Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 7 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

chaPter 7

Introduction to quantum theory

It was once thought that the motion of atoms and subatomic

particles could be expressed using ‘classical mechanics’, the

laws of motion introduced in the seventeenth century by Isaac

Newton, for these laws were very successful at explaining the

motion of everyday objects and planets However, a proper

description of electrons, atoms, and molecules requires a

differ-ent kind of mechanics, ‘quantum mechanics’, which we

intro-duce in this chapter and then apply throughout the remainder

of the text

Experimental evidence accumulated towards the end of the

nineteenth century showed that classical mechanics failed

when it was applied to particles as small as electrons More

spe-cifically, careful measurements led to the conclusion that

par-ticles may not have an arbitrary energy and that the classical

concepts of particle and wave blend together In this Topic we

see how these observations set the stage for the development of

the concepts and equations of quantum mechanics through the

early twentieth century

In quantum mechanics, all the properties of a system are

expressed in terms of a wavefunction which is obtained by

solving the ‘Schrödinger equation’ In this Topic we see how to interpret wavefunctions

7C the principles of quantum theory

This Topic introduces some of the mathematical techniques

of quantum mechanics in terms of operators We also see that quantum theory introduces the ‘uncertainty principle’, one of the most profound departures from classical mechanics

What is the impact of this material?

In Impact I7.1 we highlight an application of quantum

mechan-ics that still requires much research before it becomes a useful technology It is based on the speculation that through ‘quan-tum computing’ calculations can be carried out on many states

of a system simultaneously, leading to a new generation of very fast computers

To read more about the impact of this material, scan the QR code, or go to bcs.whfreeman.com/webpub/chemistry/pchem10e/impact/pchem-7-1.html

7A the origins of quantum mechanics

The basic principles of classical mechanics are reviewed

in Foundations B In brief, they show that classical physics

(1)  predicts a precise trajectory for particles, with precisely specified locations and momenta at each instant, and (2) allows the translational, rotational, and vibrational modes of motion

to be excited to any energy simply by controlling the forces that are applied These conclusions agree with everyday experience Everyday experience, however, does not extend to individ-ual atoms, and careful experiments have shown that classical mechanics fails when applied to the transfers of very small energies and to objects of very small mass

We also investigate the properties of light The classical view,

discussed in Foundations C, is of light as an oscillating

electro-magnetic field that spreads as a wave through empty space with

a wavelength, λ (lambda), a frequency, ν (nu), and a constant speed, c (Fig C.1) Again, a number of experimental results are

not consistent with this interpretation

This Topic describes the experiments that revealed tions of classical physics The remaining Topics of the Chapter show how a new picture of light and matter led to the formu-lation of an entirely new and hugely successful theory called

limita-quantum mechanics.

Here we outline three experiments conducted near the end of the nineteenth century and which drove scientists to the view that energy can be transferred only in discrete amounts

(a) Black-body radiation

A hot object emits electromagnetic radiation At high tures, an appreciable proportion of the radiation is in the visible region of the spectrum and a higher proportion of short-wave-length blue light is generated as the temperature is raised This behaviour is seen when a heated metal bar glowing red hot becomes white hot when heated further The dependence is illustrated in Fig 7A.1, which shows how the energy output varies with wavelength at several temperatures The curves are

tempera-those of an ideal emitter called a black body, which is an object

capable of emitting and absorbing all wavelengths of radiation uniformly A good approximation to a black body is a pinhole

in an empty container maintained at a constant temperature: any radiation leaking out of the hole has been absorbed and re-emitted inside so many times as it reflected around inside

Contents

(a) Black-body radiation 282

example 7a.1: using the Planck distribution 284

(b) Heat capacities 285

brief illustration 7a.1: the debye formula 286

(c) Atomic and molecular spectra 286

brief illustration 7a.2: the bohr frequency condition 287

(a) The particle character of electromagnetic radiation 287

example 7a.2: calculating the number of photons 288

example 7a.3: calculating the maximum

wavelength capable of photoejection 289

(b) The wave character of particles 289

example 7a.4: estimating the de broglie wavelength 290

➤

➤ Why do you need to know this material?

You should know how experimental results motivated

the development of quantum theory, which underlies

all descriptions of the structure of atoms and molecules

and pervades the whole of spectroscopy and chemistry

in general.

