Questions on Multi-dimensional Potentials with Answers

Practice Questions on Multi-dimensional Potentials with Answers.

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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) = 1/22(x2 + y2 + z2). 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 = √(/) and Hn(ξ) is the nth Hermite polynomial)

(a) H2(αx)exp(-α(y2 + z2))
(b) H2(αx)exp(-α(x2 + y2 + z2))
(c) H1(αy)H1(αz)exp(-α(x2 + y2 + z2))
(d) H1(αx)H1(αz)exp(-α(x2 + y2 + z2))

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

V(x,y,z) = 1/22(x2 + y2 + z2).

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) 49h2/288mL2
(b) 9h2/32mL2
(c) 61h2/288mL2
(d) 11h2/36mL2

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 3h2/8mL2 . 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 mn = 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 = 25π22/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 E1 and ground state eigenfunction φo are true?

(a) E1 = ℏ2π2/mL2
(b) E1 = 5ℏ2π2/8mL2
(c) φo = √2/L sin πx/L sin πy/2L
(d) φo = √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) ℏ2/mL2
(b) 3ℏ2/2mL2
(c) 5ℏ2/2mL2
(d) 7ℏ2/2mL2

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) = 3x3 + 2x2y + 2xy2 + y3. 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 ψnlm(r) (n, l, m are the usual quantum numbers)

ψ(r) = √2/√7 ψ100(r) – 3/√14 ψ210(r) + 1/√14 ψ322(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 12ℏ2/7
(b) 101/504 and 12ℏ2/7
(c) 101/504 and ℏ2
(d) 229/504 and ℏ2

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 ψ543(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

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

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

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