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Physics

Alan Giambattista, Betty McCarthy Richardson, Robert C. Richardson

Chapter 28

Quantum Physics - all with Video Answers

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Chapter Questions

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Problem 1

What is the de Broglie wavelength of a basketball of mass $0.50 \mathrm{kg}$ when it is moving at $10 \mathrm{m} / \mathrm{s} ?$ Why don't we see diffraction effects when a basketball passes through the circular aperture of the hoop?

Kathleen Tatem
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Problem 2

A fly with a mass of $1.0 \times 10^{-4} \mathrm{kg}$ crawls across a table at a speed of $2 \mathrm{mm} / \mathrm{s} .$ Compute the de Broglie wavelength of the fly and compare it with the size of a proton (about $\left.1 \mathrm{fm}, 1 \mathrm{fm}=10^{-15} \mathrm{m}\right)$.

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Problem 3

An 81 -kg student who has just studied matter waves is concerned that he may be diffracted as he walks through a doorway that is $81 \mathrm{cm}$ across and $12 \mathrm{cm}$ thick. (a) If the wavelength of the student must be about the same size as the doorway to exhibit diffraction, what is the fastest the student can walk through the doorway to exhibit diffraction? (b) At this speed, how long would it take the student to walk through the doorway?

Kathleen Tatem
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Problem 4

What is the magnitude of the momentum of an electron with a de Broglie wavelength of $0.40 \mathrm{nm} ?$

Kathleen Tatem
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Problem 5

What is the de Broglie wavelength of an electron moving at speed $\frac{3}{5} c ?$

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01:55

Problem 6

The distance between atoms in a crystal of $\mathrm{NaCl}$ is $0.28 \mathrm{nm} .$ The crystal is being studied in a neutron diffraction experiment. At what speed must the neutrons be moving so that their de Broglie wavelength is $0.28 \mathrm{nm} ?$

Vipender Yadav
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Problem 7

An x-ray diffraction experiment using 16 -keV x-rays is repeated using electrons instead of $x$ -rays. What should the kinetic energy of the electrons be in order to produce the same diffraction pattern as the x-rays (using the same crystal)?

Kathleen Tatem
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Problem 8

What are the de Broglie wavelengths of electrons with the following values of kinetic energy? (a) $1.0 \mathrm{eV}$
(b) $1.0 \mathrm{keV} .

Kathleen Tatem
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Problem 9

What is the ratio of the wavelength of a 0.100-keV photon to the wavelength of a 0.100-keV electron?

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Problem 10

Neutron diffraction by a crystal can be used to make a velocity selector for neutrons. Suppose the spacing between the relevant planes in the crystal is $d=0.20 \mathrm{nm}$ A beam of neutrons is incident at an angle $\theta=10.0^{\circ}$ with respect to the planes. The incident neutrons have speeds ranging from 0 to $2.0 \times 10^{4} \quad \mathrm{m} / \mathrm{s} .$ (a) What wavelength(s) are strongly reflected from these planes? [Hint: Bragg's law, Eq. $(25-15),$ applies to neutron diffraction as well as to x-ray diffraction.] (b) For each of the wavelength(s), at what angle with respect to the incident beam do those neutrons emerge from the crystal?

Kathleen Tatem
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Problem 11

A nickel crystal is used as a diffraction grating for x-rays. Then the same crystal is used to diffract electrons. If the two diffraction patterns are identical, and the energy of each $\mathrm{x}$ -ray photon is $E=20.0 \mathrm{keV},$ what is the kinetic energy of each electron?

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Problem 12

If diffraction were the only limitation on resolution, what would be the smallest structure that could be resolved in an electron microscope using 10-keV electrons?

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Problem 13

To resolve details of an object, you must use a wavelength that is about the same size, or smaller, than the details you want to observe. Suppose you want to study a molecule that is about $1.000 \times 10^{-10} \mathrm{m}$ in length.
(a) What minimum photon energy is required to study this molecule? (b) What is the minimum kinetic energy of electrons that could study this? (c) Through what potential difference should the electrons be accelerated to reach this energy?

Kathleen Tatem
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Problem 14

A scanning electron microscope is used to look at cell structure with 10 -nm resolution. A beam of electrons from a hot filament is accelerated with a voltage of $12 \mathrm{kV}$ and then focused to a small spot on the specimen.
(a) What is the wavelength in nanometers of the beam of incoming electrons? (b) If the size of the focal spot were determined only by diffraction, and if the diameter of the electron lens is one fifth the distance from the lens to the specimen, what would be the minimum separation resolvable on the specimen? (In practice, the resolution is limited much more by aberrations in the magnetic lenses and other factors.)