➤

➤ What is the key idea?

Experimental evidence led to the conclusions that energy

cannot be continuously varied and that the classical

concepts of a ‘particle’ and a ‘wave’ blend together when

applied to light, atoms, and molecules.

➤

➤ What do you need to know already?

You should be familiar with the basic principles of classical

mechanics, which are reviewed in Foundations B The

discussion of heat capacities of solids formally makes use

of material in Topic 2A but is introduced independently

here.

7A The origins of quantum mechanics  283

the container that it has come to thermal equilibrium with the

walls (Fig 7A.2)

The approach adopted by nineteenth-century scientists

to explain black-body radiation was to calculate the energy

density, dE, the total energy in a region of the

electromag-netic field divided by the volume of the region (units: joules

per metre-cubed, J m−3), due to all the oscillators

correspond-ing to wavelengths between λ and λ + dλ This energy density

is proportional to the width, dλ, of this range, and is written

where ρ (rho), the constant of proportionality between d E

and dλ, is called the density of states (units: joules per metre4,

J m−4) A high density of states at the wavelength λ and

tem-perature T simply means that there is a lot of energy associated

with wavelengths lying between λ and λ + dλ at that

tempera-ture The total energy density in a region is the integral over all

a region of volume V is this energy density multiplied by the

volume:

The physicist Lord Rayleigh thought of the electromagnetic field as a collection of oscillators of all possible frequencies He regarded the presence of radiation of frequency ν (and there-

fore of wavelength λ = c/ν, eqn C.3) as signifying that the

elec-tromagnetic oscillator of that frequency had been excited (Fig 7A.3) Rayleigh knew that according to the classical equipar-

tition principle (Foundations B), the average energy of each oscillator, regardless of its frequency, is kT On that basis, with

minor help from James Jeans, he arrived at the Rayleigh–Jeans

law for the density of states:

ρ λ( , )T =8πλ kT4 rayleigh–Jeans law (7A.4)

where k is Boltzmann’s constant (k = 1.381 × 10−23 J K−1)

Although the Rayleigh–Jeans law is quite successful at long wavelengths (low frequencies), it fails badly at short wave-

lengths (high frequencies) Thus, as λ decreases, ρ increases

without going through a maximum (Fig 7A.4) The equation therefore predicts that oscillators of very short wavelength (corresponding to ultraviolet radiation, X-rays, and even

γ-rays) are strongly excited even at room temperature The total

energy density in a region, the integral in eqn 7A.2, is also dicted to be infinite at all temperatures above zero This absurd result, which implies that a large amount of energy is radiated

pre-in the high-frequency region of the electromagnetic spectrum,

is called the ultraviolet catastrophe According to classical

Wavelength, λ

Figure 7A.1 The energy distribution in a black-body cavity at

several temperatures Note how the spectral density of states

increases in the region of shorter wavelength as the temperature

is raised, and how the peak shifts to shorter wavelengths

Detected radiation

Pinhole

Container

at a

temperature T

Figure 7A.2 An experimental representation of a black body

is a pinhole in an otherwise closed container The radiation

is reflected many times within the container and comes to

thermal equilibrium with the walls Radiation leaking out

through the pinhole is characteristic of the radiation within the

container

(a)

(b)

Figure 7A.3 The electromagnetic vacuum can be regarded

as able to support oscillations of the electromagnetic field When a high-frequency, short-wavelength oscillator (a) is excited, that frequency of radiation is present The presence of low-frequency, long-wavelength radiation (b) signifies that an oscillator of the corresponding frequency has been excited

284 7 Introduction to quantum theory

physics, even cool objects should radiate in the visible and

ultraviolet regions, so objects should glow in the dark; there

should in fact be no darkness

In 1900, the German physicist Max Planck found that he

could account for the experimental observations by proposing

that the energy of each electromagnetic oscillator is limited to

discrete values and cannot be varied arbitrarily This proposal

is contrary to the viewpoint of classical physics in which all

possible energies are allowed and every oscillator has a mean

energy kT The limitation of energies to discrete values is called

the quantization of energy In particular, Planck found that he

could account for the observed distribution of energy if he

sup-posed that the permitted energies of an electromagnetic

oscil-lator of frequency ν are integer multiples of hν:

where h is a fundamental constant now known as Planck’s

constant On the basis of this assumption, Planck was able to

derive what is now called the Planck distribution:

