PHYS 3310 Module 6 Self Assessment Practice Problems

Module 6 Self Assessment Practice Problems

6.1
(a) Find the first two energy levels for an electron confined to a one-dimensional box 5.0 x 10-10 m across (about the diameter of an atom).
(b) Compare your answers in part (a) to the first two energy levels for a proton confined to a one-dimensional box 1.1 x 10-14 m across (the width of a medium-sized atomic nucleus).
(c) Finally, discuss the energy levels for a billiard ball (m = 0.2 kg) confined to a box 1.3 m across — the width of a billiard table.
Answer: (a) 1.5 eV, 6.0 eV (b) 1.7 MeV, 6.8 MeV
6.2
(a) Show that , where A and B are constants, is a solution of the Schrödinger equation for an E = 0 energy level of a particle in a box.
(b) What constraints do the boundary conditions at x = 0 and x = L place on the constants A and B?
Answer:
6.3
An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.
Answer: 24.7 nm
6.4
An electron is confined to a box of width 0.25 nm.
(a) Draw an energy-level diagram representing the first five states of the electron.
(b) Calculate the wavelengths of the emitted photons when the electron makes transitions between the fourth and the second excited states, between the second excited state and the ground state, and between the third and the second excited states.
Answer: 𝜆5→3 = 12.9 nm, 𝜆3→1 = 25.8 nm, 𝜆4→3 = 29.4 nm
6.5
A sodium atom of mass 3.82 x 10-26 kg vibrates within a crystal. The potential energy increases by 0.0075 eV when the atom is displaced 0.014 nm from its equilibrium position. Treat the atom as a harmonic oscillator.
(a) Find the angular frequency of the oscillations according to Newtonian mechanics.
(b) Find the spacing (in electron volts) of adjacent vibrational energy levels according to quantum mechanics.
(c) What is the wavelength of a photon emitted as the result of a transition from one level to the next lower level? In what region of the electromagnetic spectrum does this lie?
Answer: (a) 1.79 x 1013 rad/s (b) 0.0118 eV (c) 105 µm, IR
6.6
A particle with mass 0.030 kg oscillates back-and-forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.
Answer: 𝑛 ≈ 2.037 × 1030
6.7
A 5.0-eV electron impacts on a barrier of width 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is
(a) 7.0 eV
(b) 9.0 eV
(c) 13.0 eV
Answer: (a) 0.00055 or 0.055% (b) 0.000018 (c) 1.1×10–7
6.8
A quantum particle with initial kinetic energy 32.0 eV encounters a square barrier with height 41.0 eV and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is
(a) an electron
(b) a proton
Answer: (a) 0.13% (b) close to 0%
6.9
Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of 9.4 × 10−26 kg, what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a 525-µm photon?
Answer: 1.2 N/m
6.10
A proton with initial kinetic energy 50.0 eV encounters a barrier of height 70.0 eV. What is the width of the barrier if the probability of tunneling is 8.0 x 10-3? How does this compare with the barrier width for an electron with the same energy tunneling through a barrier of the same height with the same probability?
Answer: 3.05 pm, 0.131 nm