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Problem 15

An image of a biological sample is to have a resolution of $5 \mathrm{nm} .$ (a) What is the kinetic energy of a beam of electrons with a de Broglie wavelength of $5.0 \mathrm{nm} ?$
(b) Through what potential difference should the electrons be accelerated to have this wavelength? (c) Why not just use a light microscope with a wavelength of
5 nm to image the sample?

Kathleen Tatem
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Problem 16

The phenomenon of Brownian motion is the random motion of microscopically small particles as they are buffeted by the still smaller molecules of a fluid in which they are suspended. For a particle of mass $1.0 \times 10^{-16} \mathrm{kg},$ the fluctuations in velocity are of the order of $0.010 \mathrm{m} / \mathrm{s} .$ For comparison, how large is the change in this particle's velocity when the particle absorbs a photon of light with a wavelength of $660 \mathrm{nm}$ such as might be used in observing its motion under a microscope?

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Problem 17

If the momentum of the basketball in Problem 1 has a fractional uncertainty of $\Delta p / p=10^{-6},$ what is the uncertainty in its position?

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02:03

Problem 18

An electron passes through a slit of width $1.0 \times 10^{-8} \mathrm{m}$ What is the uncertainty in the electron's momentum component in the direction perpendicular to the slit but in the plane containing the slit?

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Problem 19

At a baseball game, a radar gun measures the speed of a 144-g baseball to be $137.32 \pm 0.10 \mathrm{km} / \mathrm{h} .$ (a) What is the minimum uncertainty of the position of the baseball? (b) If the speed of a proton is measured to the same precision, what is the minimum uncertainty in its position?

Kathleen Tatem
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Problem 20

A hydrogen atom has a radius of about $0.05 \mathrm{nm}$
(a) Estimate the uncertainty in any component of the momentum of an electron confined to a region of this size. (b) From your answer to (a), estimate the electron's kinetic energy. (c) Does the estimate have the correct order of magnitude? (The ground-state kinetic energy predicted by the Bohr model is $13.6 \mathrm{eV} .$ )

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Problem 21

A bullet with mass $10.000 \mathrm{g}$ has a speed of $300.00 \mathrm{m} / \mathrm{s}$ the speed is accurate to within $0.04 \% .$ (a) Estimate the minimum uncertainty in the position of the bullet, according to the uncertainty principle. (b) An electron has a speed of $300.00 \mathrm{m} / \mathrm{s}$, accurate to $0.04 \% .$ Estimate the minimum uncertainty in the position of the electron.
(c) What can you conclude from these results?

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Problem 22

A radar pulse has an average wavelength of $1.0 \mathrm{cm}$ and lasts for $0.10 \mu \mathrm{s}$. (a) What is the average energy of the photons? (b) Approximately what is the least possible uncertainty in the energy of the photons?

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Problem 23

A beam of electrons passes through a single slit $40.0 \mathrm{nm}$ wide. The width of the central fringe of a diffraction pattern formed on a screen $1.0 \mathrm{m}$ away is $6.2 \mathrm{cm} .$ What is the kinetic energy of the electrons passing through the slit?

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Problem 24

Electrons are accelerated through a potential difference of $38.0 \mathrm{V} .$ The beam of electrons then passes through a single slit. The width of the central fringe of a diffraction pattern formed on a screen 1.00 m away is $1.13 \mathrm{mm} .$ What is the width of the slit?

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Problem 25

The omega particle $(\Omega)$ decays on average about $0.1 \mathrm{ns}$ after it is created. Its rest energy is 1672 MeV. Estimate the fractional uncertainty in the $\Omega$ 's rest energy $\left(\Delta E_{0} / E_{0}\right)$ [Hint: Use the energy-time uncertainty principle, Eq. $(28-3) .]$

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Problem 26

Nuclei have energy levels just as atoms do. An excited nucleus can make a transition to a lower energy level by emitting a gamma-ray photon. The lifetime of a typical nuclear excited state is about 1 ps. What is the uncertainty in the energy of the gamma-rays emitted by a typical nuclear excited state? [Hint: Use the energy-time uncertainty principle, Eq. $(28-3) .]$

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Problem 27

What is the minimum kinetic energy of an electron confined to a region the size of an atomic nucleus $(1.0 \mathrm{fm}) ?$

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Problem 28

An electron is confined to a box of length $1.0 \mathrm{nm} .$ What is the magnitude of its momentum in the $n=4$ state?

