Problem 18
Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width 0.360 nm. (b) The electron makes a transition from the \(n\) = 1 to \(n\) = 4 level by absorbing a photon. Calculate the wavelength of this photon.
Problem 19
An electron is in a box of width 3.0 \(\times\) 10\(^{-10}\) m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the \(n\) = 1 level; (b) the \(n\) = 2 level; (c) the \(n\) = 3 level? In each case how does the wavelength compare to the width of the box?
Problem 20
When an electron in a one-dimensional box makes a transition from the \(n\) = 1 energy level to the \(n\) = 2 level, it absorbs a photon of wavelength 426 nm. What is the wavelength of that photon when the electron undergoes a transition (a) from the \(n\) = 2 to the \(n\) = 3 energy level and (b) from the n = 1 to the \(n\) = 3 energy level? (c) What is the width \(L\) of the box?
Problem 24
An electron is in the ground state of a square well of width \(L = 4.00 \times 10^{-10}\) m. The depth of the well is six times the ground-state energy of an electron in an infinite well of the same width. What is the kinetic energy of this electron after it has absorbed a photon of wavelength 72 nm and moved away from the well?
Problem 30
An electron with initial kinetic energy 5.0 eV encounters a barrier with height \(U_0\) and width 0.60 nm. What is the transmission coefficient if (a) \(U_0\) = 7.0 eV; (b) \(U_0\) = 9.0 eV; (c) \(U_0\) = 13.0 eV?
Problem 32
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 \(\times\) 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?
Problem 35
Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8 \(\mu\)m is absorbed when a molecule makes a transition from its ground harmonic oscillator level to its first excited level. (a) Find the energy of this transition. (b) If the molecule can be treated as a harmonic oscillator with mass 5.6 \(\times\) 10\(^{-26}\) kg, find the force constant.
Problem 36
A harmonic oscillator absorbs a photon of wavelength 6.35 \(\mu\)m when it undergoes a transition from the ground state to the first excited state. What is the ground-state energy, in electron volts, of the oscillator?
Problem 37
The ground-state energy of a harmonic oscillator is 5.60 eV. If the oscillator undergoes a transition from its \(n\) = 3 to \(n\) = 2 level by emitting a photon, what is the wavelength of the photon?
Problem 42
Consider the wave packet defined by $$\psi(x) = \int ^\infty_0 B(k)cos kx dk$$ Let \(B(k) = e^{-a^2k^2}\). (a) The function \(B(k)\) has its maximum value at \(k\) = 0. Let \(k_h\) be the value of \(k\) at which \(B(k)\) has fallen to half its maximum value, and define the width of \(B(k)\) as \(w_k = k_h\) . In terms of \(\alpha\), what is \(w_k\) ? (b) Use integral tables to evaluate the integral that gives \(\psi(x)\). For what value of \(x\) is \(\psi(x)\) maximum? (c) Define the width of \(\psi(x)\) as \(w_x = x_h\) , where \(x_h\) is the positive value of \(x\) at which \(\psi(x)\) has fallen to half its maximum value. Calculate \(w_x\) in terms of \(\alpha\). (d) The momentum \(p\) is equal to \(hk/2\pi\), so the width of \(B\) in momentum is \(w_p = hw_k /2\pi\). Calculate the product \(w_p w_x\) and compare to the Heisenberg uncertainty principle.
