Chapter 40

A particle of mass \(m\) in a one-dimensional box has the following wave function in the region \(x\) = 0 to \(x = L\) : $$\Psi(x, t) = {1\over \sqrt2} \psi_1(x)e^{-iE_1t/\hslash} + {1\over \sqrt 2} \psi_3(x)e^{-iE_3t/\hslash}$$ Here \(\psi_1(x)\) and \(\psi_3(x)\) are the normalized stationary-state wave functions for the \(n\) = 1 and \(n\) = 3 levels, and \(E_1\) and \(E_3\) are the energies of these levels. The wave function is zero for \(x <\) 0 and for \(x > L\). (a) Find the value of the probability distribution function at \(x = L\)/2 as a function of time. (b) Find the angular frequency at which the probability distribution function oscillates.

Consider a beam of free particles that move with velocity \(v = p/m\) in the \(x\)-direction and are incident on a potentialenergy step \(U(x)\) = 0, for \(x <\) 0, and \(U(x) = U_0 < E\), for \(x >\) 0. The wave function for \(x <\) 0 is \(\psi(x) = Ae^{ik_1x} + Be^{-ik_1x}\), representing incident and reflected particles, and for \(x >\) 0 is \(\psi(x) = Ce^{ik_2x}\), representing transmitted particles. Use the conditions that both \(\psi\) and its first derivative must be continuous at \(x\) = 0 to find the constants \(B\) and \(C\) in terms of \(k_1\), \(k_2\), and \(A\).

The \(penetration\) \(distance\) \(\eta\) in a finite potential well is the distance at which the wave function has decreased to 1/\(e\) of the wave function at the classical turning point: $$\psi(x = L + \eta) = {1\over e} \psi(L)$$ The penetration distance can be shown to be $$\eta = {\hslash \over \sqrt{2m(U_0 - E)}}$$ The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find \(\eta\) for an electron having a kinetic energy of 13 eV in a potential well with \(U_0\) = 20 eV. (b) Find \(\eta\) for a 20.0-MeV proton trapped in a 30.0-MeV-deep potential well.

An electron with initial kinetic energy 5.5 eV encounters a square potential barrier of height 10.0 eV. What is the width of the barrier if the electron has a 0.50\(\%\) probability of tunneling through the barrier?

A harmonic oscillator consists of a 0.020-kg mass on a spring. The oscillation frequency is 1.50 Hz, and the mass has a speed of 0.480 m/s as it passes the equilibrium position. (a) What is the value of the quantum number n for its energy level? (b) What is the difference in energy between the levels \(E_n\) and \(E_{n+1}\)? Is this difference detectable?

(a) Show by direct substitution in the Schr\(\ddot{o}\)dinger equation for the one-dimensional harmonic oscillator that the wave function \(\psi_1(x) = A_1xe^{-a^2x^2/2}\), where \(\alpha^2 = m\omega/\hslash\), is a solution with energy corresponding to \(n\) = 1 in Eq. (40.46). (b) Find the normalization constant A1. (c) Show that the probability density has a minimum at \(x\) = 0 and maxima at \(x = \pm1/\alpha\), corresponding to the classical turning points for the ground state \(n\) = 0.

(a) The wave nature of particles results in the quantum-mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls. Use this to show that an electron confined in a one-dimensional box of length \(L\) will have energy levels given by $$E_n = {n^2h^2 \over8mL^2}$$ (\(Hint\): Recall that the relationship between the de Broglie wavelength and the speed of a nonrelativistic particle is \(mv = h/\lambda\). The energy of the particle is \({1\over2} mv^2\).) (b) If a hydrogen atom is modeled as a one-dimensional box with length equal to the Bohr radius, what is the energy (in electron volts) of the lowest energy level of the electron?

In your research on new solid-state devices, you are studying a solid-state structure that can be modeled accurately as an electron in a one-dimensional infinite potential well (box) of width \(L\). In one of your experiments, electromagnetic radiation is absorbed in transitions in which the initial state is the \(n\) = 1 ground state. You measure that light of frequency \(f\) = 9.0 \(\times\) 10\(^{14}\) Hz is absorbed and that the next higher absorbed frequency is 16.9 \(\times\) 10\(^{14}\) Hz. (a) What is quantum number \(n\) for the final state in each of the transitions that leads to the absorption of photons of these frequencies? (b) What is the width \(L\) of the potential well? (c) What is the longest wavelength in air of light that can be absorbed by an electron if it is initially in the \(n\) = 1 state?

When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) \(T\) equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of \(T\) = 1 occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width \(L\) of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies \(E\) that will result in \(T\) = 1. You assume a barrier height of 10 eV and a width of 1.8 \(\times\) 10\(^{-10}\) m. Calculate the three lowest values of \(E\) for which \(T\) = 1.

Protons, neutrons, and many other particles are made of more fundamental particles called \(quarks\) and \(antiquarks\) (the antimatter equivalent of quarks). A quark and an antiquark can form a bound state with a variety of different energy levels, each of which corresponds to a different particle observed in the laboratory. As an example, the \(\psi\) particle is a low-energy bound state of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the \(\psi\)(2S) particle is an excited state of this same quark- antiquark combination, with a rest energy of 3686 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is \(U(x) = A\mid x \mid\) , where \(A\) is a positive constant and \(x\) represents the distance between the quark and the antiquark. You can use the WKB approximation (see Challenge Problem 40.64) to determine the bound-state energy levels for this potential-energy function. In the WKB approximation, the energy levels are the solutions to the equation $$\int ^b _a \sqrt{2m[E - U(x)]} dx = {nh \over 2} \space (n = 1, 2, 3, . . .)$$ Here \(E\) is the energy, \(U(x)\) is the potential-energy function, and \(x\) = a and \(x\) = b are the classical turning points (the points at which \(E\) is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for the potential \(U(x) = A \mid x \mid\) and for an energy \(E\). (b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are given by $$E_n = {1 \over2m} ( {3mAh \over 4} ) ^{2/3} n^{2/3} \space (n = 1, 2, 3, . . .)$$ (\(Hint\): The integrand is even, so the integral from \(-x\) to \(x\) is equal to twice the integral from 0 to \(x\).) (c) Does the difference in energy between successive levels increase, decrease, or remain the same as \(n\) increases? How does this compare to the behavior of the energy levels for the harmonic oscillator? For the particle in a box? Can you suggest a simple rule that relates the difference in energy between successive levels to the shape of the potential-energy function?