ρ λ( , )T =λ5(e8hc kTπ/λ hc−1) Planck distribution (7A.6)

This expression fits the experimental curve very well at all

wavelengths (Fig 7A.5), and the value of h, which is an

unde-termined parameter in the theory, may be obtained by varying

its value until a best fit is obtained The currently accepted value

for h is 6.626 × 10−34 J s

As usual, it is a good idea to ‘read’ the content of an equation:

• The Planck distribution resembles the Rayleigh–

Jeans law (eqn 7A.4) apart from the all-important

exponential factor in the denominator For short

wavelengths, hc/νkT ≫ 1 and e hc/λkT → ∞ faster than

λ5 → 0; therefore ρ → 0 as λ → 0 or ν → ∞ Hence, the

energy density approaches zero at high frequencies,

in agreement with observation

• For long wavelengths, hc/λkT ≪ 1, and the denominator in the Planck distribution can be

replaced by (see Mathematical background 1)

we find that the Planck distribution reduces to the Rayleigh–Jeans law

• As we should infer from the graph in Fig 7A.5, the total energy density (the integral in eqn 7A.2 and therefore the area under the curve) is no longer infinite, and in fact

0

5 4 3

π

πλ

λ λ

Example 7A.1 Using the Planck distribution

Compare the energy output of a black-body radiator (such as

an incandescent lamp) at two different wavelengths by lating the ratio of the energy output at 450 nm (blue light) to that at 700 nm (red light) at 298 K

calcu-Method Use eqn 7A.6 At a temperature T, the ratio of the spectral density of states at a wavelength λ1 to that at λ2 is

ρ λ

ρ λ λ λ

λ λ

( , )( , ) (( ))

/ / 1

2

2 1

5 2 1

11

T T

Wavelength, λ

Rayleigh–Jeans law

Experimental

Figure 7A.4 The Rayleigh–Jeans law (eqn 7A.4) predicts an

infinite spectral density of states at short wavelengths This

approach to infinity is called the ultraviolet catastrophe

7A The origins of quantum mechanics  285

It is easy to see why Planck’s approach was successful whereas

Rayleigh’s was not The thermal motion of the atoms in the

walls of the black body excites the oscillators of the

electromag-netic field According to classical mechanics, all the oscillators

of the field share equally in the energy supplied by the walls,

so even the highest frequencies are excited The excitation of

very high frequency oscillators results in the ultraviolet

catas-trophe According to Planck’s hypothesis, however, oscillators

are excited only if they can acquire an energy of at least hν

This energy is too large for the walls to supply in the case of the

very high frequency oscillators, so the latter remain unexcited

The effect of quantization is to reduce the contribution from

the high frequency oscillators, for they cannot be significantly

excited with the energy available

(b) Heat capacities

In the early nineteenth century, the French scientists

Pierre-Louis Dulong and Alexis-Thérèse Petit determined the heat

capacities, C V = (∂U/∂T) V (Topic 2A), of a number of

mona-tomic solids On the basis of some somewhat slender

experi-mental evidence, they proposed that the molar heat capacities

of all monatomic solids are the same and (in modern units)

close to 25 J K−1 mol−1

Dulong and Petit’s law is easy to justify in terms of

classi-cal physics in much the same way as Rayleigh attempted to

explain black-body radiation If classical physics were valid,

the equipartition principle could be used to infer that the

mean energy of an atom as it oscillates about its mean

posi-tion in a solid is kT for each direcposi-tion of displacement As

each atom can oscillate in three dimensions, the average

energy of each atom is 3kT; for N atoms the total energy is 3NkT The contribution of this motion to the molar internal

energy is therefore

Um=3N kTA =3RT (7A.8a)

because NAk = R, the gas constant The molar constant volume

heat capacity is then predicted to be

monatomic solids are lower than 3R at low temperatures, and that the values approach zero as T → 0 To account for these

observations, Einstein (in 1905) assumed that each atom lated about its equilibrium position with a single frequency ν

oscil-He then invoked Planck’s hypothesis to assert that the energy

of oscillation is confined to discrete values, and specifically to

nhν, where n is an integer Einstein discarded the

equiparti-tion result, calculated the vibraequiparti-tional contribuequiparti-tion of the atoms

to the total molar internal energy of the solid (by a method described in Topic 15E), and obtained the expression now

known as the Einstein formula:

