The probability distributions for the harmonic oscillator

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

A free particle moving in one dimension has wave function where k and are positive real constants. (a) At what are the two smallest positive values of x for which the probability function is a maximum? (b) Repeat part (a) for time . (c) Calculate as the distance the maxima have moved divided by the elapsed time. Compare your result to the expression from Example 40.1.

Consider the free-particle wave function of Example 40.1. Let . At the probability distribution function has a maximum at . (a) What is the smallest positive value of x for which the probability distribution function has a maximum at time , where . (b) From your result in part (a), what is the average speed with which the probability distribution is moving in the x-direction? Compare your result to the expression from Example 40.1.

Consider the free particle of Example 40.1. Show that can be written as , where p . av = 1Uk 2 + Uk 12>2

Consider a wave function given by where 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 for where is time independent and is a real constant. Is this a wave function for a stationary state? Why or why not?

CALC Let and be two solutions of Eq. (40.23) with energies and respectively, where Is where A and B are nonzero constants, a solution to Eq. (40.23)? Explain your answer

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.

Linear Combinations of Wave Functions. Let \(\psi_{1}\) and \(\psi_{2}\) be two solutions of Eq. (40.23) with the same energy E. Show that \(\psi=B \psi_{1}+C \psi_{2}\) is also a solution with energy E, for any values of the constants B and C.

CALC A particle moving in one dimension (the x-axis) is described by the wave function where and the 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 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 and

Ground-Level Billiards. (a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (m = 0.20 kg) and the box has a width of 1.3 m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.) (b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other? (c) What is the difference in energy between the n = 2 and n = 1 levels? (d) Are quantum-mechanical effects impor-tant for the game of billiards?

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 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 \(10^{-14} \mathrm{\ m}\). Text Transcription: 10^-14 m

Find the width L of a one-dimensional box for which the ground-state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom.

When a hydrogen atom undergoes a transition from the n = 2 to n = 1 the 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.)

A certain atom requires 3.0 eV of energy to excite an electron from the ground level to the first excited level. Model the atom as an electron in a box and find the width of the box.

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?

Show that the time-dependent wave function given by Eq. (40.35) is a solution to the one-dimensional Schrödinger equation, Eq. (40.23).

Recall that \(|\psi|^{2} d x\) is the probability of finding the particle that has normalized wave function \(\psi(x)\) in the interval x to x + dx. Consider a particle in a box with rigid walls at x = 0 and x = L. Let the particle be in the ground level and use \(\psi_{n}\) as given in Eq. (40.35). (a) For which values of if x, any, in the range from 0 to L is the probability of finding the particle zero? (b) For which values of x is the probability highest? (c) In parts (a) and (b) are your answers consistent with Fig. 40.12? Explain. Text Transcription: |psi|^2 dx psi(x) psi_n

Repeat Exercise 40.18 for the particle in the first excited level.

CALC (a) Show that is a solution to Eq. (40.25) if (b) Explain why this is an acceptable wave function for a particle in a box with rigid walls at and only if is an integer multiple of

(a) Repeat Exercise 40.20 for \(\psi = A cos kx\). (b) Explain why this cannot be an acceptable wave function for a particle in a box with rigid walls at x = 0 and x = L no matter what the value of k. Text Transcription: Psi = A cos kx

(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 nm. (b) The electron makes a transition from the to level by absorbing a photon. Calculate the wavelength of this photon.

An electron is in a box of width What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the level; (b) the level; (c) the 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.

CALC (a) Show that where is a real (not complex) constant, is not a solution of Eq. (40.23) for and (b) Is this a solution for

An electron is moving past the square well shown in Fig. 40.13. The electron has energy \(E=3 U_{0}\). What is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?

An electron is bound in a square well of depth What is the width of the well if its ground-state energy is 2.00 eV?

An electron is bound in a square well of width 1.50 nm and depth \(U_{0}=6 E_{1-\mathrm{IDW}}\). If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well? Text Transcription: U_0=6E_1-IDW

CALC Calculate for the wave function of Eq. (40.38), and show that the function is a solution of Eq. (40.37)

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

A proton is bound in a square well of width \(4.0 \mathrm{fm}=4.0 \times 10^{-15} \mathrm{~m}\). The depth of the well is six times the ground-level energy \(E_{1-\text { IDW }}\) of the corresponding infinite well. If the proton makes a transition from the level with energy \(E_1\) to the level with energy \(E_3\) by absorbing a photon, find the wavelength of the photon.

Alpha Decay. In a simple model for a radioactive nucleus, an alpha particle is trapped by a square barrier that has width 2.0 fm and height 30.0 MeV. (a) What is the tunneling probability when the alpha particle encounters the barrier if its kinetic energy is 1.0 MeV below the top of the barrier (Fig. E40.32)? (b) What is the tunneling probability if the energy of the alpha particle is 10.0 MeV below the top of the barrier?

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

An electron with initial kinetic energy encounters a barrier with height and width What is the transmission coefficient if (a) (b) (c)

An electron is moving past the square barrier shown in Fig. 40.19, but the energy of the electron is greater than the barrier height. If \(E=2 U_{0}\), what is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?

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 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?

(a) An electron with initial kinetic energy 32 eV encounters a square barrier with height 41 eV and width 0.25 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?

Show that given by Eq. (40.47) is a solution to Eq. (40.44) with energy

A wooden block with mass 0.250 kg is oscillating on the end of a spring that has force constant Calculate the ground-level energy and the energy separation between adjacent levels. Express your results in joules and in electron volts. Are quantum effects important?

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

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 transi-tion 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 \(5.6\times10^{-26}\mathrm{\ kg}\), find the force constant.

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?

In Section 40.5 it is shown that for the ground level of a harmonic oscillator, \(\Delta x \Delta p_{x}=\hbar / 2\). Do a similar analysis for an excited level that has quantum number n. How does the uncertainty product \(\Delta x \Delta p_{x}\) depend on n?

For the ground-level harmonic oscillator wave function given in Eq. (40.47), has a maximum at (a) Compute the ratio of at to at where is given by Eq. (40.48) with for the ground level. (b) Compute the ratio of at to at In each case is your result consistent with what is shown in Fig. 40.27?

