Assignment PhysicsII 02
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Department of Physics Indian Institute of Technology Kharagpur Kharagpur-721302, West Bengal, India Subject No. PH20003(PHYSICS II) Assignment Due date : 14 th August 2015 Total Marks: 100 Thursday 6 th August, 2015 Assignment # 2 1. Answer the following questions (a) A hydrogen atom is 0.53 ˚ A in radius. Estimate the minimum energy an electron can have in this atom. [use Heisenberg uncertainty principle] (b) Estimate the lowest energy of a neutron confined in one-dimensional infinitely high potential box of length 10 -14 m. 2. Consider the wave function Ψ(x)= C √ xe -(x/2a) , where ‘C ’ is normalization constant, describes a particle in the range (0, ∞). Find the expectation value for x and x 2 , and most probable value of x where probability density is maximum. 3. A particle is located in a 1-D square potential well of width ‘L’ with absolutely impenetrable walls (0 <x<L). Find the probability of staying within a region L/3 ≤ x ≤ 2L/3, for the particle (a) in the ground state, (b) in the first excited state. 4. A particle of mass m moves in a potential well of width 2a. Its potential energy is infinite for x< -a & x> +a. Inside the region, -a<x<a, its potential energy is given by U (x)= -¯ h 2 a 2 ma 2 (a 2 -x 2 ) . In addition, the particle is in a stationary state that is described by the wave- function Ψ(x)= A(1 - x 2 /a 2 ) for -a<x< +a and by Ψ(x) = 0 elsewhere. (a) Show that A = (15/16a) 1/2 (b) Determine the energy of the particle in terms of ¯ h, m and a. (c) Find the probability that the particle is located between x = -a/3& x = a/3. 5. A photon with wavelength λ is absorbed by an electron confined to a box having infinite potential wall. As a result, the electron moves from state n = 1 to n = 4. (a) Find the width of the box. (b) What is the wavelength of the photon emitted in the transition of that electron from the state n = 4 to the state n = 2? 6. In a 1-D potential barrier (potential height V 0 , and length L) problem, A particle of mass m having energy E>V 0 coming from -∞. Using boundary condition solve for R & T which will be a function of length of the potential barrier. R = (k 2 1 -k 2 2 ) 2 sin 2 k 2 L 4k 2 1 k 2 2 +(k 2 1 -k 2 2 ) 2 sin 2 k 2 L where k 1 = q 2mE ¯ h 2 ; and k 2 = q 2m(E-V 0 ) ¯ h 2 T = 4k 2 1 k 2 2 4k 2 1 k 2 2 +(k 2 1 -k 2 2 ) 2 sin 2 k 2 L
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Assignment PhysicsII 02
Transcript of Assignment PhysicsII 02
Department of PhysicsIndian Institute of Technology Kharagpur
Kharagpur-721302, West Bengal, India
Subject No. PH20003(PHYSICS II)
Assignment Due date : 14th
August 2015 Total Marks: 100
Thursday 6th August, 2015
Assignment # 2
§1. Answer the following questions(a) A hydrogen atom is 0.53 A in radius. Estimate the minimum energy an electron can have in thisatom. [use Heisenberg uncertainty principle](b) Estimate the lowest energy of a neutron confined in one-dimensional infinitely high potential boxof length 10−14 m.
§2. Consider the wave function Ψ(x) = C√x e−(x/2a) , where ‘C’ is normalization constant, describes a
particle in the range (0, ∞). Find the expectation value for x and x2, and most probable value of xwhere probability density is maximum.
§3. A particle is located in a 1-D square potential well of width ‘L’ with absolutely impenetrable walls(0 < x < L).Find the probability of staying within a region L/3 ≤ x ≤ 2L/3, for the particle (a) in the groundstate, (b) in the first excited state.
§4. A particle of mass m moves in a potential well of width 2a. Its potential energy is infinite forx < −a & x > +a. Inside the region, −a < x < a, its potential energy is given byU(x) = −h2a2
m a2(a2−x2). In addition, the particle is in a stationary state that is described by the wave-
function Ψ(x) = A(1− x2/a2) for −a < x < +a and by Ψ(x) = 0 elsewhere.(a) Show that A = (15/16a)1/2
(b) Determine the energy of the particle in terms of h, m and a.(c) Find the probability that the particle is located between x = −a/3 & x = a/3.
§5. A photon with wavelength λ is absorbed by an electron confined to a box having infinite potentialwall. As a result, the electron moves from state n = 1 to n = 4.(a) Find the width of the box.(b) What is the wavelength of the photon emitted in the transition of that electron from the staten = 4 to the state n = 2?
§6. In a 1-D potential barrier (potential height V0, and length L) problem, A particle of mass m havingenergy E > V0 coming from −∞. Using boundary condition solve for R & T which will be a functionof length of the potential barrier.
R =(k21−k22)2 sin2 k2L
4k21k22+(k21−k22)2 sin2 k2L
where k1 =√
2mEh2 ; and k2 =
√2m(E−V0)
h2
T =4k21k
22
4k21k22+(k21−k22)2 sin2 k2L
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