T T

,

/ /

The Einstein temperature, θE = hν/k, is a way of expressing

the frequency of oscillation of the atoms as a temperature and allows us to be quantitative about what we mean by ‘high tem-

perature’ (T ≫ θE) and ‘low temperature’ (T ≪ θE) in this text Note that a high vibrational frequency corresponds to a high Einstein temperature

con-As before, we now ‘read’ this expression:

• At high temperatures (when T ≫ θE) the exponentials

in fE can be expanded as 1 + θE/T + … and higher

terms ignored The result is

Consequently, the classical result (C V,m = 3R) is

obtained at high temperatures

• At low temperatures (when T ≪ θE) and eθE/T1,

E E

Insert the data to evaluate this ratio

Answer With λ1 = 450 nm and λ2 = 700 nm:

.

e

At room temperature, the proportion of short wavelength

radiation is insignificant

Self-test 7A.1 Repeat the calculation for a temperature of

13.6 MK, which is close to the temperature at the core of the

Sun

Answer: 5.85

286 7 Introduction to quantum theory

The strongly decaying exponential function goes to zero

more rapidly than 1/T goes to infinity; so fE → 0 as T → 0,

and the heat capacity therefore approaches zero too

We see that Einstein’s formula accounts for the decrease of

heat capacity at low temperatures The physical reason for this

success is that at low temperatures only a few oscillators possess

enough energy to oscillate significantly so the solid behaves as

though it contains far fewer atoms than is actually the case At

higher temperatures, there is enough energy available for all

the oscillators to become active: all 3N oscillators contribute,

many of their energy levels are accessible, and the heat capacity

approaches its classical value

Figure 7A.6 shows the temperature dependence of the heat

capacity predicted by the Einstein formula The general shape

of the curve is satisfactory, but the numerical agreement is in

fact quite poor The poor fit arises from Einstein’s assumption

that all the atoms oscillate with the same frequency, whereas in

fact they oscillate over a range of frequencies from zero up to a

maximum value, νD This complication is taken into account by

averaging over all the frequencies present, the final result being

the Debye formula:

x x

where θD = hνD/k is the Debye temperature The integral in eqn

7A.11 has to be evaluated numerically, but that is simple with

mathematical software The details of this modification, which,

as Fig 7A.7 shows, gives improved agreement with

experi-ment, need not distract us at this stage from the main

conclu-sion, which is that quantization must be introduced in order to

explain the thermal properties of solids

(c) Atomic and molecular spectra

The most compelling and direct evidence for the

quantiza-tion of energy comes from spectroscopy, the detecquantiza-tion and

analysis of the electromagnetic radiation absorbed, emitted,

or scattered by a substance The record of the intensity of light intensity transmitted or scattered by a molecule as a function

of frequency (ν), wavelength (λ), or wavenumber ( = c is / )

called its spectrum (from the Latin word for appearance).

A typical atomic spectrum is shown in Fig 7A.8, and a typical molecular spectrum is shown in Fig 7A.9 The obvious feature of both is that radiation is emitted or absorbed at a series of discrete frequencies This observation can be understood if the energy

of the atoms or molecules is also confined to discrete values, for then energy can be discarded or absorbed only in discrete amounts (Fig 7A.10) Then, if the energy of an atom decreases by

ΔE, the energy is carried away as radiation of frequency ν, and an

emission ‘line’, a sharply defined peak, appears in the spectrum

We say that a molecule undergoes a spectroscopic transition, a change of state, when the Bohr frequency condition

Brief illustration 7A.1 The Debye formula

The Debye temperature for lead is 105 K, corresponding to a vibrational frequency of 2.2 × 1012 Hz As we see from Fig 7A.7,

fD ≈ 1 for T > θD and the heat capacity is almost classical For

lead at 25 °C, corresponding to T/θD = 2.8, fD = 0.99 and the heat capacity has almost its classical value

Self-test 7A.2 Evaluate the Debye temperature for diamond (νD = 4.6 × 1013 Hz) What fraction of the classical value of the heat capacity does diamond reach at 25 °C?

