PHYS 3310 Module 6 Self Assessment Practice Problems

Module 6 Self Assessment Practice Problems

6.1
One of the most important insights from quantum mechanics is that quantization of energy is universal — it applies to electrons in atoms, protons in nuclei, and billiard balls on tables. The reason we never notice it for everyday objects is that the energy level spacing becomes vanishingly small at macroscopic scales.
(a) Find the first two energy levels for an electron confined to a one-dimensional box 5.0 × 10⁻¹⁰ m across — approximately the diameter of an atom.
(b) Find the first two energy levels for a proton confined to a one-dimensional box 1.1 × 10⁻¹⁴ m across — approximately the width of a medium-sized atomic nucleus. How do these compare to your answers in part (a), and why are they so different despite both being quantum systems?
(c) Estimate the energy level spacing for a billiard ball of mass 0.2 kg confined to a box 1.3 m wide — the width of a billiard table. Is the quantization of energy detectable in this case? What does this tell you about why quantum effects are invisible in everyday life?
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
The particle-in-a-box model is a surprisingly good approximation for real quantum systems. Quantum dots — nanoscale semiconductor crystals used in modern display screens, solar cells, and medical imaging — behave much like electrons confined to a small box, and their color is determined by exactly this kind of energy level transition. An electron is confined to a one-dimensional box of width 0.15 nm by infinite potential energy barriers. When the electron transitions from the first excited state to the ground state, it emits a photon. Find the wavelength of the emitted photon and identify what region of the electromagnetic spectrum it falls in.
Answer: 24.7 nm

6.3

The particle-in-a-box model is a surprisingly good approximation for real quantum systems. Quantum dots — nanoscale semiconductor crystals used in modern display screens, solar cells, and medical imaging — behave much like electrons confined to a small box, and their color is determined by exactly this kind of energy level transition.

An electron is confined to a one-dimensional box of width 0.15 nm by infinite potential energy barriers. When the electron transitions from the first excited state to the ground state, it emits a photon. Find the wavelength of the emitted photon and identify what region of the electromagnetic spectrum it falls in.

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: 𝜆_5→3 = 12.9 nm, 𝜆_3→1 = 25.8 nm, 𝜆_4→3 = 29.4 nm

6.5
Atoms in a crystal lattice don’t sit perfectly still — they vibrate around their equilibrium positions, held in place by the forces from neighboring atoms. To a good approximation, these vibrations behave like a quantum harmonic oscillator, with equally spaced energy levels separated by ℏω. A sodium atom of mass 3.82 × 10⁻²⁶ kg vibrates within a crystal. When displaced 0.014 nm from its equilibrium position, its potential energy increases by 0.0075 eV.
(a) Using classical mechanics, find the angular frequency of the atom’s oscillations.
(b) Using your answer from part (a), find the spacing between adjacent vibrational energy levels according to quantum mechanics.
(c) When the atom transitions from one vibrational level to the next lower level, it emits a photon. What is the wavelength of this photon, and what region of the electromagnetic spectrum does it fall in?
Does it surprise you that a classical calculation in part (a) feeds directly into a quantum prediction in parts (b) and (c)?
Answer: (a) 1.79 x 10¹³ rad/s (b) 0.0118 eV (c) 105 µm, IR

6.6
Quantum mechanics applies to all oscillating systems — from atoms vibrating in a crystal lattice to a mass on a spring sitting on a lab bench. The difference is in the quantum number. A 0.030-kg particle oscillates on a spring with a frequency of 4.0 Hz. At its equilibrium position it has a speed of 0.60 m/s.
(a) Find the total energy of the particle.
(b) If the particle is in a state of definite energy, find its quantum number n.
(c) Your answer to part (b) should be an extraordinarily large number. What does this tell you about why the quantization of energy is completely undetectable for macroscopic oscillators? How does this connect to what you found for the marble in problem 5.5?
Answer: 𝑛 ≈ 2.037 × 10³⁰