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Problem 29

A marble of mass $10 \mathrm{g}$ is confined to a box $10 \mathrm{cm}$ long and moves at a speed of $2 \mathrm{cm} / \mathrm{s} .$ (a) What is the marble's quantum number $n ?$ (b) Why can we not observe the quantization of the marble's energy? [Hint: Calculate the energy difference between states $n$ and $n+1 .$ How much does the marble's speed change?]

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Problem 30

Suppose the electron in a hydrogen atom is modeled as an electron in a one-dimensional box of length equal to the Bohr diameter, $2 a_{0} .$ What would be the ground-state energy of this "atom"? How does this compare with the actual ground-state energy?

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Problem 31

The particle in a box model is often used to make rough estimates of energy level spacings. For a metal wire
$10 \mathrm{cm}$ long, treat a conduction electron as a particle confined to a one-dimensional box of length $10 \mathrm{cm} .$
(a) Sketch the wave function $\psi$ as a function of position for the electron in this box for the ground state and each of the first three excited states. (b) Estimate the spacing between energy levels of the conduction electrons by finding the energy spacing between the ground state and the first excited state.

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Problem 32

The particle in a box model is often used to make rough estimates of ground-state energies. Suppose that you have a neutron confined to a one-dimensional box of length equal to a nuclear diameter (say $10^{-14} \mathrm{m}$ ). What is the ground-state energy of the confined neutron?

Kathleen Tatem
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Problem 33

An electron confined to a one-dimensional box has a ground-state energy of $40.0 \mathrm{eV} .$ (a) If the electron makes a transition from its first excited state to the ground state, what is the wavelength of the emitted photon? (b) If the box were somehow made twice as long, how would the photon's energy change for the same transition (first excited state to ground state)?

Kathleen Tatem
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Problem 34

An electron is confined to a one-dimensional box. When the electron makes a transition from its first excited state to the ground state, it emits a photon of energy 1.2 eV. (a) What is the ground-state energy (in electronvolts) of the electron? (b) List all energies (in electronvolts) of photons that could be emitted when the electron starts in its second excited state and makes transitions downward to the ground state either directly or through intervening states. Show all these transitions on an energy level diagram. (c) What is the length of the box (in nanometers)?

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04:21

Problem 35

What is the ground state electron configuration of a $\mathrm{K}^{+}$ ion?

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Problem 36

How many electron states of the H atom have the quantum numbers $n=3$ and $\ell=1 ?$ Identify each state by listing its quantum numbers.

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Problem 37

What are the possible values of $L_{z}$ (the component of angular momentum along the $z$ -axis) for the electron in the second excited state $(n=3)$ of the hydrogen atom?

Kathleen Tatem
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Problem 38

What is the largest number of electrons with the same pair of values for $n$ and $\ell$ that an atom can have?

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Problem 39

List the number of electron states in each of the subshells in the $n=7$ shell. What is the total number of electron states in this shell?

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01:40

Problem 40

What is the ground-state electron configuration of nickel (Ni, atomic number 28 )?

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01:24

Problem 41

What is the ground-state electron configuration of bromine (Br, atomic number 35)?

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04:03

Problem 42

What is the maximum possible value of the angular momentum for an outer electron in the ground state of a bromine atom?

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02:32

Problem 43

(a) What are the electron configurations of the ground states of lithium $(Z=3),$ sodium $(Z=11),$ and potassium $(Z=19) ?$ (b) Why are these elements placed in the same column of the periodic table?

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02:22

Problem 44

(a) What are the electron configurations of the ground states of fluorine $(Z=9)$ and chlorine $(Z=17) ?$ (b) Why are these elements placed in the same column of the periodic table?

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04:20

Problem 45

What is the electronic configuration of the ground state of the carbon atom? Write it in the following ways:
(a) using spectroscopic notation $\left(1 s^{2}, \ldots\right) ;$ (b) listing the four quantum numbers for each of the electrons. Note that there may be more than one possibility in (b).

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Problem 46

(a) Find the magnitude of the angular momentum $\overrightarrow{\mathbf{L}}$ for an electron with $n=2$ and $\ell=1$ in terms of $\hbar .$ (b) What are the allowed values for $L_{2} ?$ (c) What are the angles between the positive z-axis and $\overline{\mathbf{L}}$ so that the quantized components, $L_{z},$ have allowed values?