Answer: 2230 K; 15 per cent

Figure 7A.6 Experimental low-temperature molar heat

capacities and the temperature dependence predicted on the

basis of Einstein’s theory His equation (eqn 7A.10) accounts for

the dependence fairly well, but is everywhere too low

C V,m

T/θE or T/ θD

Debye Einstein

Figure 7A.7 Debye’s modification of Einstein’s calculation (eqn 7A.11) gives very good agreement with experiment For

copper, T/ θD= 2 corresponds to about 170 K, so the detection of deviations from Dulong and Petit’s law had to await advances

in low-temperature physics

(7A.11)

debye formula

7A The origins of quantum mechanics  287

is fulfilled We develop the principles and applications of atomic spectroscopy in Topics 9A–9C and of molecular spectroscopy

in Topics 12A–14D

At this stage we have established that the energies of the tromagnetic field and of oscillating atoms are quantized In this section we see the experimental evidence that led to the revision of two other basic concepts concerning natural phe-nomena One experiment shows that electromagnetic radia-tion—which classical physics treats as wave-like—actually also displays the characteristics of particles Another experiment shows that electrons—which classical physics treats as parti-cles—also display the characteristics of waves

elec-(a) The particle character of electromagnetic radiation

The observation that electromagnetic radiation of frequency

ν can possess only the energies 0, hν, 2hν, … suggests (and

at this stage it is only a suggestion) that it can be thought of

as consisting of 0, 1, 2, … particles, each particle having an

energy hν Then, if one of these particles is present, the energy

is hν, if two are present the energy is 2hν, and so on These

particles of electromagnetic radiation are now called photons

The observation of discrete spectra from atoms and molecules can be pictured as the atom or molecule generating a photon

Brief illustration 7A.2 The Bohr frequency condition

Atomic sodium produces a yellow glow (as in some street lamps) resulting from the emission of radiation of 590 nm The spectroscopic transition responsible for the emission involves electronic energy levels that have a separation given

−−19JThis energy difference can be expressed in a variety of ways For instance, multiplication by Avogadro’s constant results in

an energy separation per mole of atoms, of 203 kJ mol−1, parable to the energy of a weak chemical bond The calculated

com-value of ΔE also corresponds to 2.10 eV (Foundations B).

Self-test 7A.3 Neon lamps emit red radiation of wavelength

736 nm What is the energy separation of the levels in joules, kilojoules per mole, and electronvolts responsible for the emission?

Answer: 2.70 × 10 −19 J, 163 kJ mol −1 , 1.69 eV

Wavelength, λ/nm

Figure 7A.8 A region of the spectrum of radiation emitted by

excited iron atoms consists of radiation at a series of discrete

wavelengths (or frequencies)

Wavelength, λ/nm

Rotational transitions Vibrational

transitions

Figure 7A.9 When a molecule changes its state, it does so by

absorbing radiation at definite frequencies This spectrum is

part of that due to the electronic, vibrational, and rotational

excitation of sulfur dioxide (SO2) molecules This observation

suggests that molecules can possess only discrete energies,

not an arbitrary energy

Figure 7A.10 Spectroscopic transitions, such as those shown

above, can be accounted for if we assume that a molecule

emits electromagnetic radiation as it changes between

discrete energy levels Note that high-frequency radiation is

emitted when the energy change is large

288 7 Introduction to quantum theory

of energy hν when it discards an energy of magnitude ΔE, with

ΔE = hν.

So far, the existence of photons is only a suggestion

Experimental evidence for their existence comes from the

measurement of the energies of electrons produced in the

pho-toelectric effect This effect is the ejection of electrons from

metals when they are exposed to ultraviolet radiation The

experimental characteristics of the photoelectric effect are as

follows:

• No electrons are ejected, regardless of the intensity of the

radiation, unless its frequency exceeds a threshold value

characteristic of the metal

• The kinetic energy of the ejected electrons increases

linearly with the frequency of the incident radiation but

is independent of the intensity of the radiation

• Even at low light intensities, electrons are ejected

immediately if the frequency is above the threshold

Figure 7A.11 illustrates the first and second characteristics.These observations strongly suggest that the photoelectric effect depends on the ejection of an electron when it is involved

in a collision with a particle-like projectile that carries enough energy to eject the electron from the metal If we suppose that

the projectile is a photon of energy hν, where ν is the frequency

of the radiation, then the conservation of energy requires that the kinetic energy of the ejected electron (Ek=1me 2)

