Chapter 12: Quantum Mechanics — Review and Exercises

Chapter contents: Chapter 12: Quantum Mechanics

Glossary

blackbody

an ideal radiator, which emits thermal radiation ideally and absorbs all radiation incident on it at all wavelengths

blackbody radiation

the thermal electromagnetic radiation emitted by an ideal blackbody, with a spectrum determined by its temperature

Bohr radius

the ground-state orbital radius of hydrogen in the Bohr model, also a characteristic atomic length scale

cathode-ray tube

a vacuum tube containing a source of electrons and a screen to view images

correspondence principle

the requirement that a newer theory reproduce the successful predictions of an older theory in the conditions where the older theory applies

de Broglie wavelength

the wavelength possessed by a particle of matter, calculated by

λ = h / p

energy-level diagram

a diagram used to analyze the energy level of electrons in the orbits of an atom

Heisenberg’s uncertainty principle

the quantum restriction on the simultaneous spreads of certain pairs of quantities, such as position and momentum

hydrogen-like ion

an ion with only one electron remaining, such as He⁺ or Li²⁺. Like hydrogen, it has a single electron interacting with the nucleus, so Bohr’s model can describe its allowed energy levels.

photoelectric effect

the phenomenon whereby some materials eject electrons when light is shined on them

photon

a quantum, or particle, of electromagnetic radiation

photon energy

the amount of energy a photon has;

E = h f

Planck law

the law describing the spectrum of blackbody radiation as a function of temperature and wavelength or frequency

Planck’s constant

the constant that relates a photon’s energy to its frequency,

E = h f

;

h \approx 6.626 \times 10^{−34} J \cdot s

planetary model of the atom

a historical model in which electrons orbit a small, massive nucleus

probability distribution

the overall spatial distribution of probabilities to find a particle at a given location

probability wave

an informal name for a wave function, whose squared magnitude gives a probability density for position

quantized

the fact that certain physical entities exist only with particular discrete values and not every conceivable value

quantum mechanics

the framework describing matter and radiation in terms of quantum states, with probabilistic predictions for measurement outcomes. Its distinctive effects are especially apparent at atomic and subatomic scales

Rydberg constant

a physical constant related to the atomic spectra with an established value of

1.097 \times 10^{7} m^{−1}

ultraviolet catastrophe

failure of classical theory to correctly describe blackbody radiation at short wavelengths

uncertainty in momentum

the spread of momentum measurement outcomes for systems prepared in the same quantum state

uncertainty in position

the spread of position measurement outcomes for systems prepared in the same quantum state

work function

the minimum energy required to remove an electron from a material’s surface into vacuum

Section Summary

12.2: Blackbody Radiation

The first indication that energy is sometimes quantized came from blackbody radiation, which is the emission of EM radiation by an object that absorbs all radiation incident on it.
The failure of classical theory to successfully explain blackbody radiation is called “ultraviolet catastrophe”, after the large intensity of short-wavelength radiation incorrectly predicted.
Planck derived the Planck Law by assuming that energy of oscillators is quantized and can change only by amount $Δ E = h f$ , where $h$ , Planck’s constant is,
$h = 6 . 626 \times 10^{−34} J \cdot s$ .
(12.2.1)

12.3: The Photoelectric Effect

The photoelectric effect is the process in which EM radiation ejects electrons from a material.
Einstein proposed photons to be quanta of EM radiation having energy $E = h f$ , where $f$ is the frequency of the radiation.
All EM radiation is composed of photons. As Einstein explained, all characteristics of the photoelectric effect are due to the interaction of individual photons with individual electrons.
The maximum kinetic energy ${KE}_{e}$ of ejected electrons (photoelectrons) is given by ${KE}_{e} = h f − BE$ , where $h f$ is the photon energy and BE is the binding energy (or work function) of the electron to the particular material.

12.4: The Wave Nature of Matter

Particles of matter also have a wavelength, called the de Broglie wavelength, given by $λ = \frac{h}{p}$ , where $p$ is momentum.
The de Broglie relationship between wavelength and momentum is universally applicable to all particles and waves.
Matter is found to have the same interference characteristics as any other wave.

12.5: Uncertainty Principle

Matter is found to have the same interference characteristics as any other wave.
There is now a probability distribution for the location of a particle rather than a definite position.
Another consequence of the wave character of all particles is the Heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. For position and momentum, the uncertainty principle is $Δ x Δ p \geq \frac{h}{4π}$ , where $Δ x$ is the uncertainty in position and $Δ p$ is the uncertainty in momentum.
These small limits are fundamentally important on the quantum-mechanical scale.

