Chapter 40

An electron is moving as a free particle in the \(-x\) -direction with momentum that has magnitude \(4.50 \times 10^{-24} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) What is the one-dimensional time-dependent wave function of the electron?

Consider a wave function given by \(\psi(x)=A \sin k x,\) where \(k=2 \pi / \lambda\) and \(A\) is a real constant. (a) For what values of \(x\) is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of \(x\) is the probability zero? Explain.

Compute \(|\Psi|^{2}\) for \(\Psi=\psi \sin \omega t,\) where \(\psi\) is time independent and \(\omega\) is a real constant. Is this a wave function for a stationary state? Why or why not?

A particle is described by a wave function \(\psi(x)=A e^{-\alpha x^{2}}\) where \(A\) and \(\alpha\) are real, positive constants. If the value of \(\alpha\) is increased, what effect does this have on (a) the particle's uncertainty in position and (b) the particle's uncertainty in momentum? Explain your answers.

A particle moving in one dimension (the \(x\) -axis) is described by the wave function \(\psi(x)=\left\\{\begin{array}{ll}{A e^{-b x},} & {\text { for } x \geq 0} \\ {A e^{b x},} & {\text { for } x<0}\end{array}\right.\) where \(b=2.00 \mathrm{m}^{-1}, A>0,\) and the \(+x\) -axis points toward the right. (a) Determine \(A\) so that the wave function is normalized. (b) Sketch the graph of the wave function. (c) Find the probability of finding this particle in each of the following regions: (i) within 50.0 \(\mathrm{cm}\) of the origin, (ii) on the left side of the origin (can you first guess the answer by looking at the graph of the wave function? (iii) between \(x=0.500 \mathrm{m}\) and \(x=1.00 \mathrm{m} .\)

A proton is in a box of width \(L .\) What must the width of the box be for the ground-level energy to be \(5.0 \mathrm{MeV},\) a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus- that is, on the order of \(10^{-14} \mathrm{m} .\)

When a hydrogen atom undergoes a transition from the \(n=2\) to the \(n=1\) level, a photon with \(\lambda=122 \mathrm{nm}\) is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n=2\) to \(n=1\) transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of \(n . )\)

An electron in a one-dimensional box has ground-state energy 1.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of 0.125 \(\mathrm{nm}\) . (b) The electron makes a transition from the \(n=1\) to \(n=4\) level by absorbing a photon. Calculate the wave-length of this photon.

An electron is in a box of width \(3.0 \times 10^{-10} \mathrm{m} .\) 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?

Normalization of the Wave Function. Consider a particle moving in one dimension, which we shall call the \(x\) -axis. (a) What does it mean for the wave function of this particle to be normalized? (b) Is the wave function \(\psi(x)=e^{a x},\) where \(a\) is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function \(\psi(x)=A e^{b x},\) where \(A\) and \(b\) are positive real numbers, is confined to the range \(x \geq 0\) , determine \(A\) (including its units) so that the wave function is normalized.

An electron is bound in a square well with a depth equal to six times the ground-level energy \(E_{1-\mathrm{LDW}}\) of an infinite well of the same width. The longest-wavelength photon that is absorbed by the electron has a wavelength of 400.0 \(\mathrm{nm} .\) Determine the width of the well.

An electron with initial kinetic energy 6.0 \(\mathrm{eV}\) encounters a barrier with height 11.0 \(\mathrm{eV}\) . What is the probability of tunneling if the width of the barrier is (a) 0.80 \(\mathrm{nm}\) and (b) 0.40 \(\mathrm{nm} ?\)

An electron with initial kinetic energy 5.0 eV encounters a barrier with height \(U_{0}\) and width 0.60 \(\mathrm{nm} .\) What is the transmission coefficient if (a) \(U_{0}=7.0 \mathrm{eV} ;\) (b) \(U_{0}=9.0 \mathrm{eV} ;\) (c) \(U_{0}=\) 13.0 \(\mathrm{eV} ?\)

(a) An electron with initial kinetic energy 32 eV encounters a square barrier with height 41 \(\mathrm{eV}\) and width 0.25 \(\mathrm{nm}\) . What is the probability that the electron will tunnel through the barrier? (b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

A harmonic oscillator absorbs a photon of wavelength \(8.65 \times 10^{-6} \mathrm{m}\) 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?

Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8\(\mu \mathrm{m}\) 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 \(5.6 \times 10^{-26} \mathrm{kg},\) find the force constant.

The ground-state energy of a harmonic oscillator is 5.60 \(\mathrm{eV} .\) 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?

A particle of mass \(m\) in a one-dimensional box has the following wave function in the region \(x=0\) to \(x=L :\) $$\Psi(x, t)=\frac{1}{\sqrt{2}} \psi_{1}(x) e^{-i E_{1} t / \hbar}+\frac{1}{\sqrt{2}} \psi_{3}(x) e^{-i E_{3} t / \hbar}$$ 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 the wave packet defined by $$\psi(x)=\int_{0}^{\infty} B(k) \cos k x d k$$ Let \(B(k)=e^{-\alpha^{2} k^{2}} .\) (a) The function \(B(k)\) has its maximum value at \(k=0 .\) Let \(k_{\mathrm{h}}\) be the value of \(k\) at which \(B(k)\) has fallen to half its maximum value, and define the width of \(B(k)\) as \(w_{k}=k_{\mathrm{h}}\) . 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 \(w_{x}=x_{\mathrm{h}},\) where \(x_{\mathrm{h}}\) 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 \(h k / 2 \pi\) so the width of \(B\) in momentum is \(w_{p}=h w_{k} / 2 \pi .\) Calculate the product \(w_{p} w_{x}\) and compare to the Heisenberg uncertainty principle.