2 v should obey

Ek=1mev2= −h Φ Photoelectric effect (7A.13)

In this expression, Φ (uppercase phi) is a characteristic of the

metal called its work function, the energy required to remove

an electron from the metal to infinity (Fig 7A.12), the analogue

of the ionization energy of an individual atom or molecule We

Example 7A.2 Calculating the number of photons

Calculate the number of photons emitted by a 100 W yellow

lamp in 1.0 s Take the wavelength of yellow light as 560 nm

and assume 100 per cent efficiency

Method Each photon has an energy hν, so the total number

of photons needed to produce an energy E is E/hν To use this

equation, we need to know the frequency of the radiation

(from ν = c/λ) and the total energy emitted by the lamp The

latter is given by the product of the power (P, in watts) and the

time interval for which the lamp is turned on (E = PΔt).

Answer The number of photons is

N E=h=h c( / )P t∆λ =λ P t hc∆

Substitution of the data gives

N =( 6 626 10( 5 60 10×× −−734mJs) () ( ××1002 998 10Js−×1) ( )×81 0msss−1)=2 8 10 × 20

Note that it would take the lamp nearly 40 min to produce

1 mol of these photons

A note on good practice To avoid rounding and other

numerical errors, it is best to carry out algebraic

calcu-lations first, and to substitute numerical values into a

single, final formula Moreover, an analytical result may

be used for other data without having to repeat the entire

calculation

Self-test 7A.4 How many photons does a monochromatic

(sin-gle frequency) infrared rangefinder of power 1 mW and

Increasing work function

Figure 7A.11 In the photoelectric effect, it is found that

no electrons are ejected when the incident radiation has a frequency below a value characteristic of the metal, and, above that value, the kinetic energy of the photoelectrons varies linearly with the frequency of the incident radiation

Figure 7A.12 The photoelectric effect can be explained if

it is supposed that the incident radiation is composed of photons that have energy proportional to the frequency of the radiation (a) The energy of the photon is insufficient to drive

an electron out of the metal (b) The energy of the photon is more than enough to eject an electron, and the excess energy

is carried away as the kinetic energy of the photoelectron

7A The origins of quantum mechanics  289

can now see that the existence of photons accounts for the three

observations we have summarized:

• Photoejection cannot occur if hν < Φ because the photon

brings insufficient energy

• Equation 7A.13 predicts that the kinetic energy of an

ejected electron should increase linearly with frequency

• When a photon collides with an electron, it gives up all

its energy, so we should expect electrons to appear as

soon as the collisions begin, provided the photons have

sufficient energy

A practical application of eqn 7A.13 is that it provides a

tech-nique for the determination of Planck’s constant, for the slopes

of the lines in Fig 7A.11 are all equal to h.

(b) The wave character of particles

Although contrary to the long-established wave theory of light, the view that light consists of particles had been held before, but discarded No significant scientist, however, had taken the view that matter is wave-like Nevertheless, experi-ments carried out in 1925 forced people to consider that possibility The crucial experiment was performed by the American physicists Clinton Davisson and Lester Germer, who observed the diffraction of electrons by a crystal (Fig

7A.13) Diffraction is the interference caused by an object in

the path of waves Depending on whether the interference is constructive or destructive, the result is a region of enhanced

or diminished intensity of the wave Davisson and Germer’s success was a lucky accident, because a chance rise of temper-ature caused their polycrystalline sample to anneal, and the ordered planes of atoms then acted as a diffraction grating

At almost the same time, G.P Thomson, working in Scotland, showed that a beam of electrons was diffracted when passed through a thin gold foil

The Davisson–Germer experiment, which has since been repeated with other particles (including α particles and molecular hydrogen), shows clearly that particles have wave-like properties, and the diffraction of neutrons is a well-established technique for investigating the structures and dynamics of condensed phases (Topic 18A) We have also seen that waves of electromagnetic radiation have particle-like properties Thus we are brought to the heart of modern