For the sodium atom of Example 40.8, find (a) the ground-state energy, (b) the wavelength of a photon emitted when the n = 4 to n = 3 transition occurs; (c) the energy difference for any \(\Delta n=1\) transition.

The discussion in Section 40.1 shows that the wave function is a stationary state, where is time independent and is a real (not complex) constant. Consider the wave function where and are different timeindependent functions and and are different real constants. Assume that and are real-valued functions, so that and Is this a wave function for a stationary state? Why or why not?

A particle of mass m in a one-dimensional box has the following wave function in the region x = 0 to x = L: \(\Psi(x,\ t)=\frac{1}{\sqrt{2}}\psi_1(x)e^{-iE_1t/\hbar}+\frac{1}{\sqrt{2}}\psi_3(x)e^{-iE_3t/\hbar}\) 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.

CALC Consider the wave packet defined by Let (a) The function has its maximum value at Let be the value of k at which has fallen to half its maximum value, and define the width of as In terms of what is (b) Use integral tables to evaluate the integral that gives For what value of x is maximum? (c) Define the width of as where is the positive value of x at which has fallen to half its maximum value. Calculate in terms of (d) The momentum p is equal to so the width of B in momentum is Calculate the product and compare to the Heisenberg uncertainty principle.

(a) Using the integral in Problem 40.48, determine the wave function \(\psi(x)\) for a function B(k) given by \(B(k)=\left\{\begin{array}{ll} 0 & k<0 \\ 1 / k_{0}, & 0 \leq k \leq k_{0} \\ 0, & k>k_{0} \end{array}\right.\) This represents an equal combination of all wave numbers between 0 and \(k_{0}\). Thus \(\psi(x)\) represents a particle with average wave number \(k_{0} / 2\), with a total spread or uncertainty in wave number of \(k_{0}\). We will call this spread the width \(w_{k}\) of B(k), so \(w_{k}=k_{0}\). (b) Graph B(k) versus k and \(\psi(x)\) versus x for the case \(k_{0}=2 \pi / L\), where L is a length. Locate the point where \(\psi(x)\) has its maximum value and label this point on your graph. Locate the two points closest to this maximum (one on each side of it) where \(\psi(x)=0\), and define the distance along the x-axis between these two points as \(w_{x}\), the width of \(\psi(x)\). Indicate the distance \(w_{x}\) on your graph. What is the value of \(w_{x}\) if \(k_{0}=2 \pi / L\). (c) Repeat part (b) for the case \(k_0=\pi/L\). (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 \(w_{p} w_{x}\) for each of the cases \(k_{0}=2 \pi / L\) and \(k_0=\pi/L\). Discuss your results in light of the Heisenberg uncertainty principle.

CALC Show that the wave function is a solution of Eq. (40.23) for a particle of mass m, in a region where the potential energy is a constant Find an expression for k, and relate it to the particles momentum and to its de Broglie wavelength.

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 \(U(x)=0 \text {, for } x<0 \text {, and } U(x)=U_{0}<E\), 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 constant B and C in terms of \(k_{1}, k_{2}, \text { and } A\)

Let \(\Delta E_{n}\) be the energy difference between the adjacent energy levels \(E_{n} \text { and } E_{n+1}\) for a particle in a box. The ratio \(R_{n}=\Delta E_{n} / E_{n}\) compares the energy of a level to the energy separation of the next higher energy level. (a) For what value of n is \(R_{n}\) largest, and what is this largest \(R_{n}\)? (b) What does \(R_{n}\) approach as n becomes very large? How does this result compare to the classical value for this quantity? Text Transcription: Delta E_n E_n and E_n+1 R_n=Delta E_n/E_n R_n

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?

A particle is in the ground level of a box that extends from to (a) What is the probability of finding the particle in the region between 0 and Calculate this by integrating where is normalized, from to (b) What is the probability of finding the particle in the region to (c) How do the results of parts (a) and (b) compare? Explain. (d) Add the probabilities calculated in parts (a) and (b). (e) Are your results in parts (a), (b), and (d) consistent with Fig. 40.12b? Explain

CALC What is the probability of finding a particle in a box of length in the region between and when the particle is in (a) the ground level and (b) the first excited level? (Hint: Integrate where is normalized, between and ) (c) Are your results in parts (a) and (b) consistent with Fig. 40.12b? Explain

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 \(|\psi|^2\ dx\) that the particle will be found in the interval x to x + dx for (a) x = L/4; (b) x = L/2; (c) x = 3L/4.

Repeat Problem 40.56 for a particle in the first excited level.

A particle is confined within a box with perfectly rigid walls at x = 0 and x = L. Although the magnitude of the instantaneous force exerted on the particle by the walls is infinite and the time over which it acts is zero, the impulse (that involves a product of force and time) is both finite and quantized. Show that the impulse exerted by the wall at x = 0 is \((n h / L) \hat{\imath}\) and that the impulse exerted by the wall at x = L is \(-(n h / L) \hat{\imath}\). (Hint: You may wish to review Section 8.1.)

CALC A fellow student proposes that a possible wave function for a free particle with mass (one for which the potential-energy function is zero) is where is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schrdinger equation for if the energy is that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schrdinger equation for with the same energy as in part (b). (d) Explain why the proposed wave function is nonetheless not an acceptable solution of the Schrdinger equation for a free particle. (Hint: What is the behavior of the function at It is in fact impossible for a free particle (one for which to have an energy less than zero.

The penetration distance in a finite potential well is the distance at which the wave function has decreased to of the wave function at the classical turning point: The penetration distance can be shown to be The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find for an electron having a kinetic energy of 13 eV in a potential well with (b) Find for a proton trapped in a -deep potential well

CALC (a) For the finite potential well of Fig. 40.13, what relationships among the constants and of Eq. (40.38) and and of Eq. (40.40) are obtained by applying the boundary condition that be continuous at and at (b) What relationships among and are obtained by applying the boundary condition that be continuous at and at

An electron with initial kinetic energy encounters a square potential barrier with height What is the width of the barrier if the electron has a 0.10% probability of tunneling through the barrier?