12.6: Discovery of the Atomic Nucleus

Atoms are composed of negatively charged electrons, first proved to exist in cathode-ray-tube experiments, and a positively charged nucleus.
All electrons are identical and have a charge-to-mass ratio of
$\frac{q_{e}}{m_{e}} = - 1.76 \times 10^{11} C/kg$ .
(12.6.1)
The positive charge in the nuclei is carried by particles called protons, which have a charge-to-mass ratio of
$\frac{q_{p}}{m_{p}} = 9 . 58 \times 10^{7} C/kg$ .
(12.6.2)
Mass of electron,
$m_{e} = 9 . 11 \times 10^{- 31} kg$ .
(12.6.4)
Mass of proton,
$m_{p} = 1 . 67 \times 10^{- 27} kg$ .
(12.6.5)
The planetary model of the atom pictures electrons orbiting the nucleus in the same way that planets orbit the sun.

12.7: Bohr’s Theory of the Hydrogen Atom

The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
$\frac{1}{λ} = R (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$ ,
(12.7.1)
where $λ$ is the wavelength of the emitted EM radiation and $R$ is the Rydberg constant, which has the value
$R = 1.097 \times 10^{7} m^{−1}$ .
(12.7.2)
The constants $n_{i}$ and $n_{f}$ are positive integers, and $n_{i}$ must be greater than $n_{f}$ .
Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with the energy of a photon emitted in a downward transition given by
$h f = E_{i} - E_{f}$ ,
(12.7.4)
where $h f$ is the emitted photon’s energy, equal to the initial atomic energy minus the final atomic energy. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
$L = n ℏ$ ,
(12.7.3)
where $L$ is the orbital angular momentum, $n$ is a positive integer, and $ℏ = \frac{h}{2 π}$ is the reduced Planck constant.
Additional quantized orbital quantities—orbital radius, orbital speed, and orbital energy—can be derived starting from Bohr’s assumption, and they yield predictions consistent with the experimental Rydberg formula.
While Bohr’s semiclassical model of the atom does not account for all experimental facts about the atom, it is an important stepping stone to fully-quantum-mechanical models of the atom.

12.8: The Wave Nature of Matter Causes Quantization

Quantization of orbital energy is caused by the wave nature of matter. Allowed orbits in atoms occur for constructive interference of electrons in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that is,
$n λ_{n} = 2 π r_{n}$ ( $n = 1, 2, 3, \dots$ ),
(12.8.1)
where $λ_{n}$ is the electron’s de Broglie wavelength.
Owing to the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits; rather, there are clouds of probability.
Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with the energy of a photon emitted in a downward transition given by
$h f = E_{i} - E_{f}$ ,
(12.7.4)
where $h f$ is the emitted photon’s energy, equal to the initial atomic energy minus the final atomic energy.
It is useful to plot orbit energies on a vertical graph called an energy-level diagram.
The allowed orbits are circular, Bohr proposed, and must have quantized orbital angular momentum given by
$L = m_{e} v r_{n} = n \frac{h}{2 π} = n ℏ$ ( $n = 1, 2, 3, \dots$ ),
(12.8.2)
where $L$ is the orbital angular momentum, $r_{n}$ is the radius of orbit $n$ , and $ℏ = \frac{h}{2 π}$ is the reduced Planck constant.

Questions and Exercises

12.2: Blackbody Radiation

Exercise 1

Give two examples of a physical property that is quantized. For each property, describe its allowed values.

Exercise 2

Give two examples of a physical property that is not quantized. For each property, describe the continuous range of values it can take.

Exercise 3

What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in the thermal oscillators?

Exercise 4

Why don’t we notice quantization in everyday events?

Exercise 5

A LiBr molecule oscillates with a frequency of

1 . 7 \times 10^{13} Hz

. (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of

n

for a state having an energy of 1.0 eV?

Exercise 6

The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?

12.3: The Photoelectric Effect

Exercise 7

What determines whether electromagnetic radiation can eject electrons from a particular material? Explain how this affects which parts of the electromagnetic spectrum can produce the photoelectric effect in that material.

Exercise 8

Monochromatic light ejects electrons from a metal surface. (a) If the light intensity increases while its frequency stays fixed, what happens to the number of electrons emitted per second and their maximum kinetic energy? (b) If the frequency increases, what happens to the maximum kinetic energy? Explain both answers using the photon model.

Exercise 9

Which aspects of the photoelectric effect cannot be explained without photons? Which can be explained without photons? Are the latter inconsistent with the existence of photons?

Exercise 10

What is the longest wavelength of EM radiation that can eject a photoelectron from a silver surface with a work function of 4.73 eV? Is this in the visible range?