Example 7A.3 Calculating the maximum wavelength

capable of photoejection

A photon of radiation of wavelength 305 nm ejects an electron

from a metal with a kinetic energy of 1.77 eV Calculate the

maximum wavelength of radiation capable of ejecting an

elec-tron from the metal

Method Use eqn 7A.13 rearranged into Φ = hν − Ek with ν = c/λ

to calculate the work function of the metal from the data The

threshold for photoejection, the frequency able to remove the

electron but not give it any excess energy, then corresponds

to radiation of frequency νmin = Φ/h Use this value of the

frequency to calculate the maximum wavelength capable of

photoejection

Answer From the expression for the work function Φ = hν − Ek

the minimum frequency for photoejection is

Self-test 7A.5 When ultraviolet radiation of wavelength

165 nm strikes a certain metal surface, electrons are ejected with a speed of 1.24 Mm s−1 Calculate the speed of electrons ejected by radiation of wavelength 265 nm

Answer: 735 km s −1

Electron beam

Diffracted electrons

Ni crystal

Figure 7A.13 The Davisson–Germer experiment The scattering of an electron beam from a nickel crystal shows a variation of intensity characteristic of a diffraction experiment

in which waves interfere constructively and destructively in different directions

290 7 Introduction to quantum theory

physics When examined on an atomic scale, the classical

concepts of particle and wave melt together, particles taking

on the characteristics of waves, and waves the characteristics

of particles

Some progress towards coordinating these properties had

already been made by the French physicist Louis de Broglie

when, in 1924, he suggested that any particle, not only

pho-tons, travelling with a linear momentum p = mv (with m the

mass and v the speed of the particle) should have in some

sense a wavelength given by what is now called the de Broglie

relation:

That is, a particle with a high linear momentum has a short

wavelength (Fig 7A.14) Macroscopic bodies have such high

momenta even when they are moving slowly (because their

mass is so great), that their wavelengths are undetectably small,

and the wavelike properties cannot be observed This

unde-tectability is why, in spite of its deficiencies, classical mechanics

can be used to explain the behaviour of macroscopic bodies

It is necessary to invoke quantum mechanics only for

micro-scopic systems, such as atoms and molecules, in which masses

are small

We now have to conclude that not only has electromagnetic radiation the character classically ascribed to particles, but electrons (and all other particles) have the characteristics clas-sically ascribed to waves This joint particle and wave character

of matter and radiation is called wave − particle duality.

Example 7A.4 Estimating the de Broglie wavelength

Estimate the wavelength of electrons that have been ated from rest through a potential difference of 40 kV

acceler-Method To use the de Broglie relation, we need to know the

linear momentum, p, of the electrons To calculate the linear

momentum, we note that the energy acquired by an electron

accelerated through a potential difference Δφ is eΔφ, where

e is the magnitude of its charge At the end of the period of

acceleration, all the acquired energy is in the form of kinetic

energy, Ek me 2 p m2

e

/2

=1 =

2 v , so we can determine p by setting

p2/2me equal to eΔφ As before, carry through the calculation

algebraically before substituting the data

Answer The expression p2/2me = eΔφ solves to p = (2meeΔφ)1/2;

then, from the de Broglie relation λ = h/p,

λ=(2m e h φ)1 2/

e ∆Substitution of the data and the fundamental constants (from inside the front cover) gives

λ ={ ( 2 9 109 10× × −31kg) ( 6 626 10×. 1 609 10× ×−34−Js19C) ( ×4 0 10× 4V))}

/

1 2 12

6 1 10

= × − mFor the manipulation of units we have used 1 V C = 1 J and

1 J = 1 kg m2 s−2 The wavelength of 6.1 pm is shorter than typical bond lengths in molecules (about 100 pm) Electrons acceler-ated in this way are used in the technique of electron diffraction

for the visualization of biological systems (Impact I7.1) and the

determination of the structures of solid surfaces (Topic 22A)

Self-test 7A.6 Calculate the wavelength of (a) a neutron with a

translational kinetic energy equal to kT at 300 K, (b) a tennis

ball of mass 57 g travelling at 80 km h−1

Answer: (a) 178 pm, (b) 5.2 × 10 −34 m

Checklist of concepts

☐ 1 A black body is an object capable of emitting and

absorbing all wavelengths of radiation uniformly ☐ 2 The vibrations of atoms can take up energy only in

dis-crete amounts

Short wavelength,

high momentum Long wavelength,low momentum

Figure 7A.14 An illustration of the de Broglie relation

between momentum and wavelength The wave is

associated with a particle A particle with high momentum

corresponds to a wave with a short wavelength, and

vice versa

7A The origins of quantum mechanics  291

☐ 3 Atomic and molecular spectra show that atoms and

molecules can take up energy only in discrete amounts

☐ 4 The photoelectric effect establishes the view that

elec-tromagnetic radiation, regarded in classical physics as

wavelike, consists of particles (photons)

☐ 5 The diffraction of electrons establishes the view that trons, regarded in classical physics as particles, are wave-

elec-like with a wavelength given by the de Broglie relation.