Kathleen Tatem
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03:36

Problem 47

(a) Show that the ground-state energy of the hydrogen atom can be written $E_{1}=-k e^{2} /\left(2 a_{0}\right),$ where $a_{0}$ is the Bohr radius. (b) Explain why, according to classical physics, an electron with energy $E_{1}$ could never be found at a distance greater than $2 a_{0}$ from the nucleus.

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03:04

Problem 48

A light-emitting diode (LED) has the property that electrons can be excited into the conduction band by the electrical energy from a battery; a photon is emitted when the electron drops back to the valence band. (a) If the band gap for this diode is $2.36 \mathrm{eV},$ what is the wavelength of the light emitted by the LED? (b) What color is the light emitted? (c) What is the minimum battery voltage required in the electrical circuit containing the diode?

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01:34

Problem 49

A photoconductor (see Conceptual Question 13 ) allows charge to flow freely when photons of wavelength $640 \mathrm{nm}$ or less are incident on it. What is the band gap for this photoconductor?

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01:38

Problem 50

What is the wavelength of the light usually emitted by a helium-neon laser? (See Fig. 28.24.)

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02:36

Problem 51

Many lasers, including the helium-neon, can produce beams at more than one wavelength. Photons can stimulate emission and cause transitions between the $20.66-\mathrm{eV}$ metastable state and several different states of lower energy. One such state is 18.38 eV above the ground state. What is the wavelength for this transition? If only these photons leave the laser to form the beam, what color is the beam?

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02:14

Problem 52

In a ruby laser, laser light of wavelength $694.3 \mathrm{nm}$ is emitted. The ruby crystal is $6.00 \mathrm{cm}$ long, and the index of refraction of ruby is $1.75 .$ Think of the light in the ruby crystal as a standing wave along the length of the crystal. How many wavelengths fit in the crystal? (Standing waves in the crystal help to reduce the range of wavelengths in the beam.)

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03:47

Problem 53

The beam emerging from a ruby laser passes through a circular aperture $5.0 \mathrm{mm}$ in diameter. (a) If the spread of the beam is limited only by diffraction, what is the angular spread of the beam? (b) If the beam is aimed at the Moon, how large a spot would be illuminated on the Moon's surface?

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06:00

Problem 54

A proton and a deuteron (which has the same charge as the proton but 2.0 times the mass) are incident on a barrier of thickness $10.0 \mathrm{fm}$ and "height" $10.0 \mathrm{MeV} .$ Each particle has a kinetic energy of $3.0 \mathrm{MeV} .$ (a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.

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02:33

Problem 55

Refer to Example $28.6 .$ Estimate the percentage change in the tunneling current if the distance between the sample surface and the STM tip increases $2.0 \%$

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02:29

Problem 56

Mitch drops a 2.0 -g coin into a 3.0 -m-deep wishing well. What is the de Broglie wavelength of the coin just before it hits the bottom of the well?

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04:52

Problem 57

A magnesium ion $\mathrm{Mg}^{2+}$ is accelerated through a potential difference of $22 \mathrm{kV}$. What is the de Broglie wavelength of this ion?

Guilherme Barros
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03:52

Problem 58

The energy-time uncertainty principle allows for the creation of virtual particles, that appear from a vacuum for a very brief period of time $\Delta t,$ then disappear again. This can happen as long as $\Delta E \Delta t=\hbar / 2,$ where $\Delta E$ is the rest energy of the particle. (a) How long could an electron created from the vacuum exist according to the uncertainty principle? (b) How long could a shot put with a mass of $7 \mathrm{kg}$ created from the vacuum exist according to the uncertainty principle?

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02:58

Problem 59

An electron moving in the positive $x$ -direction passes through a slit of width $\Delta y=85 \mathrm{nm} .$ What is the minimum uncertainty in the electron's velocity in the $y$ -direction?

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05:21

Problem 60

In Fig. $28.4 \mathrm{b}$, the $\mathrm{x}$ -rays had a frequency of $1.0 \times 10^{19} \mathrm{Hz}$. Through what potential difference were the electrons in Fig. 28.4 a accelerated?

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03:12

Problem 61

The neutrons produced in fission reactors have a wide range of kinetic energies. After the neutrons make several collisions with atoms, they give up their excess kinetic energy and are left with the same average kinetic energy as the atoms, which is $\frac{3}{2} k_{\mathrm{B}} T .$ If the temperature of the reactor core is $T=400.0 \mathrm{K},$ find (a) the average kinetic energy of the thermal neutrons, and (b) the de Broglie wavelength of a neutron with this kinetic energy.

