_{0}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 ψ _{0} (x) and y_{1} (x) of a one dimensional system. They satisfy ψ_{0} (x) = ψ^{*}_{0} (x) and ψ_{1}(x) = **

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) = (mω ^{2}x^{2}**)

(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 = mv

^{2}/ 2 = p^{2}/ (2m)

or,de broglie wavelength is

For an electron, h

^{2}/2m = 1.5eV nm^{2}.

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 E_{1}= -5 , E_{2}= 3 , E_{3}= 5 ; the respective (orthonormal) eigenvectors of these values areThe probabilities of finding the values E

_{1}= -5 , E_{2}= 3 , E_{3}= 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 haveThe 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

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