PHYS241 Modern Physics –I (Take Home Exam –IAT-2)Submission time: Monday 4th April, 2016 (10:30 a.m.)

Answer any Six ( 5 Marks x 6 =30 Marks)1. Consider a rectangular potential well given by a potential

V (x) = 5V0 for x ≤ 0; 0 for 0 < x < a; 2V0 for x ≥ a;

Let the energy of the first excited state be E2 = V0. (a) Make a qualitative plot of the wave-function ψ2(x). Indicate the relative rates of decrease of |ψ2(x)| with x in region I and III. Does the node of the wave-function occur to the left or right of the center of the well? (b) Comparing your qualitative plot of ψ2(x) with the wave-function for second energy level of infinite square well potential, we can infer that the mass of the particle m is less than a certain quantity involving a, V0 and ħ. What is this quantity?

2. Consider a particle bound in a region with one side rigid wall, given by the potentialV (x) = ∞ for x ≤ 0;

0 for 0 < x < a; V0 for x ≥ a;

(a) For E < V0, solve Schrodinger equation for region inside and outside the well. (b) Apply boundary conditions at x = 0 and x = a to obtain an equation that defines allowed values of energy. (c) For V0 very large, the permitted energies approach those for particle in a box. (d) Is there a minimum value of V0 below which no bound state exists?

3. Consider a particle in the ground state of a one dimensional box [0, a]. (a) What is the probability that the particle is in the interval [0, a/4]? (b) What is the probability that the particle is in the interval [a/4, a/2] (without evaluating the integral).

4. Using H= ∮ pdx=nħ, find the allowed energies of a particle confined to a one dimensional box [0, a].

5. Given the one dimensional potential V (x) = V0 xq using H= ∮ pdx=nħfind the dependence of energy on the quantum number n.

6. Show that for a particle of mass m which moves in a one dimensional infinite potential well of length a, the uncertainties product ∆xn ∆pn is given by ∆xn ∆pn ≅ n π ħ√12

7. Consider a particle in an infinite square well whose wave function is given by (x) = A(a2 –x2) 0<x<a

0 elsewhere where A is real constant. Find A so that (x) is normalized8. A particle is initially in its ground state in a one-dimensional harmonic oscillator

potential V(x) =1/2 kx2. If the spring constant is suddenly doubled, calculate the probability of finding the particle in the ground state of the new potential.