A particle with mass and total energy tunnels through a square barrier of height and width When the transmission coefficient is not much less than unity, it is given by where is the hyperbolic sine of (a) Show that if this expression for approaches Eq. (40.42). (b) Explain why the restriction in part (a) implies either that the barrier is relatively wide or that the energy is relatively low compared to (c) Show that as the particles incident kinetic energy approaches the barrier height approaches where is the wave number of the incident particle. (Hint: If then

P A harmonic oscillator consists of a 0.020-kg mass on a spring. Its frequency is 1.50 Hz, and the mass has a speed of as it passes the equilibrium position. (a) What is the value of the quantum number for its energy level? (b) What is the difference in energy between the levels and Is this difference detectable?

For small amplitudes of oscillation the motion of a pendulum is simple harmonic. For a pendulum with a period of 0.500 s, find the ground-level energy and the energy difference between adjacent energy levels. Express your results in joules and in electron volts. Are these values detectable?

Some 164.9-nm photons are emitted in a \(\Delta n=1\) transition within a solid-state lattice. The lattice is modeled as electrons in a box having length 0.500 nm. What transition corresponds to the emitted light?

Show that for \(\psi(x)\) given by Eq. (40.47), the probability distribution function has a maximum at x = 0. Text Transcription: psi(x)

CALC (a) Show by direct substitution in the Schrdinger equation for the one-dimensional harmonic oscillator that the wave function where is a solution with energy corresponding to in Eq. (40.46). (b) Find the normalization constant (c) Show that the probability density has a minimum at and maxima at corresponding to the classical turning points for the ground state

(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 wavelength 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 for for and for (Fig. P40.70). Consider a particle with mass and kinetic energy that is trapped in the well. (a) The boundary condition at the infinite wall is What must the form of the function for be in order to satisfy both the Schrdinger equation and this boundary condition? (b) The wave function must remain finite as What must the form of the function for be in order to satisfy both the Schrdinger equation and this boundary condition at infinity? (c) Impose the boundary conditions that and are continuous at Show that the energies of the allowed levels are obtained from solutions of the equation where and

Section 40.2 considered a box with walls at and Now consider a box with width but centered at so that it extends from to (Fig. P40.71). Note that this box is symmetric about (a) Consider possible wave functions of the form Apply the boundary conditions at the wall to obtain the allowed energy levels. (b) Another set of possible wave functions are functions of the form Apply the boundary conditions at the wall to obtain the allowed energy levels. (c) Compare the energies obtained in parts (a) and (b) to the set of energies given in Eq. (40.31). (d) An odd function satisfies the condition and an even function satisfies Of the wave functions from parts (a) and (b), which are even and which are odd?

CALC The WKB Approximation. It can be a challenge to solve the Schrdinger equation for the bound- state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the WKB approximation (named for the physicists Gregor Wentzel, Hendrik Kramers, and Lon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum of a quantum-mechanical particle is (ii) The magnitude of momentum is related to the kinetic energy by the relationship (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy for a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point: where is the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate can be written as Thus we envision a quantum-mechanical particle in a potential well as being like a free particle, but with a wavelength that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a classical turning point, since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amplitude moves back and forth between the points and each of these is a classical turning point, since there the potential energy equals the total energy In the WKB expression for what is the wavelength at a classical turning point? (d) For a particle in a box with length the walls of the box are classical turning points (see Fig. 40.8). Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig. 40.10), so that and hence where [Note that this is a restatement of Eq. (40.29).] The WKB scheme for finding the allowed bound-state energy levels of an arbitrary potential well is an extension of these observations. It demands that for an allowed energy there must be a half-integer number of wavelengths between the classical turning points for that energy. Since the wavelength in the WKB approximation is not a constant but depends on the number of wavelengths between the classical turning points and for a given value of the energy is the integral of between those points: Using the expression for you found in part (a), show that the WKB condition for an allowed bound-state energy can be written as (e) As a check on the expression in part (d), apply it to a particle in a box with walls at and Evaluate the integral and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.31). (Hint: Since the walls of the box are infinitely high, the points and are classical turning points for any energy Inside the box, the potential energy is zero.) (f ) For the finite square well shown in Fig. 40.13, show that the WKB expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. (Hint: Assume Then the classical turning points are at and ) This shows that the WKB approximation does a poor job when the potential-energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potential-energy function changes gradually and the WKB approximation is much more useful.

The WKB approximation (see Challenge Problem 40.72) can be used to calculate the energy levels for a harmonic oscillator. In this approximation, the energy levels are the solutions to the equation Here is the energy, is the potential-energy function, and x = a and are the classical turning points (the points at which is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for a harmonic oscillator with energy and force constant (b) Carry out the integral in the WKB approximation and show that the energy levels in this approximation are where and (Hint: Recall that A useful standard integral is where arcsin denotes the inverse sine function. Note that the integrand is even, so the integral from to is equal to twice the integral from 0 to (c) How do the approximate energy levels found in part (b) compare with the true energy levels given by Eq. (40.46)? Does the WKB approximation give an underestimate or an overestimate of the energy levels?

CALC 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 particle is a low-energy bound state of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the particle is an excited state of this same quarkantiquark combination, with a rest energy of 3686 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is , where is a positive constant and 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 Here is the energy, is the potential-energy function, and and are the classical turning points (the points at which is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for the potential and for an energy (b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are given by (Hint: The integrand is even, so the integral from to is equal to twice the integral from 0 to ) (c) Does the difference in energy between successive levels increase, decrease, or remain the same as 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?

Problem 33E An electron with initial kinetic energy 6.0 eV encounters a barrier with height 11.0 eV. What is the probability of tunneling if the width of the barrier is (a) 0.80 nm and (b) 0.40 nm?

An electron with initial kinetic energy 5.0 eV encounters a barrier with height U0 and width 0.60 nm. What is the transmission coefficient if (a) U0 = 7.0 eV; (b) U0 = 9.0 eV; (c) U0 = 13.0 eV?

