An electron is trapped in a 1.00-nm-wide rigid

The neutrons in a parallel beam, each having kinetic energy 0.030 eV, are directed through two slits 0.60 mm apart. How far apart will the interference peaks be on a screen

Pellets of mass 3.0 g are fired in parallel paths with speeds of 150 m/s through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of pellets?

A proton is traveling with a speed of (7.560 + 0.012) X 105 m/s. With what maximum precision can its position be ascertained?

An electron remains in an excited state of an atom for typically 10-8 s. What is the minimum uncertainty in the energy of the state (in eV)?

If an electrons position can be measured to a precision of 2.6 X 10 8 m, how precisely can its speed be known?

The lifetime of a typical excited state in an atom is about 10 ns. Suppose an atom falls from one such excited state and emits a photon of wavelength about 500 nm. Find the fractional energy uncertainty A E/E and wavelength uncertainty A A/A of this photon.

A radioactive element undergoes an alpha decay with a lifetime of 12 /as. If alpha particles are emitted with 5.5-keV kinetic energy, find the uncertainty A E/E in the particle energy.

A 12-g bullet leaves a rifle horizontally at a speed of 180 m/s. (a) What is the wavelength of this bullet? (b) If the position of the bullet is known to a precision of 0.65 cm (radius of the barrel), what is the minimum uncertainty in its vertical momentum?

An electron and a 140-g baseball are each traveling 95 m/s measured to a precision of 0.085%. Calculate and compare the uncertainty in position of each

What is the uncertainty in the mass of a muon (ra = 105.7 MeV/c2), specified in eV/c2, given its lifetime of 2.20 fi s?

A free neutron (ra = 1.67 X 10 27 kg) has a mean life of 900 s. What is the uncertainty in its mass (in kg)?

Use the uncertainty principle to show that if an electron were present in the nucleus (r 10-15 m), its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: Assume a particle can have energy as large as its uncertainty.]

An electron in the n = 2 state of hydrogen remains there on average about 10-8 s before jumping to the n = 1 state. (a) Estimate the uncertainty in the energy of the n = 2 state, (b) What fraction of the transition energy is this? (c) What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?

How accurately can the position of a 3.50-keV electron be measured assuming its energy is known to 1.00%?

In a double-slit experiment on electrons (or photons), suppose that we use indicators to determine which slit each electron went through (Section 38-2). These indicators must tell us the y coordinate to within d/2, where d is the distance between slits. Use the uncertainty principle to show that the interference pattern will be destroyed. [Note: First show that the angle 6 between maxima and minima of the interference pattern is given by \\ /d ,FIGURE 38-19 Problem 15.

Show that the superposition principle holds for the timedependent Schrodinger equation. That is, show that if ^ ( x , t) and 0 are solutions, then y4^r1(x, t) + B ^ 2(x, 0 is also a solution where A and B are arbitrary constants.

(a) Show that ( x , t) = Ae^kx~a*) is a solution to the time-dependent Schrodinger equation for a free particle [/(*) = U0= constant] but that ^(.x, t) = A cos(A:x - (ot) and W(x, t) = A sin k x cot) are not. (b) Show that the valid solution of part (a) satisfies conservation of energy if the de Broglie relations hold; A = h/p, w = E/h. That is, show that direct substitution into Eq. 38-7 gives

A free electron has a wave function if/(x) = A sin(2.0 X IO10*), where x is given in meters. Determine the electrons (a) wavelength, (b) momentum, (c) speed, and ((d) kinetic energy.

Write the wave function for (a) a free electron and (b) a free proton, each having a constant velocity v = 3.0 X 105 m/s.

Show that the uncertainty principle holds for a wave packet that is formed by two waves of similar wavelength A1 and A2. To do so, follow the argument leading up to Eq. 16-8, but use as the two waves fa = A s in k x and fa = A sin k2 x. Then show that the width of each wave packet is Ajc = 27r/(A:1 - k2) = 2it/ Ak (from t = 0.05 s to t = 0.15 s in Fig. 16-17). Finally, show that Ax Ap = h for this simple situation.

What is the minimum speed of an electron trapped in a 0.20-nm-wide infinitely deep square well?

Show that for a particle in a perfectly rigid box, the wavelength of the wave function for any state is the de Broglie wavelength.

An electron trapped in an infinitely deep square well has a ground-state energy E = 9.0 eV. (a) What is the longest wavelength photon that an excited state of this system can emit? (b) What is the width of the well?

An n = 4 to n = 1 transition for an electron trapped in a rigid box produces a 340-nm photon. What is the width of the box?

For a particle in a box with rigid walls, determine whether our results for the ground state are consistent with the uncertainty principle by calculating the product Ap Ax. Take Ax , since the particle is somewhere within the box. For Ap, note that although p is known (= hk), the direction of p is not known, so the x component could vary from p to +p; hence take Ap ~ 2p.