Exercise 11

Find the longest wavelength of EM radiation that can eject an electron from a potassium surface with a work function of 2.30 eV. Is this visible EM radiation?

Exercise 12

What is the work function in eV of a magnesium surface if the longest wavelength that can eject electrons is 337 nm?

Exercise 13

Calculate the work function in eV of an aluminum surface if the longest wavelength that can eject electrons is 304 nm.

Exercise 14

(a) What is the maximum kinetic energy in eV of electrons ejected by 450-nm EM radiation from a sodium surface with a work function of 2.28 eV? (b) What magnitude of retarding potential is just sufficient to prevent even the most energetic photoelectrons from reaching the collector?

Exercise 15

UV radiation with a wavelength of 120 nm falls on a gold surface with a work function of 4.82 eV. What is the maximum kinetic energy in eV of the ejected photoelectrons?

Exercise 16

Violet light of wavelength 400 nm ejects electrons with a maximum kinetic energy of 0.820 eV from a sodium surface. What is the work function of this surface?

Exercise 17

UV radiation with a wavelength of 300 nm falls on a uranium surface, ejecting electrons with a maximum kinetic energy of 0.500 eV. What is the work function of this surface?

Exercise 18

What is the wavelength of EM radiation that ejects electrons with a maximum kinetic energy of 2.00 eV from a calcium surface with a work function of 2.87 eV? What type of EM radiation is this?

Exercise 19

Find the wavelength of photons that eject electrons with a maximum kinetic energy of 0.100 eV from a potassium surface with a work function of 2.30 eV. Are these photons visible?

Exercise 20

A laser beam delivers 2.00 mW of power at a wavelength of 400 nm to a calcium surface with a work function of 2.87 eV. (a) Find the ideal upper limit on the number of electrons ejected per second, assuming that every incident photon ejects one electron. (b) Find the ideal upper limit on the power carried away as electron kinetic energy, additionally assuming that every emitted electron has the maximum kinetic energy.

12.4: The Wave Nature of Matter

Exercise 21

(a) How is the interference pattern produced by electrons similar to that produced by water waves? (b) Compare how the two patterns are observed. When electrons are sent through the apparatus one at a time, what is detected for each electron, and how does the interference pattern emerge after many electrons have been detected?

Exercise 22

(a) Describe an experimental observation that supports the wave nature of matter. (b) Describe an experimental observation that supports the particle nature of electromagnetic radiation. For each observation, explain how it supports the corresponding description.

Exercise 23

(a) At what speed would an electron have a de Broglie wavelength of 1.00 mm? (b) Calculate the speed of an electron with a de Broglie wavelength of 1.00 μm. Through what potential difference must an electron be accelerated from rest to reach the speed in part (b)?

Exercise 24

What is the wavelength of an electron moving at 3.00% of the speed of light?

Exercise 25

(a) Using the nonrelativistic relation

p = m v

, calculate the speed of a proton with a de Broglie wavelength of 6.00 fm, approximately a nuclear length scale. (b) Repeat the calculation for an electron with the same wavelength. Compare both results with the speed of light. For which particle is the nonrelativistic approximation reasonable? Explain. (

1 fm = 10^{−15} m

Exercise 26

A pool ball has a mass of 0.170 kg and a radius of 2.86 cm. At what speed would its de Broglie wavelength equal its radius?

Exercise 27

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b) What is its kinetic energy in eV?

Exercise 28

What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?

Exercise 29

What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?

Exercise 30

The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

Exercise 31

An electron accelerated in an X-ray tube has a kinetic energy of 100 keV. (a) Using the nonrelativistic approximation, estimate its de Broglie wavelength. (b) What is the shortest wavelength of an X-ray photon that can be produced if all of the electron’s kinetic energy is transferred to a single photon? Compare this wavelength with the electron’s wavelength from part (a).

12.5: Uncertainty Principle

Exercise 32

A student says that an electron has a wavelength because it moves back and forth like a vibrating object. (a) Explain why this is not what the matter-wave description means. (b) What does the electron’s wave function allow us to predict about a position measurement?

Exercise 33

(a) State the Heisenberg uncertainty principle for position and momentum, and explain what each uncertainty represents. (b) Does this principle impose a universal lower limit on position uncertainty, such as

10^{−10} m

(an atomic scale),

10^{−15} m

(a nuclear scale), or some other fixed length? Explain. (c) If neither position uncertainty nor momentum uncertainty has a fixed lower limit by itself, what does the principle constrain about how precisely both can be known simultaneously?