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04:47

Problem 62

A double-slit interference experiment is performed with 2.0-ev photons. The same pair of slits is then used for an experiment with electrons. What is the kinetic energy of the electrons if the interference pattern is the same as for the photons (i.e., the spacing between maxima is the same)?

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01:57

Problem 63

An electron is confined in a one-dimensional box of length $L$. Another electron is confined in a box of length
2 $L$. Both are in the ground state. What is the ratio of their energies $E_{2 l} / E_{L} ?$

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06:17

Problem 64

Before the discovery of the neutron, one theory of the nucleus proposed that the nucleus contains protons and electrons. For example, the helium-4 nucleus would contain 4 protons and 2 electrons instead of - as we now know to be true- 2 protons and 2 neutrons.
(a) Assuming that the electron moves at nonrelativistic speeds, find the ground-state energy in mega-electron-
volts of an electron confined to a one-dimensional box of length $5.0 \mathrm{fm}$ (the approximate diameter of the $^{4} \mathrm{He}$ nucleus). (The electron actually does move at relativistic speeds. See Problem $80 .)$ (b) What can you conclude about the electron-proton model of the nucleus? The binding energy of the $^{4} \mathrm{He}$ nucleus - the energy that would have to be supplied to break the nucleus into its constituent particles-is about $28 \mathrm{MeV} .$ (c) Repeat
(a) for a neutron confined to the nucleus (instead of an electron). Compare your result with (a) and comment on the viability of the proton-neutron theory relative to the electron-proton theory.

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Problem 65

What is the ground-state electron configuration of tellurium (Te, atomic number 52 )?

Susan Hallstrom
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03:13

Problem 66

A free neutron (that is, a neutron on its own rather than in a nucleus) is not a stable particle. Its average lifetime is 15 min, after which it decays into a proton, an electron, and an antineutrino. Use the energy-time uncertainty principle $[\mathrm{Eq} .(28-3)]$ and the relationship between mass and rest energy to estimate the inherent uncertainty in the mass of a free neutron. Compare with the average neutron mass of $1.67 \times 10^{-27} \mathrm{kg} .$ (Although the uncertainty in the neutron's mass is far too small to be measured, unstable particles with extremely short lifetimes have marked variation in their measured masses.)

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07:16

Problem 67

A beam of electrons is accelerated across a potential of $15 \mathrm{kV}$ before passing through two slits. The electrons form a interference pattern on a screen $2.5 \mathrm{m}$ in front of the slits. The first-order maximum is $8.3 \mathrm{mm}$ from the central maximum. What is the distance between the slits?

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02:59

Problem 68

A bullet leaves the barrel of a rifle with a speed of $300.0 \mathrm{m} / \mathrm{s} .$ The mass of the bullet is $10.0 \mathrm{g} .$ (a) What is the de Broglie wavelength of the bullet? (b) Compare $\lambda$ with the diameter of a proton (about $1 \mathrm{fm}$ ). (c) Is it possible to observe wave properties of the bullet, such as diffraction? Explain.

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07:43

Problem 69

A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of $16.4 \mathrm{m}$ downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected?
(b) What is the de Broglie wavelength of the neutrons?
(c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.

Guilherme Barros
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07:14

Problem 70

The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about $10^{-14} \mathrm{m}$ ).
(a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie?
(c) Sketch the wave function $\psi$ as a function of position for the proton in this box for the ground state and each of the first three excited states.

Guilherme Barros
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09:03

Problem 71

An electron is confined to a one-dimensional box of length $L$. When the electron makes a transition from its first excited state to the ground state, it emits a photon of energy 0.20 eV. (a) What is the ground-state energy (in electron-volts) of the electron in this box? (b) What are the energies (in electron-volts) of the photons that can be emitted when the electron starts in its third excited state and makes transitions downwards to the ground state (either directly or through intervening states)? (c) Sketch the wave function of the electron in the third excited state. (d) If the box were somehow made longer, how would the electron's new energy level spacings compare with its old ones? (Would they be greater, smaller, or the same? Or is more information needed to answer this question? Explain.)

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06:46

Problem 72

An electron in an atom has an angular momentum quantum number of $2 .$ (a) What is the magnitude of the angular momentum of this electron in terms of $\hbar ?$ (b) What are the possible values for the $z$ -components of this electron's angular momentum? (c) Draw a diagram showing possible orientations of the angular momentum vector $\overrightarrow{\mathbf{L}}$ relative to the z-axis. Indicate the angles with respect to the z-axis.

