Quantum Mechanics | Week 2

Quantum Mechanics Week 2 Answers

Course Link: Click Here

Q1. A particle of mass m is confined to move in a potential of the form
V(x)={0, 0≤x≤a
∞, otherwise
The wavefunction of the particle at time t = 0 is given by
(x,0)=Asin(5πх/a)cos(2πx/a)
Normalized (x, 0) can be written as
a) 2/√a(sin(5πх/a)+sin(2πx/a))
b) 2/√a(sin(7πх/a)+sin(7πx/a))
c) 1/√a(sin(5πх/a)+sin(2πx/a))
d) 1/√a(sin(7πх/a)+sin(7πx/a))

Q2.

Answer: True

Q3. A particle of mass m is trapped in the potential
V={0, -a≤x≤a
{∞, otherwise
The probability of finding the trapped particle between x = 0 and x = a/n, when it is in the nth state is 1/n.
a) True
b) False

Answer: b) False

Q4. An electron is confined in the ground state of a one-dimensional harmonic oscillator such that the uncertainty in position Δ(x) = 10^-10m. If the expectation values of kinetic and potential energy are same, i.e. (T) = (V), then the energy required to excite the electron to its first excited state is ______eV. (Write upto two decimal places.)
(Hint : Use h =1.05 x 10^-34 Js, m = 9.1 x 10-³¹ kg)

Answer: 3.97

Q5.

Answer: b) <E> =2hw

Q6.

Answer: a)

Q7. For a potential barrier of height Up and finite width b, the position space wavefunction (x) will be oscillatory for any x if and only if energy E of the particle is
a) greater than Uo
b) zero
c) less than Uo
d) negative

Q8. A 1eV electron got trapped inside the surface of a metal. If the potential barrier is 4.0eV and the width of the barrier is 2Å, then the transmission co-efficient T is
a) 1
b) 0.25
c) 0.85
d) 0.085
(Hint : Use h = 1.05 x 10^-34 Js, m= 9.1 x 10^-31 kg)

Answer: d) 0.085

Q9. Consider a potential step of form,
V(x) ={0, x < 0
Vo, x ≥0
A beam of particles with energy E approach this potential step from left. Then the statement that best describes the reflection coefficient (R) is
a) R is non-zero when E > Vo, R = 1 when E < Vo
b) R = 0 when E > Vo, R = 1 when E < Vo.
c) R = 1 when E > Vo, R = 0 when E < Vo.
d) R is non-zero when E > Vo, R = 0 when E < Vo.

Q10. For an infinite square well potential, if the width of the well decreases, what happens to the zero
point energy (Eo) and momentum uncertainty (Δp)?
a) both Eo and Δp decrease
b) both Eo and Δp increase
c) Eo decreases and Δp increases
d) Eo increases and Δp decreases