The longest-wavelength line in the spectrum emitted by an electron trapped in an infinitely deep square well is 610 nm. What is the width of the well?

Determine the lowest four energy levels and wave functions for an electron trapped in an infinitely deep potential well of width 2.0 nm.

Write a formula for the positions of (a) the maxima and (ib) the minima in \if/\2 for a particle in the nth state in an infinite square well.

Consider an atomic nucleus to be a rigid box of width 2.0 X 10-14 m. What would be the ground-state energy for (a) an electron, (b) a neutron, and (c) a proton in this nucleus?

A proton in a nucleus can be roughly modeled as a particle in a box of nuclear dimensions. Calculate the energy released when a proton confined in a nucleus of width 1.0 X 10-14 m makes a transition from the first excited state to the ground state.

Consider a single oxygen molecule confined in a onedimensional rigid box of width 4.0 mm. (a) Treating this as a particle in a rigid box, determine the ground-state energy, (b) If the molecule has an energy equal to the onedimensional average thermal energy \k T at T = 300 K, what is the quantum number nl (c) What is the energy difference between the th state and the next higher state?

An electron is trapped in a 1.00-nm-wide rigid box. Determine the probability of finding the electron within 0.15 nm of the center of the box (on either side of center) for (a) n = 1, (b) n = 5, and (c) n = 20. (d) Compare to the classical prediction.

If an infinitely deep well of width is redefined to be located from x = \ l to x = \ l (as opposed to x = 0 to x = ), speculate how this will change the wave function for a particle in this well. Investigate your speculation(s) by determining the wave functions and energy levels for this newly defined well. [Hint: Try if/ = A sin (kx +

An electron with 180 eV of kinetic energy in free space passes over a finite potential well 56 eV deep that stretches from x = 0 to x = 0.50 nm. What is the electrons wavelength (a) in free space, (b) when over the well? (c) Draw a diagram showing the potential energy and total energy as a function of x, and on the diagram sketch a possible wave function.

Sketch the wave functions and the probability distributions for the n = 4 and n = 5 states for a particle trapped in a finite square well.

Suppose that a particle of mass ra is trapped in a finite potential well that has a rigid wall at x = 0 (U = oo for x < 0) and a finite wall of height U = U0 at x = , Fig. 38-20. (a) Sketch the wave functions for the lowest three states. (b) What is the form of the wave function in the ground state in the three regions x < 0, 0 < jc < I, x > 17 FIGURE 38-20 Problem 36.

An electron is trapped in a 0.16-nm-wide finite square well of height U0 = 2.0 keV. Estimate at what distance outside the walls of the well the ground state wave function drops to 1.0% of its value at the walls.

A potential barrier has a height U0 = 14 eV and thickness = 0.85 nm. If the transmission coefficient for an incident electron is 0.00050, what is the electrons energy?

An electron approaches a potential barrier 18 eV high and 0.55 nm wide. If the electron has a 1.0% probability of tunneling through the barrier, what is the electrons energy?

A proton and a helium nucleus approach a 25-MeV potential energy barrier. If each has a kinetic energy of 5.0 MeV, what is the probability of each to tunnel through the barrier, assuming it is rectangular and 3.6 fm thick?

An electron with an energy of 8.0 eV is incident on a potential barrier which is 9.2 eV high and 0.25 nm wide. (a) What is the probability that the electron will pass through the barrier? (b) What is the probability that the electron will be reflected?

A 1.0-mA current of 1.6-MeV protons strikes a 2.6-MeV-high potential barrier 2.8 X 10-13 m thick. Estimate the transmitted current.

For part (b) of Example 38-11, what effect will there be on the transmission coefficient if (a) the barrier height is raised 2.0%, (b) the barrier thickness is increased by 2.0%?

Show that the transmission coefficient is given roughly by Eqs. 38-17 for a high or thick barrier, by calculating \ij/(x = )|2/|^(0)|2. [Hint: Assume that if/ is a decaying exponential inside the barrier.]

A uranium-238 nucleus (Q = +92e) lasts about 4.5 X 109 years before it decays by emission of an alpha particle (q = + 2e, M = 4Mproton). (a) Assuming that the a particle is a point, and the nucleus is roughly 8 fm in radius, estimate the height of the Coulomb barrier (the peak in Fig. 38-17). (b) The alpha particle, when free, has kinetic energy ~ 4 MeV. Estimate the width of the barrier, (c) Assuming that the square well has U = 0 inside (and U = 0 far from the nucleus), calculate the speed of the alpha particle and how often it hits the barrier inside, and from this (and Eqs. 38-17) estimate the uranium lifetime. [Hint: Replace the llr Coulomb barrier with an averaged rectangular barrier (as in Fig. 38-15) of width equal to | that calculated in (b).]