Exercise 34

An electron is localized with a position uncertainty of 0.100 nm, approximately an atomic length scale. (a) Using the Heisenberg uncertainty principle, calculate the minimum uncertainty in its velocity. (b) Estimate the kinetic-energy scale associated with this localization using

K = \frac{1}{2} m_{e} {(Δ v)}^{2}

. Express your answer in electron volts. (c) Compare this energy scale with 4.48 eV, the energy required to separate a hydrogen molecule into two hydrogen atoms. Is it small enough to neglect when considering energies involved in molecular bonding?

12.6: Discovery of the Atomic Nucleus

Exercise 35

In Millikan’s oil-drop experiment, one looks at a small oil drop held motionless between two plates. Take the voltage between the plates to be 2033 V, and the plate separation to be 2.00 cm. The oil drop (of density

0 . 81 g / {cm}^{3}

) has a diameter of

4 . 0 \times 10^{- 6} m

. Find the charge on the drop, in terms of electron units.

Exercise 36

What two measured quantities allowed the first calculation of

m_{e}

, the mass of the electron?

(a) The ratios

q_{e} / m_{e}

and

q_{p} / m_{p}

(b) The values of

q_{e}

and

E_{B}

q_{e} / m_{e}

and

q_{e}

Explain how the two quantities you selected can be combined to calculate the electron’s mass. Which of these quantities did Thomson measure in his cathode-ray experiments?

Exercise 37

In Rutherford’s experiment, most alpha particles passed through a thin gold foil with little deflection, while a small fraction were deflected through large angles. (a) What does the first observation suggest about how an atom’s mass and positive charge are distributed? (b) What does the second observation reveal? Explain why these results were inconsistent with positive charge spread throughout the atom.

Exercise 38

Rutherford found the size of the nucleus to be about

10^{- 15} m

. This implied a huge density. What would this density be for gold?

Exercise 39

An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus?

12.7: Bohr’s Theory of the Hydrogen Atom

Exercise 40

(a) Calculate the wavelengths of the first and third lines in the Lyman series. (b) In which region of the electromagnetic spectrum do these wavelengths lie?

Exercise 41

What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

Exercise 42

Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.

Exercise 43

Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

Exercise 44

How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

Exercise 45

Which feature of Bohr’s model is classical, and what quantum assumption does it add?

Exercise 46

Look up the values of the quantities in

a_{B} = \frac{h^{2}}{4 π^{2} m_{e} k q_{e}^{2}}

, and verify that the Bohr radius

a_{B}

0.529 \times 10^{- 10} m

Exercise 47

Find the radius of a hydrogen atom in the

n = 2

state according to Bohr’s theory.

Exercise 48

Verify that the ground state energy

E_{0}

is 13.6 eV by using

E_{0} = \frac{2 π^{2} q_{e}^{4} m_{e} k^{2}}{h^{2}}

Exercise 49

If a hydrogen atom has its electron in the

n = 4

state, how much energy in eV is needed to ionize it?

Exercise 50

A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is

n

for a hydrogen atom if 0.850 eV of energy can ionize it?

Exercise 51

For hydrogen,

E_{n} = - \frac{13.6 eV}{n^{2}}

. (a) Calculate the energies of the

n = 2

and

n = 4

states and sketch their positions on an energy-level diagram. Mark the transition from

n = 4

n = 2

. (b) Find the energy of the emitted photon. (c) Find its frequency and wavelength. Explain why the photon’s energy equals the difference between the two levels rather than either level’s energy alone.

Exercise 52

Show that

\frac{13.6 eV}{h c} = 1.097 \times 10^{7} m^{−1} = R

(Rydberg’s constant), as discussed in the text.

Exercise 53

A wavelength of

4 . 653 μm

is observed in a hydrogen spectrum for a transition that ends in the

n_{f} = 5

level. What was

n_{i}

for the initial level of the electron?

12.8: The Wave Nature of Matter Causes Quantization

Exercise 54

How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?

Exercise 55

In the circular standing-wave model, a proposed orbit has a circumference equal to (a) three electron wavelengths or (b) three and a half electron wavelengths. Which satisfies the condition for an allowed orbit? Explain how the wave matches up after one complete circuit.

Exercise 56

Bohr’s model describes an electron following a definite circular orbit. The fuller quantum description uses a probability cloud. (a) What does that cloud represent? (b) If an electron’s position is measured once, is it detected at one location or spread throughout the cloud? Explain.

Source sections: Introduction to Quantum Mechanics, Blackbody Radiation, The Photoelectric Effect, The Wave Nature of Matter, Uncertainty Principle, Discovery of the Atomic Nucleus, Bohr’s Theory of the Hydrogen Atom, The Wave Nature of Matter Causes Quantization. Credits and licenses appear with each source section.