Exam 39: Quantum Mechanics

An electron is confined in a harmonic oscillator potential well. A photon is emitted when the electron undergoes a 3→1 quantum jump. What is the wavelength of the emission if the net force on the electron behaves as though it has a spring constant of 9.6 N/m? (^mel = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

One fairly crude method of determining the size of a molecule is to treat the molecule as an infinite square well (a box) with an electron trapped inside, and to measure the wavelengths of emitted photons. If the photon emitted during the n = 2 to n = 1 transition has wavelength 1940 nm, what is the width of the molecule? (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s, ^mel = 9.11 × 10^-31 kg)

An electron is confined in a one-dimensional box (an infinite well). Two adjacent allowed energies of the electron are 1.068 × 10^-18 J and 1.352 × 10^-18 J. What is the length of the box? (h = 6.626 × 10-³⁴ J ∙ s, ^mel = 9.11 × 10^-31 kg)

A 3.10-eV electron is incident on a 0.40-nm barrier that is 5.67 eV high. What is the probability that this electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, ^mel = 9.11 × 10^-31 kg, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

The square of the wave function of a particle, |ψ(x)|², gives the probability of finding the particle at the point x.

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) = sin , with 0 ≤ x ≤ L. What is the probability of finding the particle between x = 0 and x = L/3?

An electron is in an infinite square well (a box) that is 2.0 nm wide. The electron makes a transition from the n = 8 to the n = 7 state, what is the wavelength of the emitted photon? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

The energy of a proton is 1.0 MeV below the top of a 1.2-MeV-high energy barrier that is 6.8 fm wide. What is the probability that the proton will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, ^mproton = 1.67 × 10^-27 kg, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

The atoms in a nickel crystal vibrate as harmonic oscillators with an angular frequency of 2.3 × 10¹³ rad/s. The mass of a nickel atom is 9.75 × 10^-26 kg. What is the difference in energy between adjacent vibrational energy levels of nickel? (h = 6.626 × 10-³⁴ J ∙ s, = 1.055 × 10^-34 J ∙ s, 1 eV = 1.60 × 10^-19 J)

An electron is trapped in an infinite square well (a box) of width 6.88 nm. Find the wavelength of photons emitted when the electron drops from the n = 5 state to the n = 1 state in this system. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s, ^mel = 9.11 × 10^-31 kg)

The wave function for an electron that is confined to x ≥ 0 nm is ψ(x) = (a) What must be the value of b? (b) What is the probability of finding the electron in a 0.010 nm-wide region centered at x = 1.0 nm?

The wave function for a particle must be normalizable because

An electron is in the ground state of an infinite well (a box) where its energy is 5.00 eV. In the next higher level, what would its energy be? (1 eV = 1.60 × 10^-19 J)

An electron in an infinite potential well (a box) makes a transition from the n = 3 level to the ground state and in so doing emits a photon of wavelength 20.9 nm. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10⁻³⁴J ∙ s, ^mel = 9.11 × 10⁻³¹ kg) (a) What is the width of this well? (b) What wavelength photon would be required to excite the electron from its original level to the next higher one?

The lowest energy level of a particle confined to a one-dimensional region of space (a box, or infinite well) with fixed length L is E₀. If an identical particle is confined to a similar region with fixed length L/6, what is the energy of the lowest energy level that the particles have in common? Express your answer in terms of E₀.

You want to have an electron in an energy level where its speed is no more than 66 m/s. What is the length of the smallest box (an infinite well) in which you can do this? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

If an atom in a crystal is acted upon by a restoring force that is directly proportional to the distance of the atom from its equilibrium position in the crystal, then it is impossible for the atom to have zero kinetic energy.

The smallest kinetic energy that an electron in a box (an infinite well) can have is zero.

An electron is in an infinite square well that is 2.6 nm wide. What is the smallest value of the state quantum number n for which the energy level exceeds 100 eV? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

Calculate the ground state energy of a harmonic oscillator with a classical frequency of 3.68 × 10¹⁵ Hz. (1 eV = 1.60 × 10^-19 J, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)