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 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 wavelength 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? Equation Transcription: Text Transcription: E_n=n^2h^2 over 8mL^2 mv=h/lambda 1 over 2 mv^2

Consider a potential well defined as \(U(x)=\infty\) for \(x<0\) for \(0<x<L\), and \(U(x)=U_{0}>0\) for \(x>L\) (Fig. P40.70). Consider a particle with mass 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 Schrödinger 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 Schrödinger 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 \(\kappa=\sqrt{2 m\left(U_{0}-E\right) / \hbar}\). Equation Transcription: ? ? ? ? ? Text Transcription: U(x)=infinity x<0 U(x)=0 0<x<L U(x)=U_0>0 x>L E<U_0 (x=0) psi(0)=0 psi(x) 0<x<L x rt arow infinity psi (x) x>L psi dpsi/dx x=L kcotkL=-kappa k=sqrt 2me/hbar kappa=2m(U_0-E/hbar) U(x) infinity U0

Problem 1DQ If quantum mechanics replaces the language of Newtonian mechanics, why don’t we have to use wave functions to describe the motion of macroscopic bodies such as baseballs and cars?

Problem 1E An electron is moving as a free particle in the –x-direction with momentum that has magnitude 4.50 × 10-24 kg ? m/s. What is the one-dimensional time-dependent wave function of the electron?

Problem 2DQ A student remarks that the relationship of ray optics to the more general wave picture is analogous to the relationship of Newtonian mechanics, with well-defined particle trajectories, to quantum mechanics. Comment on this remark.

Consider the free-particle wave function of Example 40.1. Let \(k_{2}=3 k_{1}=3 k\) At \(t=0\) the probability distribution function \(|\Psi(x, t)|^{2}\) has a maximum at . (a) What is the smallest positive value of for which the probability distribution function has a maximum at time \(t=2 \pi / \omega\), where \(\omega=h k^{2} / 2 m\) (b) From your result in part (a), what is the average speed with which the probability distribution is moving in the -direction? Compare your result to the expression \(v_{a v}=\left(\omega_{2}-\omega_{1}\right) /\left(k_{2}-k_{1}\right)\) from Example 40.1. Equation transcription: Text transcription: k{2}=3 k{1}=3 k t=0 |Psi(x, t)|^{2} t=2 pi / omega omega=h k^{2} 2 m v{a v}=(omega{2}-omega_{1}) /(k{2}-k{1})

A free particle moving in one dimension has wave function \(\Psi(x, t)=A\left[e^{i(k x-\omega t)}-e^{l(2 k x-4 \omega t)}\right]\) where and are positive real constants. (a) At \(t=0\) what are the two smallest positive values of for which the probability function \(|\Psi(x, f)|^{2}\) is a maximum? (b) Repeat part (a) for time \(t=2 \pi / \omega\). (c) Calculate \(v_{a v}\) as the distance the maxima have moved divided by the elapsed time. Compare your result to the expression \(v_{a v}=\left(\omega_{2}-\omega_{1}\right) /\left(k_{2}-k_{1}\right)\) from Example 40.1. Equation transcription: Text transcription: \Psi(x, t)=A[e^{i(k x-omega t)}-e^{l(2 k x-4 \omega t)}] t=0 |\Psi(x, f)|^{2} t=2 pi / omega v{a v} v{a v}=(omega{2}-omega{1}) (k{2}-k{1})

Why must the wave function of a particle be normalized?

Consider the free particle of Example 40.1. Show that \(v_{\mathrm{av}}=\left(\omega_{2}-\omega_{1}\right) /\left(k_{2}-k_{1}\right)\) can be written as \(p_{\mathrm{av}}=\left(\hbar k_{2}+\hbar k_{1}\right) / 2\). Text Transcription: v_av=(omega_2-omega_1)/(k_2-k_1) p_av=(hbar k_2+hbar k_1)/2

If a particle is in a stationary state, does that mean that the particle is not moving? If a particle moves in empty space with constant momentum \(\vec{p}\) and hence constant energy \(E=p^{2} / 2 m\), is it in a stationary state? Explain your answers.\ Equation transcription: Text transcription: vec{p} E=p^{2} / 2 m

Problem 5E Consider a wave function given by ?(x) = A sin kx, where k = 2?/? 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.

For the particle in a box, we chose \(k=n \pi / L \text { with } n=1,2,3, \ldots\) to fit the boundary condition that \(\psi=0\) at x = L. However, n = 0, -1, -2, -3, . . . also satisfy that boundary condition. Why didn’t we also choose those values of n?

Problem 6E Compute |?|2 for ? = ? sin ?t, where c is time independent and v is a real constant. Is this a wave function for a stationary state? Why or why not?

If ? is normalized, what is the physical significance of the area under a graph of |?|2 versus x between x1 and x2? What is the total area under the graph of |?|2 when all x are included? Explain.

For a particle in a box, what would the probability distribution function |?|2 look like if the particle behaved like a classical (Newtonian) particle? Do the actual probability distributions approach this classical form when n is very large? Explain.

A particle is described by a wave function \(\psi(x) = Ae^{-\alpha x^{2}}\) where A and \(\alpha\) are real, positive constants. If the value of 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. Text Transcription: psi(x) = Ae^-a^x^2 a

Problem 9DQ In Chapter 15 we represented a standing wave as a superposition of two waves traveling in opposite directions. Can the wave functions for a particle in a box also be thought of as a combination of two traveling waves? Why or why not? What physical interpretation does this representation have? Explain.

CALC Linear Combinations of Wave Functions. Let \(\Psi_{1}\) and \(\Psi_{2}\) be two solutions of Eq. with the same energy . Show that \(\Psi=B \Psi_{1}+C \Psi_{2}\) is also a solution with energy , for any values of the constants and . Equation transcription: Text transcription: Psi{1} Psi{2} Psi=B Psi{1}+C Psi{2}

A particle in a box is in the ground level. What is the probability of finding the particle in the right half of the box? (Refer to Fig. 40.12, but don’t evaluate an integral.) Is the answer the same if the particle is in an excited level? Explain.