Wave functions like the one in Problem 40.50 can represent free particles moving with velocity \(v=p / m\) in the \(x\) -direction. Consider a beam of such particles incident on a potential-energy step \(U(x)=0,\) for \(x < 0,\) and \(U(x)=U_{0} < E,\) for \(x > 0 .\) The wave function for \(x < 0\) is \(\psi(x)=A e^{i k_{1} x}+B e^{-i k_{1} x}\) representing incident and reflected particles, and for \(x > 0\) is \(\psi(x)=C e^{i k_{2} x},\) 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 \(A .\)

Photon in a Dye Laser. An electron in a long, organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.18 nm. What is the wavelength of the photon emitted when the electron undergoes a transition (a) from the first excited level to the ground level and (b) from the second excited level to the first excited level?

Consider a particle in a box with rigid walls at \(x=0\) and \(x=L .\) Let the particle be in the ground level. Calculate the probability \(|\psi|^{2} d x\) that the particle will be found in the interval \(x\) to \(x+d x\) for \((\) a \() x=L / 4 ;(\) b) \(x=L / 2 ;(\) c) \(x=3 L / 4\)

A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero) is $$\psi(x)=\left\\{\begin{array}{ll}{e^{+\kappa x},} & {x<0} \\ {e^{-\kappa x},} & {x \geq 0}\end{array}\right.$$ where \(\kappa\) is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schrodinger equation for \(x < 0\) if the energy is \(E=-\hbar^{2} \kappa^{2} / 2 m-\) that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schrodinger equation for \(x \geq 0\) with the same energy as in part (b). (d) Explain why the proposed wave function is nonetheless not an acceptable solution of the Schrodinger equation for a free particle. (Hint: What is the behavior of the function at \(x=0 ?\) ) It is in fact impossible for a free particle (one for which \(U(x)=0 )\) to have an energy less than zero.

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: $$\psi(x=L+\eta)=\frac{1}{e} \psi(L)$$ The penetration distance can be shown to be $$\eta=\frac{\hbar}{\sqrt{2 m\left(U_{0}-E\right)}}$$ 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 \(U_{0}=20 \mathrm{eV} .\) (b) Find \(\eta\) for a 20.0 -MeV proton trapped in a 30.0 -Me \(\mathrm{V}\) -deep potential well.

A particle with mass \(m\) and total energy \(E\) tunnels through a square barrier of height \(U_{0}\) and width \(L .\) When the trans- mission coefficient is not much less than unity, it is given by $$T=\left[1+\frac{\left(U_{0} \sinh \kappa L\right)^{2}}{4 E\left(U_{0}-E\right)}\right]^{-1}$$ where \(\sinh \kappa L=\left(e^{\kappa L}-e^{-\kappa L}\right) / 2\) is the hyperbolic sine of \(\kappa L .\) (a) Show that if \(\kappa L \gg 1\) , this expression for \(T\) approaches Eq. \((40.42) .\) (b) Explain why the restriction \(\kappa L>1\) in part (a) implies either that the barrier is relatively wide or the energy \(E\) is relatively low compared to \(U_{0 .}\) (c) Show that as the particle's incident kinetic energy \(E\) approaches the barrier height \(U_{0}\) . \(T\) approaches \(\left[1+(k L / 2)^{2}\right]^{-1},\) where \(k=\sqrt{2 m E / \hbar}\) is the wave number of the incident particle. (Hint: If \(|z|<1,\) then \(\sinh z \approx z . )\)

(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 $$E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}$$ (Hint: Recall that the relationship between the de Broglie wave-length and the speed of a nonrelativistic particle is \(m v=h / \lambda\) . The energy of the particle is \(\frac{1}{2} m v^{2} . )\) (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?

Consider a potential well defined as \(U(x)=\infty\) for \(x < 0, U(x)=0\) for \(0 < x < L,\) and \(U(x)=U_{0} > 0\) for \(x > L\) (Fig. \(\mathrm{P} 40.70 ) .\) Consider a particle with mass \(m\) and kinetic energy \(E < U_{0}\) that is trapped in the well. (a) The boundary condition at the infinite wall \((x=0)\) is \(\psi(0)=0 .\) What must the form of the function \(\psi(x)\) for \(0 < x < L\) be in order to satisfy both the Schrodinger equation and this boundary condition (b) The wave function must remain finite as \(x \rightarrow \infty .\) What must the form of the function \(\psi(x)\) for \(x>L\) be in order to satisfy both the Schrodinger equation and this boundary condition at infinity? (c) Impose the boundary conditions that \(\psi\) and \(d \psi / d x\) are continuous at \(x=L .\) Show that the energies of the allowed levels are obtained from solutions of the equation \(k \cot k L=-\kappa,\) where \(k=\sqrt{2 m E} / \hbar\) and \(\kappa=\sqrt{2 m\left(U_{0}-E\right) / \hbar}\)

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 \(\psi(2 S)\) 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 \(U(x)=A|x|,\) 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 40.72 ) 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 $$\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \ldots)$$ 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 \(U(x)=A|x|\) 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 $$E_{n}=\frac{1}{2 m}\left(\frac{3 m A h}{4}\right)^{2 / 3} n^{2 / 3} \quad(n=1,2,3, \ldots)$$ (Hint: The integrand is even, so the integral from \(-x\) to \(x\) is equal to twice the integral from 0 to \(x\) . \((\mathrm{c})\) 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?