Practice Questions on One dimensional Potential with Answers.

**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} ψ_{1}(x) + i√(^{2}/_{3}) ψ_{2}(x) and ψ_{2}(x) are the ground and the first excited state, respectively. Identify the correct statement**

(a) <x> = ^{L}/_{2} ; <E> = ℏ^{2}/_{2m} 3π^{2}/L^{2}

(b) <x> = ^{2L}/_{3} ; <E> = ℏ^{2}/_{2m} π^{2}/L^{2}

(c) <x> = ^{L}/_{2} ; <E> = ℏ^{2}/_{2m} 8π^{2}/L^{2}

(d) <x> = ^{2L}/_{3} ; <E> = ℏ^{2}/_{2m} 4π^{2}/3L^{2}

**Q3. A quantum mechanical particle in a harmonic oscillator potential has the initial wavefunction ψ _{0}(x) + ψ_{1}(x), where ψ_{0} and ψ_{1} 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) [ψ_{0}(x) – ψ_{1}(x)]^{2}

(b) ψ_{0}(x) + ψ_{1}(x)

(c) [ψ_{0}(x)]^{2} – [ψ_{1}(x)]^{2}

(d) [ψ_{0}(x)]2 + [ψ_{1}(x)]^{2}

**Q4. Consider a square well of depth -V _{o}, and width a with V_{o} , a fixed. Let V_{o} → ∞ 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 E _{0}, E_{1} or E_{2} 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|H^{2}|ψ> = 11ℏ^{2}ω^{2}/_{4}, then the probability that the energy measurement yields E_{0} 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{-(b ^{2}x^{2})/_{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 = ℏ^{2}b^{4}/_{2M}

(b) V = ℏ^{2}b^{4}x^{2}/_{2M}

(c) V = ℏ^{2}b^{6}x^{4}/_{2M}

(d) E = ℏ^{2}b^{2}(1-b^{2}x^{2})

**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 U _{o}(U_{o} < E)**

(a) is always 1

(b) is never 1

(c) varies from 0 to 1 periodically with increasing ^{E}/U_{o}

(d) becomes 1 periodically with increasing ^{E}/U_{o} 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 × 10^{13} Hz

(b) 2.8 × 10^{13} Hz

(c) 3.9 × 10^{14} Hz

(d) 4.2 × 10^{14} Hz

**Q14. If an electron (m = 9.1 × 10 ^{31} 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)

*Answer Key*

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

02. | (a) | 07. | (b) | 12. | (d) |

03. | (a) | 08. | (c) | 13. | (c) |

04. | (b) | 09. | (a) | 14. | (c) |

05. | (b) | 10. | (c) | 15. | (b) |