CALC A particle moving in one dimension (the -axis) is described by the wave function \(\Psi(x)=\left\{A e^{b x} \text { forx } \geq A e^{b x} \text { for } x<0\right.\) where \(b=2.00 \mathrm{~m}^{-1}, A>0\), and the -axis points toward the right. (a) Determine 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 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 and . Equation transcription: Text transcription: Psi(x)={A e^{b x} text { forx } \geq A e^{b x} text { for } x<0. b=2.00{~m}^{-1}, A>0

The wave functions for a particle in a box (see Fig. 40.12a) are zero at certain points. Does this mean that the particle can’t move past one of these points? Explain.

Problem 11E Ground-Level Billiards. (a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (m = 0.20 kg) and the box has a width of 1.3 m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.) (b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other? (c) What is the difference in energy between the n = 2 and n = 1 levels? (d) Are quantum-mechanical effects important for the game of billiards?

For a particle confined to an infinite square well, is it correct to say that each state of definite energy is also a state of definite wavelength? Is it also a state of definite momentum? Explain. (Hint: Remember that momentum is a vector.)

Problem 12E 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 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 m.

Problem 13DQ For a particle in a finite potential well, is it correct to say that each bound state of definite energy is also a state of definite wavelength? Is it a state of definite momentum? Explain.

Problem 13E Find the width L of a one-dimensional box for which the ground-state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom.

In Fig. 40.12b The probability function is zero at the points x = 0 and x = L, the "walls" of the box. Does this mean that the particle never strikes the walls? Explain.

Problem 14E When a hydrogen atom undergoes a transition from the n = 2 to the n = 1 level, a photon with ? = 122 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.)

Problem 15E A certain atom requires 3.0 eV of energy to excite an electron from the ground level to the first excited level. Model the atom as an electron in a box and find the width L of the box.

Problem 16E 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 excited state?

CALC Show that the time-dependent wave function given by Eq. (40.35) is a solution to the one-dimensional Schrödinger equation, Eq. (40.23).

Figure a shows that the higher the energy of a bound state for a finite potential well, the more the wave function extends outside the well (into the intervals \(x<0\) and \(x>L\)). Explain why this happens. Equation transcription: Text transcription: x<0 x>L

Recall that |\(\Psi\)|\(^{2} dx\) is the probability of finding the particle that has normalized wave function \(\Psi (x)\) in the interval x to x + dx. Consider a particle in a box with rigid walls at x = 0 and x = L. Let the particle be in the ground level and use \(\Psi_{x}\) as given in Eq. (40.35). (a) For which values of x, if any, in the range from 0 to L is the probability of finding the particle zero? (b) For which values of x is the probability highest? (c) In parts (a) and (b) are your answers consistent with Fig. 40.12? Explain.

Repeat Exercise 40.18 for the particle in the first excited level.

CALE (a) Show that \(\psi=A \sin k x\) is a solution to Eq. (40.25) if \(k=\sqrt{2 m E} / \hbar\). (b) Explain why this is an acceptable wave function for a particle in a box with rigid walls at \(x=0\)and \(x=L\) only if k is an integer multiple of \(\pi / L\). Equation Transcription: ? Text Transcription: Psi - Asinkx k = sqrt 2mE/h x = 0 x = L pi/L

Qualitatively, how would you expect the probability for a particle to tunnel through a potential barrier to depend on the height of the barrier? Explain.

CALC (a) Repeat Exercise 40.20 for \(\psi=A \cos k x\). (b) Explain why this cannot be an acceptable wave function for a particle in a box with rigid walls at \(x=0\) and \(x=L\) no matter what the value of k. Equation Transcription: Text Transcription: Psi = Acoskx x = 0 x = L

The wave function shown in Fig. 40.20 is nonzero for both x < 0 and x > L. Does this mean that the particle splits into two parts when it strikes the barrier, with one part tunneling through the barrier and the other part bouncing off the barrier? Explain.

Problem 22E (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 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.

The probability distributions for the harmonic oscillator wave functions (see Figs. 40.27 and 40.28) begin to resemble the classical (Newtonian) probability distribution when the quantum number becomes large. Would the distributions become the same as in the classical case in the limit of very large Explain.

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 level; (b) the level; (c) the level? In each case how does the wavelength compare to the width of the box? Equation transcription: Text transcription: 3.0 times 10^{-10} m

In Fig. 40.28, how does the probability of finding a particle in the center half of the region -A < x < A compare to the probability of finding the particle in the outer half of the region? Is this consistent with the physical interpretation of the situation?

Problem 24E CALC 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 benormalized? (b) Is the wave function ?(x) = eax, where a is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function ?(x) = Ae-bx, where A and b are positive real numbers, is confined to the range x ? 0, determine A(including its units) so that the wave function is normalized.

Problem 25 DQ Compare the allowed energy levels for the hydrogen atom, the particle in a box, and the harmonic oscillator. What are the values of the quantum number n for the ground level and the second excited level of each system?

CALC (a) Show that \(\psi=A \sin k x\), where k is a real (not complex) constant, is not a solution of Eq. 40.23 for \(U=U_{0}\) and \(E<U_{0}\). (b) Is this \(\psi\) a solution for \(E>U_{0}\)?

Sketch the wave function for the potential-energy well shown in Fig. Q40.26 when \(E_{1}\) is less than \(U_{0}\) and when \(E_{3}\) is greater than \(U_{0}\).

An electron is moving past the square well shown in Fig. 40.13. The electron has energy \(E = 3U_{0}\). What is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?

An electron is bound in a square well of depth U0 = 6E1-IDW. What is the width of the well if its ground-state energy is 2.00 eV?

Problem 28E An electron is bound in a square well of width 1.50 nm and depth U0 = 6E1-IDW. If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well?

CALC Calculate \(d^{2} \Psi / d x^{2}\) for the wave function of Eq. (40.38), and show that the function is a solution of Eq. (40.37).

Problem 30E An electron is bound in a square well with a depth equal to six times the ground-level energyE1–IDW of an infinite well of the same width. The longest-wavelength photon that is absorbed by the electron has a wavelength of 400.0 nm. Determine the width of the well.

Problem 31E A proton is bound in a square well of width 4.0 fm = 4.0 × 10-15 m. The depth of the well is six times the ground-level energy E1-IDW of the corresponding infinite well. If the proton makes a transition from the level with energy E1 to the level with energy E3 by absorbing a photon, find the wavelength of the photon.