6.7
Quantum tunneling is one of the most counterintuitive predictions of quantum mechanics — a particle can pass through a barrier even when it doesn’t have enough energy to go over it classically. The probability of tunneling depends sensitively on both the width and height of the barrier. A 5.0-eV electron encounters a barrier of width 0.60 nm. Calculate the probability that the electron tunnels through the barrier for each of the following barrier heights:
(a) 7.0 eV
(b) 9.0 eV
(c) 13.0 eV
How does the tunneling probability change as the barrier height increases? Is the relationship linear, or does the probability drop more steeply than that? What does this tell you about how sensitive tunneling is to the barrier height?
Answer: (a) 0.00055 or 0.055% (b) 0.000018 (c) 1.1×10^–7

6.8
The probability of quantum tunneling depends strongly on the mass of the particle — heavier particles have shorter de Broglie wavelengths and tunnel far less readily than lighter ones. This problem illustrates just how dramatic that difference can be. A particle with kinetic energy 32.0 eV encounters a square barrier of height 41.0 eV and width 0.25 nm. Find the tunneling probability for:
(a) an electron
(b) a proton
The proton is approximately 1836 times more massive than the electron. How does this mass difference affect the tunneling probability? Does the result surprise you, and what does it imply about which particles are most relevant in quantum tunneling phenomena in nature?
Answer: (a) 0.13% (b) close to 0%

6.9
In problem 6.5 you started with a known force constant and predicted the wavelength of light a vibrating atom would emit. Real experiments work in the opposite direction: physicists shine infrared light on a crystal and measure which wavelengths are absorbed, then work backwards to determine the force constants that hold the lattice together. This technique — infrared spectroscopy — is a powerful tool for characterizing the mechanical properties of materials at the atomic scale. A lattice atom of mass 9.4 × 10⁻²⁶ kg absorbs a photon of wavelength 525 μm, making a transition from the ground state to the first excited vibrational state. What is the force constant of the lattice?
Answer: 1.2 N/m

6.9

In problem 6.5 you started with a known force constant and predicted the wavelength of light a vibrating atom would emit. Real experiments work in the opposite direction: physicists shine infrared light on a crystal and measure which wavelengths are absorbed, then work backwards to determine the force constants that hold the lattice together. This technique — infrared spectroscopy — is a powerful tool for characterizing the mechanical properties of materials at the atomic scale.

A lattice atom of mass 9.4 × 10⁻²⁶ kg absorbs a photon of wavelength 525 μm, making a transition from the ground state to the first excited vibrational state. What is the force constant of the lattice?

Answer: 1.2 N/m

6.10
In problems 6.7 and 6.8 you calculated tunneling probabilities for given barrier dimensions. This problem reverses the question: given a desired tunneling probability, what barrier width produces it? This is the kind of calculation that matters in device engineering — for example, designing a tunnel diode where a specific tunneling rate is required. A proton with kinetic energy 50.0 eV encounters a barrier of height 70.0 eV. What barrier width gives a tunneling probability of 8.0 × 10⁻³? An electron with the same kinetic energy encounters a barrier of the same height with the same tunneling probability. What barrier width does the electron require? Compare your two answers. Given what you found in problem 6.8 about the effect of mass on tunneling probability, does the relative size of these two barrier widths make sense? What does this tell you about the length scales at which proton tunneling and electron tunneling are relevant in nature?
Answer: 3.05 pm, 0.131 nm

6.10

In problems 6.7 and 6.8 you calculated tunneling probabilities for given barrier dimensions. This problem reverses the question: given a desired tunneling probability, what barrier width produces it? This is the kind of calculation that matters in device engineering — for example, designing a tunnel diode where a specific tunneling rate is required.

A proton with kinetic energy 50.0 eV encounters a barrier of height 70.0 eV. What barrier width gives a tunneling probability of 8.0 × 10⁻³?

An electron with the same kinetic energy encounters a barrier of the same height with the same tunneling probability. What barrier width does the electron require?

Compare your two answers. Given what you found in problem 6.8 about the effect of mass on tunneling probability, does the relative size of these two barrier widths make sense? What does this tell you about the length scales at which proton tunneling and electron tunneling are relevant in nature?

Answer: 3.05 pm, 0.131 nm