Questions on One dimensional Potential with Answers

Practice Questions on One dimensional Potential with Answers.

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Questions on One dimensional Potential with Answers

Q1. A free particle of energy E collides with a one -dimensional square potential barrier of height V and width W. Which one of the following statement(s) is (are) correct?

(a) for E  > V, the transmission coefficient for the particle across the barrier will always be unity
(b) for E < V, the transmission coefficient changes more rapidly width W than with V.
(c) for E < V, if V is doubled, the transmission coefficient will also be doubled.
(d) sum of the reflection and transmission coefficients is always one.

Q2. A particle of mass m in a one dimensional potential V(x) = 0 when 0 < x < L otherwise V(x) = ∞ . At some instant its wave function is given ψ(x) = 1/√3 ψ1(x) + i√(2/3) ψ2(x) and ψ2(x) are the ground and the first excited state, respectively. Identify the correct statement

(a) <x> = L/2 ; <E> = ℏ2/2m2/L2
(b) <x> = 2L/3 ; <E> = ℏ2/2m π2/L2
(c) <x> = L/2 ; <E> = ℏ2/2m2/L2
(d) <x> = 2L/3 ; <E> = ℏ2/2m2/3L2

Q3. A quantum mechanical particle in a harmonic oscillator potential has the initial wavefunction ψ0(x) + ψ1(x), where ψ0 and ψ1 are the real wavefunctions in the ground and first excited states of the harmonic oscillator Hamiltonian. For convenience we take m = ℏ = ω = 1 for the oscillator. What is the probability density of finding the particle at x at time t = π ?

(a) [ψ0(x) – ψ1(x)]2
(b) ψ0(x) + ψ1(x)
(c) [ψ0(x)]2 – [ψ1(x)]2
(d) [ψ0(x)]2 + [ψ1(x)]2

Q4. Consider a square well of depth -Vo, and width a with Vo , a fixed. Let Vo → ∞ and a → 0. This potential well has

(a) No bound states
(b) 1 bound state
(c) 2 bound states
(d) Infinitely many bound states

Q5. A one dimensional harmonic oscillator (mass m and frequency ω)is in a state ω such that the only possible outcomes of an energy measurement are E0, E1 or E2 where En is the energy is the energy of the n-th excited state. If H is the Hamiltonian of the oscillator, <ψ|H|ψ> = 3ℏω/2 and <ψ|H2|ψ> = 11ℏ2ω2/4, then the probability that the energy measurement yields E0 is

(a) 1/2
(b) 1/4
(c) 1/8
(d) 0

Q6. A free particle with kinetic energy E and de Broglie wavelength λ enters a region in which it has potential energy V. What is the particle’s new de Broglie wavelength ?

(a) λ(1-V/E)
(b) λ(1-V/E)-1
(c) λ(1+V/E)1/2
(d) λ(1-V/E)1/2

Q7. The wave function ψ(r) = A exp{-(b2x2)/2} where A and b are real constants, is a normalized eigenfunction of the Schrödinger equation for a particle of mass M and energy E in a one-dimensional potential V(x) such that V(x) = 0 at x =0. Which of the following is correct?

(a) V = ℏ2b4/2M
(b) V = ℏ2b4x2/2M
(c) V = ℏ2b6x4/2M
(d) E = ℏ2b2(1-b2x2)

Q8. In the figure blow shows one of the possible energy eigenfunctions ψ(x) for a particle bouncing freely back and forth along the x-axis between impenetrable walls located at x = -a and x = +a. The potential energy equals zero for |x| < a. If the energy of the particle is 2 electron volts when it is in the quantum state associated with this eigenfunction, what is its energy when it is in the quantum state of lowest possible energy?

(a) 0 eV
(b) 1/√2 eV
(c) 1/2 eV
(d) 1 eV

Q9. A particle is moving in a one-dimensionl potential box of infinite height. What is the probability of finding the particle in a small interval Δx at the center of the box when it is in the energy state, next to the least energy state ?

(a) 0
(b) 0.15
(c) 0.35
(d) 0.5

Q10. The wave function of a particle confined in a box of length L

ψ(x) = √(2/L) Sin (πx/L)

The probability of finding the particle in the domain 0 < x < L/2 is

(a) 1
(b) 1/4
(c) 1/2
(d) 0

Q11. The ground state energy of a particle in an one-dimensional quantum well is 4.4 eV. If the width of the well is doubled, then the new ground state energy is

(a) 1.1 eV
(b) 2.2 eV
(c) 8.8 eV
(d) 17.6 eV

Q12. Transmission probability of a particle with energy E moving through a one-dimensional rectangular potential barrier of height Uo(Uo < E)

(a) is always 1
(b) is never 1
(c) varies from 0 to 1 periodically with increasing E/Uo
(d) becomes 1 periodically with increasing E/Uo but never becomes 0

Q13. The energy of a linear harmonic oscillator in third excited state is 4.l eV. The frequency of vibration is ( μ = 6.62 × 10-34 J-sec)

(a) 3.3 × 1013 Hz
(b) 2.8 × 1013 Hz
(c) 3.9 × 1014 Hz
(d) 4.2 × 1014 Hz

Q14. If an electron (m = 9.1 × 1031 kg) is confined by an electrical fore to move between two rigid walls separated by 1.0 × 10-9 meter. The quantized energy value for the second lowest energy state is

(a) 0.38 eV
(b) 0.76 eV
(c) 1.5 eV
(d) 3.0 eV

Q15. Let |n> represent the normalized nth energy eigenstate of the one-dimensional harmonic oscillator

H|n> = ℏω (n+1/2) |n>

If |ψ> is a normalized ensemble state that can be expanded as a linear combination

|ψ> = 1/√14 |1>- 2/√14 |2> + 3/√14 |3>

of the eigenstates, what is the expectation value of the energy operator in this ensemble state ?

(a) 102/14 ℏω
(b) 43/14 ℏω
(c) 23/14 ℏω
(d) 17/14 ℏω

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Answer Key (if you find any answer wrong, feel free to Correct us)

01.(b), (d)06.(d)11.(a)

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