Alpha Decay. In a simple model for a radioactive nucleus, an alpha particle (\(m=6.64 \times 10^{-27} \mathrm{~kg}\)) is trapped by a square barrier that has width 2.0 fm and height 30.0 MeV. (a) What is the tunneling probability when the alpha particle encounters the barrier if its kinetic energy is 1.0 MeV below the top of the barrier (Fig. E40.32)? (b) What is the tunneling probability if the energy of the alpha particle is 10.0 MeV below the top of the barrier?

An electron is moving past the square barrier shown in Fig. 40.19, but the energy of the electron is greater than the barrier height. If \(E=2 U_{0}\), what is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?

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 \(3.0 \times 10^{-3}\)? 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 37E (a) An electron with initial kinetic energy 32 eV encounters a square barrier with height 41 eV and width 0.25 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?

CALC Show that \(\Psi(x)\) given by Eq. (40.47) is a solution to Eq. (40.44) with energy \(E_{0}=h \omega / 2\). Equation transcription: Text transcription: Psi(x) E{0}=h omega / 2

A wooden block with mass 0.250 kg is oscillating on the end of a spring that has force constant 110 N/m. Calculate the ground-level energy and the energy separation between adjacent levels. Express your results in joules and in electron volts. Are quantum effects important?

A harmonic oscillator absorbs a photon of wavelength 8.65 \(\times\) 10\(^{-6}\) 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 41E Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8 ?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 × 10-26 kg, find the force constant.

In Section it is shown that for the ground level of a harmonic oscillator, \(\Delta x \Delta p_{x}=\hbar / 2\). Do a similar analysis for an excited level that has quantum number n. How does the uncertainty product \(\Delta x \Delta p_{x}\) depend on n?

For the ground-level harmonic oscillator wave function \(\Psi(x)\) given in Eq. (40.47), \(\left|\Psi^{\prime}\right|^{2}\) has a maximum at . (a) Compute the ratio of \(\left|\Psi^{\prime}\right|^{2}\) at \(x=+A\) to \(\left|\Psi^{\prime}\right|^{2}\) at , where is given by Eq. (40.48) with for the ground level. (b) Compute the ratio of \(\left|\Psi^{\prime}\right|^{2}\) at \(x=+2 A\) to \(\left|\Psi^{\prime}\right|^{2}\) at In each case is your result consistent with what is shown in Fig. Equation transcription: Text transcription: \Psi(x) \|Psi^{\prime}|^{2} x=+A x=+2 A

For the sodium atom of Example , find (a) the ground-state energy, (b) the wavelength of a photon emitted when the to transition occurs; (c) the energy difference for any \(\Delta n=1\) transition. Equation transcription: Text transcription: Delta n=1

The discussion in Section 40.1 shows that the wave function \(\Psi=\psi e^{-i \omega t}\) is a stationary state, where \(\Psi\) is time independent and \(\omega\) is a real (not complex) constant. Consider the wave function \(\Psi=\psi_{1} e^{-i \omega_{1} t}+\psi_{2} e^{-i \omega_{2} t}\), where \(\Psi_{1}\) and \(\Psi_{2}\) are different time- independent functions and \(\omega_{1}\) and \(\omega_{2}\) are different real constants. Assume that \(\Psi_{1}\) and \(\Psi_{2}\) are real-valued functions, so that \(\Psi_{1}^{*}=\Psi_{1}\) and \(\Psi_{2}^{*}=\Psi_{2}\) Is this \(\Psi\) a wave function for a stationary state? Why or why not?

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.

CALC Consider the wave packet defined by \(\psi(x)=\int_{0}^{\infty} B(k) \cos k x d k\) Let \(B(k)=e^{-a^{2} k^{2}}\). (a) The function B(k) has its maximum value at k=0. Let kh be the value of k at which B(k) has fallen to half its maximum value, and define the width of B(k) as wk=kh. In terms of ?, what is wk? (b) Use integral tables to evaluate the integral that gives ?(x). For what value of x is \(\psi\)(x) maximum? (c) Define the width of \(\psi\)(x) as wx=xh, where xh is the positive value of x at which \(\psi\)(x) has fallen to half its maximum value. Calculate wx in terms of ?. (d) The momentum P is equal to hk/2? so the width of B in momentum is wp=hwk/2? Calculate the product wpwx and compare it to the Heisenberg uncertainty principle.

(a) Using the integral in Problem 40.48, determine the wave function \(\psi(x)\) for a function B(k) given by \(B(k)= \begin{cases}0 & k<0 \\ 1 / k_{0}, & 0 \leq k \leq k_{0} \\ 0, & k>k_{0}\end{cases}\) This represents an equal combination of all wave numbers between 0 and \(k_{0}\). Thus \(\psi(x)\) represents a particle with average wave number \(k_{0} / 2\), with a total spread or uncertainty in wave number of \(k_{0}\). We will call this spread the width \(w_{k}\) of B(k), so \(w_{k}=k_{0}\). (b) Graph B(k) versus k and \(\psi(x)\) versus x for the case \(k_{0}=2 \pi / L\) where L is a length. Locate the point where \(\psi(x)\) has its maximum value and label this point on your graph. Locate the two points closest to this maximum (one on each side of it) where \(\psi(x)=0\), and define the distance along the x-axis between these two points as \(w_{x}\), the width of \(\psi(x)\). Indicate the distance \(w_{x}\) on your graph. What is the value of \(w_{x}\) if \(k_{0}=2 \pi / L\)? (c) Repeat part (b) for the case \(k_{0}=\pi / L\). (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 \(w_{p} w_{x}\) for each of the cases \(k_{0}=2 \pi / L\) and \(k_{0}=\pi / L\). Discuss your results in light of the Heisenberg uncertainty principle. Text Transcription: psi(x) B(k)= {0 k<0 1/k_0, 0 leq k leq k_0 0, k>k_0 k_0 k_0/2 w_k k_0 = 2 pi/L psi(x)=0 w_x k_0 = pi/L hk/2 pi w_p = hw_k/2 pi w_pw_x

CALC Show that the wave function \(\Psi(x)=A e^{i k x}\) is a solution of Eq. (40.23) for a particle of mass , in a region where the potential energy is a constant \(U{0}<E\). Find an expression for k, and relate it to the particle's momentum and to its de Broglie wavelength.

