Practice Questions on Multi-dimensional Potentials with Answers.

**Questions on Multi dimensional Potentials with Answers**

**Q1. Consider a quantum particle of mass ‘m’ in a three-dimensional isotropic simple harmonic potential V(x,y,z) = ^{1}/_{2} mω^{2}(x^{2} + y^{2} + z^{2}). It is known that the particle is in an energy eigenstate with eigenvalue E = ^{7ℏω}/_{2} . Which of the following cannot be the wave function of the particle? (In the following α = √(^{mω}/_{ℏ}) and H_{n}(ξ) is the nth Hermite polynomial)**

(a) H_{2}(αx)exp(-α(y^{2} + z^{2}))

(b) H_{2}(αx)exp(-α(x^{2} + y^{2} + z^{2}))

(c) H_{1}(αy)H_{1}(αz)exp(-α(x^{2} + y^{2} + z^{2}))

(d) H_{1}(αx)H_{1}(αz)exp(-α(x^{2} + y^{2} + z^{2}))

**Q2. A particle of mass m is moving in a three-dimensional potential**

**V(x,y,z) = ^{1}/_{2} mω^{2}(x^{2} + y^{2} + z^{2}).**

**The energy of the particle in the ground state (lowest energy quantum state) is**

(a) ^{√7}/_{2} ℏω

(b) ^{3}/_{2} ℏω

(c) ^{7}/_{2} ℏω

(d) ^{(3 + √2)}/_{2} ℏω

**Q3. What is the energy of the second excited state of a particle of mass m moving freely inside a rectangular parallelopiped of sides L, 2L and 3L?**

(a) 49h^{2}/_{288mL2}

(b) 9h^{2}/_{32mL2}

(c) 61h^{2}/_{288mL2}

(d) 11h^{2}/_{36mL2}

**Q4. In the Larger Electron Positron (LEP) collider electrons were accelerated to an energy of 100 GeV. The ratio of the de-Broglie wavelength of these electrons to the de-Broglie wavelength of an electron in the ground state of a hydrogen atom is of the order of**

(a) 10^{-5}

(b) 10^{-7}

(c) 10^{-9}

(d) 10^{-11}

**Q5. The ground state energy of a particle of mass m in a three dimensional cubical box of side L is not zero but 3h ^{2}/_{8mL2} . This is because**

(a) The ground state has no nodes in the interior of the box.

(b) This is the most convenient choice of the zero level of potential energy.

(c) Position and momentum cannot be exactly determined simultaneously.

(d) The potential at the boundaries is not really infinite, but just very large.

**Q6. A neutron of mass m _{n} = 10^{-27} kg is moving inside a nucleus. Assume the nucleus to be a cubical box of size 10^{-14} m with impenetrable walls. Take ℏ ≈ 10^{-34} Js and 1 MeV ≈ 10^{-13} J .An estimate of the energy in MeV of the neutron is:**

(a) 80 MeV

(b) ^{1}/_{8} MeV

(c) 8 MeV

(d) ^{1}/_{80} MeV

**Q7. A particle of mass ‘m’ is confined in a two-dimensional infinite square well potential of side ‘a’. The eight Energy of the particle in a given state is E = 25π ^{2}ℏ^{2}/_{ma2}. The state is:**

(a) 4-fold degenerate

(b) 3-fold degenerate

(c) 2-fold degenerate

(d) Non-degenerate.

**Q8. A particle is moving in a two-dimensional potential well**

** V(x,y) = 0, 0 ≤ x ≤ L, 0 ≤ y ≤ 2L = ∞, elsewhere**

**Which of the following statements about the ground state energy E _{1} and ground state eigenfunction φ_{o} are true?**

(a) E_{1} = ℏ^{2}π^{2}/_{mL2}

(b) E_{1} = 5ℏ^{2}π^{2}/_{8mL2}

(c) φ_{o} = ^{√2}/_{L} sin ^{πx}/_{L} sin ^{πy}/_{2L}

(d) φ_{o} = ^{√2}/_{L} cos ^{πx}/_{L} cos ^{πy}/_{2L}

**Q9. Consider a particle of mass ‘m’ moving inside a two-dimensional square box whose sides are described by the equations x = 0, x = L, y = 0, y = L. What is the lowest eigenvalue of an eigenstate which changes sign under the exchange of x and y?**

(a) ℏ^{2}/_{mL2}

(b) 3ℏ^{2}/_{2mL2}

(c) 5ℏ^{2}/_{2mL2}

(d) 7ℏ^{2}/_{2mL2}

**Q10. The binding energy of the hydrogen atom (electron bound to proton) is 13.6 eV. The binding energy of positronium (electron bound to positron) is**

(a) ^{13.6}/_{2} eV

(b) ^{13.6}/_{180} eV

(c) 13.6 ×1810 eV

(d) 13.62 eV

**Q11. If a proton were ten times lighter, then the ground state energy of the electron in a hydrogen atom would have been**

(a) Less

(b) More

(c) The same

(d) Depends on the electron mass

**Q12. A classical particle with total energy E moves under the influence of a potential V(x,y) = 3x ^{3 }+ 2x^{2}y + 2xy^{2} + y^{3}. The average potential energy, calculated over a long time is equal to**

(a) ^{2E}/_{3}

(b) ^{E}/_{3}

(c) ^{E}/_{5}

(d) ^{2E}/_{5}

**Q13. The wavefunction of a hydrogen atom is given by the following superposition of energy eigen-functions ψ _{nlm}(r) (n, l, m are the usual quantum numbers)**

**ψ(r) = ^{√2}/_{√7} ψ_{100}(r) – ^{3}/_{√14} ψ_{210}(r) + ^{1}/_{√14} ψ_{322}(r)**

**The ratio of expectation value of the energy to the ground state energy and the expectation value of L2 are respectively:**

(a) ^{229}/_{504} and 12ℏ^{2}/_{7}

(b) ^{101}/_{504} and 12ℏ^{2}/_{7}

(c) ^{101}/_{504} and ℏ^{2}

(d) ^{229}/_{504} and ℏ^{2}

**Q14. Which of the following sets of Quantum numbers are not possible ?**

(a) n = 1, 1 = 3, m1 = 2

(b) n = 3, 1 = 2, m1 = -2

(c) n = 6, 1 = 2, m1 = 0

(d) n = 7, 1 = 3, m1 = -3

**Q15. The energy eigenvalue corresponding to the bound state ψ _{543}(r,θ,φ) for a hydrogen-like atom is**

(a) 0.544 eV

(b) 5.440 eV

(c) – 0.544 eV

(d) – 5.440 eV

NOTE:- If you need anything else like e-books, video lectures, syllabus, etc. regarding your Preparation / Examination then do 📌 mention it in the Comment Section below. |

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

*Answer Key*

01. | (a) | 06. | (c) | 11. | (b) |

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

03. | (d) | 08. | (b), (c) | 13. | (a) |

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

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