Guilherme Barros
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06:19

Problem 73

In the Davisson-Germer experiment (Section $28.2),$ the electrons were accelerated through a $54.0-\mathrm{V}$ potential difference before striking the target. (a) Find the de Broglie wavelength of the electrons. (b) Bragg plane spacings for nickel were known at the time; they had been determined through x-ray diffraction studies. The largest plane spacing (which gives the largest intensity diffraction maxima) in nickel is $0.091 \mathrm{nm} .$ Using Bragg's law [Eq. ( $25-15$ )], find the Bragg angle for the first-order maximum using the de Broglie wavelength of the electrons. (c) Does this agree with the observed maximum at a scattering angle of $130^{\circ} ?$ [Hint: The scattering angle and the Bragg angle are not the same. Make a sketch to show the relationship between the two angles.]

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04:41

Problem 74

A beam of neutrons has the same de Broglie wavelength as a beam of photons. Is it possible that the energy of each photon is equal to the kinetic energy of each neutron? If so, at what de Broglie wavelength(s) does this occur? [Hint: For the neutron, use the relativistic energy-momentum relation $\left.E^{2}=E_{0}^{2}+(p c)^{2} .\right]$

Guilherme Barros
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04:51

Problem 75

(a) Make a qualitative sketch of the wave function for the $n=5$ state of an electron in a finite box $[U(x)=0$ for $0<x<L$ and $U(x)=U_{0}>0$ elsewherel. (b) If $L=1.0 \mathrm{nm}$ and $U_{0}=1.0$ keV, estimate the number of bound states that exist.

Guilherme Barros
Guilherme Barros
Numerade Educator
06:29

Problem 76

An electron is confined to a one-dimensional box of length $L$ (a) Sketch the wave function for the third excited state. (b) What is the energy of the third excited state? (c) The potential energy can't really be infinite outside of the box. Suppose that $U(x)=+U_{0}$ outside the box, where $U_{0}$ is large but finite. Sketch the wave function for the third excited state of the electron in the finite box. (d) Is the energy of the third excited state for the finite box less than, greater than, or equal to the value calculated in part (b)? Explain your reasoning. [Hint:
Compare the wavelengths inside the box.] (e) Give a rough estimate of the number of bound states for the electron in the finite box in terms of $L$ and $U_{0}$.

Guilherme Barros
Guilherme Barros
Numerade Educator
11:57

Problem 77

An electron in a one-dimensional box has ground-state energy 0.010 eV. (a) What is the length of the box?
(b) Sketch the wave functions for the lowest three energy states of the electron. (c) What is the wavelength of the electron in its second excited state $(n=3) ?$
(d) The electron is in its ground state when it absorbs a photon of wavelength $15.5 \mu \mathrm{m}$. Find the wavelengths of the photon(s) that could be emitted by the electron subsequently.

Guilherme Barros
Guilherme Barros
Numerade Educator
01:48

Problem 78

A particle is confined to a finite box of length $L$. In the $n$ th state, the wave function has $n-1$ nodes. The wave function must make a smooth transition from sinusoidal inside the box to a decaying exponential outside - there can't be a kink at the wall. (a) Make some sketches to show that the wavelength $\lambda_{n}$ inside the box must fall in the range $2 L / n<\lambda_{n}<2 L /(n-1) .$ (b) Show that the energy levels $\quad E_{n}$ in the finite box satisfy $(n-1)^{2} E_{1}<E_{n}<n^{2} E_{1},$ where $E_{1}=h^{2} /\left(8 m L^{2}\right)$ is the
ground-state energy for an infinite box of length $L$.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:29

Problem 79

(a) Show that the number of electron states in a subshell is $4 \ell+2 .$ (b) By summing the number of states in each of the subshells, show that the number of states in a shell is $2 n^{2} .$ [Hint:
The sum of the first $n$ odd integers, from 1 to $2 n-1,$ is $n^{2} .$ That comes from regrouping the sum in pairs, starting by adding the largest to the smallest:
$1+3+5+\dots+(2 n-5)+(2 n-3)+(2 n-1)$
$=[1+(2 n-1)]+[3+(2 n-3)]+[5+(2 n-5)]+\cdots$
$=2 n+2 n+2 n+\cdots=2 n \times \frac{n}{2}=n^{2}$

Guilherme Barros
Guilherme Barros
Numerade Educator
03:35

Problem 80

Repeat Problem 64 (a), this time assuming the electron is ultra-relativistic $(E \approx p c) .$ Is the assumption justified?

Guilherme Barros
Guilherme Barros
Numerade Educator