CALC Wave functions like the one in Problem can represent free particles moving with velocity in the -direction. Consider a beam of such particles incident on a potential-energy step , for , and , for The wave function for is \(\Psi(x)=A e^{i k x}+B e^{-i k_{1} x}\), representing incident and reflected particles, and for 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 to find the constants and in terms of \(k_{1}, k_{2}\), and . Equation transcription: Text transcription: Psi(x)=A e^{i k x}+B e^{-i k{1} x} Psi(x)=C e^{i k{2} x} Psi k{1}, k{2}

Problem 52P Let ?En be the energy difference between the adjacent energy levels En and En+1 for a particle in a box. The ratio Rn = ?En/En compares the energy of a level to the energy separation of the next higher energy level. (a) For what value of n is Rn largest, and what is this largest Rn? (b) What does Rn approach as n becomes very large? How does this result compare to the classical value for this quantity?

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?

A particle is in the ground level of a box that extends from x = 0 x = L. (a) What is the probability of finding the particle in the region between 0 and L/4? Calculate this by integrating \(|\psi(x)|^2\ dx\), where \(\psi\) is normalized, from is normalized, from x = 0 to x = L/4. (b) What is the probability of finding the particle in the region x = L/4 to x = L/2? (c) How do the results of parts (a) and (b) compare? Explain. (d) Add the probabilities calculated in parts (a) and (b). (e) Are your results in parts (a), (b), and (d) consistent with Fig. 40.12b? Explain. Text Transcription: |psi(x)|^2 dx psi

CALC What is the probability of finding a particle in a box of length L in the region between x = L/4 and x = 3L/4 when the particle is in (a) the ground level and (b) the first excited level? (Hint: Integrate \(|\psi(x)|^{2} d x\), where \(\psi\) is normalized, between L/4 and 3L/4.) (c) Are your results in parts (a) and (b) consistent with Fig. 40.12b? Explain.

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 + dx for (a) x = L/4; (b) x = L/2; (c) x = 3L/4. Text Transcription: |psi|^2 dx

Problem 57P Repeat Problem 40.48 for a particle in the first excited level. 40.48 .. 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 |?|2dx that the particle will be found in the interval x to x +dx for (a) x = L/4; (b) x = L/2; (c) x = 3L/4.

CP A particle is confined within a box with perfectly rigid walls at and . Although the magnitude of the instantaneous force exerted on the particle by the walls is infinite and the time over which it acts is zero, the impulse (that involves a product of force and time) is both finite and quantized. Show that the impulse exerted by the wall at is \((n h / L) \vec{i}\) and that the impulse exerted by the wall at is \((n h / L) \vec{i}\). (Hint: You may wish to review Section 8.1.) Equation transcription: Text transcription: (n h / L) \vec{i}

CALC A fellow student proposes that a possible wave function for a free particle with mass (one for which the potential-energy function is zero) is \(\Psi(x)=\left\{e^{+k x}, x<0 e^{-k x}, x \geq 0\right.\)where is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schrödinger equation for if the energy is \(E=h^{2} k^{2} / 2 m-\) that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schrödinger 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 Schrödinger equation for a free particle. (Hint: What is the behavior of the function at ) It is in fact impossible for a free particle (one for which ) to have an energy less than zero. Equation transcription: Text transcription: \Psi(x)=\left\{e^{+k x}, x<0 e^{-k x}, x \geq 0\right. E=h^{2} k^{2} / 2 m- x \geq 0

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 \(\eta=\frac{h}{\sqrt{2 m\left(U_{0}-E\right)}}\) The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find for an electron having a kinetic energy of in a potential well with \(U_{0}=20 \mathrm{eV}\). (b) Find \(\eta\) for a 20.0-MeV proton trapped in a 30.0-MeV-deep potential well. Equation transcription: Text transcription: eta 1 / e Psi(x=L+eta)=\frac{1}{e} \Psi(L) eta=\frac{h}{sqrt{2 m(U_{0}-E)}} U_{0}=20{eV}

CALC (a) For the finite potential well of Fig. 40.13, what relationships among the constants A and B of Eq. (40.38) and C and D of Eq. (40.40) are obtained by applying the boundary condition that \(\psi\) be continuous at x = 0 and at x = L? (b) What relationships among A, B, C, and D are obtained by applying the boundary condition that \(d \psi / d x\) be continuous at x = 0 and at x = L?

Problem 64P An electron with initial kinetic energy 5.5 eV encounters a square potential barrier with height 10.0 cV. What is the width of the barrier if the electron has a 0.10% probability of tunnelling through the barrier?

A particle with mass m and total energy E tunnels through a square barrier of height \(U_{0}\) and width L. When the transmission 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 \(\kappa L\ \gg\ 1\) in part (a) implies either that the barrier is relatively wide or that 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 \(|z|\ \ll\ 1\), then \(\sinh z\ \approx\ z\).)

A harmonic oscillator consists of a 0.020-kg mass on a spring. Its frequency is 1.50 Hz, and the mass has a speed of 0.360 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?

Problem 65P For small amplitudes of oscillation the motion of a pendulum is simple harmonic. For a pendulum with a period of 0.500 s, find the ground-level energy and the energy difference between adjacent energy levels. Express your results in joules and in electron volts. Are these values detectable?

Problem 66P Some 164.9-nm photons are emitted in a ?n = 1 transition within a solid-state lattice. The lattice is modeled as electrons in a box having length 0.500 nm. What transition corresponds to the emitted light?