The Z boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.5 GeV. What is the lifetime of this particle?

Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius 1.2 X 10-15 m. [Hint: A particle can have an energy at least as large as its uncertainty.]

A neutron is trapped in an infinitely deep potential well 2.5 fm in width. Determine (a) the four lowest possible energy states and (b) their wave functions, (c) What is the wavelength and energy of a photon emitted when the neutron makes a transition between the two lowest states? In what region of the EM spectrum does this photon lie? [Note: This is a rough model of an atomic nucleus.]

Protons are accelerated from rest across 650 V. They are then directed at two slits 0.80 mm apart. How far apart will the interference peaks be on a screen 18 m away?

An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

Use the uncertainty principle to estimate the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 37-11 and assume the momentum can be anywhere between this value and zero.] How does this compare to the Bohr radius?

Simple Harmonic Oscillator. Suppose that a particle of mass ra is trapped not in a square well, but in one whose potential energy is that of a simple harmonic oscillator: U(x) = \Cx2. That is, if the particle is displaced from x = 0, a restoring force F = Cx acts on it, where C is constant. (a) Sketch this potential energy. (b) Show that if/ = Ae~B*2 is a solution to the Schrodinger equation and that the energy of this state is E = \ ho), where o) = \/C /m (as classically, Eq. 14-5) and B = raw/2/L [Note: This is the ground state, and this energy \ha) is the zero-point energy for a harmonic oscillator. The energies of higher states are En = (n + l)ha), where n is an integer.]

Estimate the kinetic energy and speed of an alpha particle (q = +2e, M = 4Mproton) trapped in a nucleus 1.5 X 10-14m wide. Assume an infinitely deep square well potential.

A small ball of mass 3.0 X 10-6 kg is dropped on a table from a height of 2.0 m. After each bounce the ball rises to 65% of its height before the bounce because of its inelastic collision with the table. Estimate how many bounces occur before the uncertainty principle plays a role in the problem. [Hint. Determine when the uncertainty in the balls speed is comparable to its speed of impact on the table.]

By how much does the tunneling current through the tip of an STM change if the tip rises 0.020 nm from some initial height above a sodium surface with a work function W0 = 2.28 eV? [Hint. Let the work function (see Section 37-2) equal the energy needed to raise the electron to the top of the barrier.]

Show that the function = Aelkx, where A is a constant and k is given by Eq. 38-11, is a solution of the time-independent Schrodinger equation for the case U = 0.

Show that the average value of the square of the position x of a particle in state n inside an infinite well of width I is x 2 ^ f x 2\if/n\2 d x = f2[| - Calculate the values of x 2 for n = 1 to 20 and make a graph of x 2 versus n. [Hint: You may want to consult a detailed Table of integrals.]

Consider a particle that can exist anywhere in space with a wave function given by if/(x) = b~^\x/bf e~(x/b^/2, where b = 1.0 nm. (a) Check that the wave function is normalized. (b) What is the most probable position for the particle in the region x > 0? (c) What is the probability of finding the particle between x = 0 nm and x = 0.50 nm?

A 7.0-gram pencil, 18 cm long, is balanced on its point. Classically, this is a configuration of (unstable) equilibrium, so the pencil could remain there forever if it were perfectly placed. A quantum mechanical analysis shows that the pencil must fall, (a) Why is this the case? (b) Estimate (within a factor of 2) how long it will take the pencil to hit the table if it is initially positioned as well as possible? [Hint: Use the uncertainty principle in its angular form to obtain an expression for the initial angle <f>0 A<f>.]

An electron is trapped in the ground state of an infinite potential well of width = 0.10 nm. The probability that the electron will be found in the central 1% of the well was estimated in Example 38-7 by \if/\2Ax. Use numerical methods to determine how large Ax could be to cause less than a 10% error in such an estimate.

Consider a particle of mass m and energy E traveling to the right where it encounters a narrow potential barrier of height U0 and width as shown in Fig. 38-21. It can be shown that: (i) for E < UQ, the transmission probability is and the reflection probability is R = 1 T. (ii) For E > Uq, the transmission probability is and R = 1 - T. Consider that the particle is an electron and it is incident on a rectangular barrier of height Uq = 10 eV and width = 1.0 X 1 0 10 m. (a) Calculate T and R for the electron from E/U0 = 0 to 10, in steps of 0.1. Make a single graph showing the two curves of T and R as a function oiE/UQ. (b) From the graph determine the energies (E/Uq) at which the electron will have transmission probabilities of 10%, 20%, 50%, and 80%. FIGURE 38-21 Problem 61.