CALC Show that for \(\Psi(x)\) given by Eq. (40.47), the probability distribution function has a maximum at . Equation transcription: Text transcription: \Psi(x)

CALC (a) Show by direct substitution in the Schrödinger equation for the one-dimensional harmonic oscillator that the wave function \(\Psi_{1}(x)=A_{1} x e^{-\epsilon^{4} x^{2} / 2}\), where \(x^{2}=m \omega / h\), is a solution with energy corresponding to in Eq. (40.46). (b) Find the normalization constant \(A_{1}\). (c) Show that the probability density has a minimum at and maxima at \(x=\pm 1 / x\), corresponding to the classical turning points for the ground state . Equation transcription: Text transcription: \Psi_{1}(x)=A_{1} x e^{-\epsilon^{4} x^{2} / 2} x^{2}=m \omega / h A_{1} x=\pm 1 / x

Section 40.2 considered a box with walls at x = 0 and x = L. Now consider a box with width L but centered at x = 0, so that it extends from x = -L/2 to x = +L/2 (Fig. P40.71). Note that this box is symmetric about x = 0. (a) Consider possible wave functions of the form \(\psi(x)=A \sin k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels. (b) Another set of possible wave functions are functions of the form \(\psi(x)=A \cos k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels. (c) Compare the energies obtained in parts (a) and (b) to the set of energies given in Eq. (40.31). (d) An odd function f satisfies the condition f(x) = -f(-x) and an even function g satisfies g(x) = g(-x) Of the wave functions from parts (a) and (b), which are even and which are odd?

CALC The WKB Approximation. It can be a challenge to solve the Schrödinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the approximation (named for the physicist Gregor Wentzel. Hendrik Kramers, and Léon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum \(p\) of a quantum-mechanical particle is \(p=h / \lambda\). (ii) The magnitude of momentum is related to the kinetic energy \(K\) by the relationship \(K=p^{2} / 2 m\). (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy for a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point:\(E=K+U(x)\), where \(x\) is the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate can be written as \(\lambda(x)=\frac{h}{\sqrt{2 m[E-U(x)}}\) Thus we envision a quantum-mechanical particle in a potential well \(U(x)\) as being like a free particle, but with a wavelength \(\lambda(x)\) that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where \(E=U(x)\), Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a classical turning point, since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amplitude A moves back and forth between the points \(x=-A\) and \(x=+A\); each of these is a classical turn. ing point, since there the potential energy \(\frac{1}{2}k^{\prime}x^2\) equals the total energy \(\frac{1}{2}k^{\prime}A^2\). In the WKB expression for \(\lambda(x)\), what is the wavelength at a classical turning point? (d) For a particle in a box with length \(L\), the walls of the box are classical turning points (see Fig. ). Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig. ), so that \(L=(n / 2) \lambda\) and hence \(L / \lambda=n / 2\), where \(n=1,2,3, \ldots\) [Note that this is a restatement of Eq. (40.29).] The WKB scheme for finding the allowed bound-state energy levels of an arbitrary potential well is an extension of these observations. It demands that for an allowed energy \(E\), there must be a half-integer number of wavelengths between the classical turning points for that cnergy. Since the wavelength in the WKB approximation is not a constant but depends on , the number of wavelengths between the classical turning points \(a\) and \(b\) for a given value of the energy is the integral of \(1 / \lambda(x)\) between those points \(\int_{\alpha}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2}(n=1,2,3, \ldots)\) Using the expression for \(\lambda(x)\) you found in part (a), show that the WKB condition for an allowed bound-state energy can be written ass \(\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2}(n=1,2,3, \ldots)\) (e) As a check on the expression in part (d), apply it to a particle in ? box with walls at \(x=0\) and \(x=L\) Evaluate the integral and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.31). (Hint: Since the walls of the box are infinitely high, the points \(x=0\) and \(x=L\) are classical turning points for any energy \(E\). Inside the box, the potential energy is zero.) (f) For the finite square well shown in Fig. , show that the WKB expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. (Hint: Assume \(E<U_{0}\). Then the classical turning points are at \(x=0\) and \(x=L\) ) This shows that the WKB approxi: mation does a poor job when the potential-energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potential-energy function changes gradually and the WKB approximation is much more useful. Equation Transcription: Text Transcription: p p=h/lamda K=p^2/2m K E=K+U(x) x lamda (x) = h/ sqrt 2m[E - U(x) U(x) lamda(x) E=U(x) x=-A x=+A 1/2k'x^2 1/2k'A^2 L L=(n/2)lamda L/lamda=n/2 n=1,2,3,... E a b 1/lamda(x) integral_alpha^b dx/alpha(x)=n/2 (n=1,2,3,...) integral_a^b sqrt 2m[E-U(x)]dx=nh/2 (n=1,2,3,...) x=0 x=L E<U_0

CALC The WKB approximation (see Challenge Problem 40.72) can be used to calculate the energy levels for a harmonic oscillator. In this 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 Neatoerian kinetic energy would be zero). (a) Determine the classical turning points for a harmonic oscillator with energy E and force constant \(k^{\prime}\). (b) Carry out the integral in the WKB approximation and show that the energy levels in this approximation are \(E_{n}=\hbar \omega\), where \(\omega=\sqrt{k^{\prime} / m}\) and n = 1, 2, 3, …. (Hint. Recall that \(\hbar=h / 2 \pi\). A useful standard integral is \(\int \sqrt{A^{2}-x^{2}} d x=\frac{1}{2}\left[x \sqrt{A^{2}-x^{2}}+A^{2} \arcsin \left(\frac{x}{|A|}\right)\right]\) where arcsin denotes the inverse sine function. Note that the integrand is even, so the integral from -x to x is equal to twice the interval from 0 to x.) (c) How do the approximate energy levels found in part (b) compare with the true energy levels given by Eq. (40.46)? Does the WKB approximation give an underestimate or an overestimate of the energy levels?

CALC 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 ? particle is a low-energy bound stane of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the W(2S) particle is an excited state of this same quark-antiquark combination, with a rest energy of 3606 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is U(x)=N|x|, where A is a positive constant and x represents the distance between the quark and the antiquark. You can see 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 \(\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2}(n=1,2,3, \ldots)\) Here E is the energy. U(x) is the potential-energy function, and x=a and x=ab re 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 \(E_{n}=\frac{1}{2 m}\left(\frac{3 m A h}{4}\right)^{2 / 3} n^{2 / 3}(n=1,2,3, \ldots)\) (Hint: The integrand is even, so the integral from ?x to x is equal to twice the integral from 0 to x ). (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?