☐ 6 Wave–particle duality is the recognition that the

con-cepts of particle and wave blend together

Checklist of equations

Photoelectric effect Ek= 1me 2 =h

2 v −Φ Φ is the work function 7A.13

7B dynamics of microscopic systems

Wave–particle duality (Topic 7A) strikes at the heart of

clas-sical physics, where particles and waves are treated as entirely

distinct entities Experiments have also shown that the

ener-gies of electromagnetic radiation and of matter cannot be

var-ied continuously, and that for small objects the discreteness of

energy is highly significant In classical mechanics, in contrast,

energies can be varied continuously Such total failure of

clas-sical physics for small objects implied that its basic concepts

are false A new mechanics—quantum mechanics—had to be

devised to take its place

A new mechanics can be constructed from the ashes of sical physics by supposing that, rather than travelling along a definite path, a particle is distributed through space like a wave This remark may seem mysterious: it will be interpreted more fully shortly The mathematical representation of the wave that

clas-in quantum mechanics replaces the classical concept of

trajec-tory is called a wavefunction, ψ (psi), a function that contains

all the dynamical information about a system, such as its tion and momentum

In 1926, the Austrian physicist Erwin Schrödinger proposed an

equation for finding the wavefunction of any system The

time-independent Schrödinger equation for a particle of mass m

moving in one dimension with energy E in a system that does

not change with time (for instance, its volume remains stant) is

con-−2m x V x2 dd2ψ2 + ( )ψ=E ψ

The factor V(x) is the potential energy of the particle at the point x; because the total energy E is the sum of potential and

kinetic energies, the first term must be related (in a manner

we explore later) to the kinetic energy of the particle; ħ = h/2π (which is read h-cross or h-bar) is a convenient modification

of Planck’s constant with the value 1.055 × 10−34 J s Three ple but important general forms of the potential energy are (the explicit forms are found in the corresponding Topics):

sim-• For a particle moving freely in one dimension the

potential energy is constant, so V(x) = V It is often convenient to write V = 0 (Topic 8A).

• For a particle free to oscillate to-and-fro near a point x0,

V(x) ∝ (x − x0)2 (Topic 8B)

• For two electric charges Q1 and Q2 separated by a

distance x, V(x) ∝ Q1Q2/x (Foundations B).

The following Justification shows that the Schrödinger

equa-tion is plausible and the discussions later in the chapter will help to overcome its apparent arbitrariness For the present, we

➤

➤ Why do you need to know this material?

Quantum theory provides the essential foundation for

understanding of the properties of electrons in atoms and

molecules.

➤

➤ What is the key idea?

All the dynamical properties of a system are contained

in the wavefunction, which is obtained by solving the

Schrödinger equation.

➤

➤ What do you need to know already?

You need to be aware of the shortcomings of classical

physics that drove the development of quantum theory

(Topic 7A).

time-independent schrödinger

Contents

7b.2 The Born interpretation of the wavefunction 293

example 7b.1: Interpreting a wavefunction 294

(a) Normalization 295

example 7b.2: normalizing a wavefunction 296

(b) Constraints on the wavefunction 296

example 7b.3: determining a probability 298

7B Dynamics of microscopic systems  293

shall treat the equation simply as a quantum-mechanical

pos-tulate that replaces Newton’s pospos-tulate of his apparently equally

arbitrary equation of motion (that force = mass × acceleration)

Various ways of expressing the Schrödinger equation, of

incor-porating the time-dependence of the wavefunction, and of

extending it to more dimensions, are collected in Table 7B.1 In

the Topics of Chapter 8 we solve the equation for a number of

important cases; in this chapter we are mainly concerned with

its significance, the interpretation of its solutions, and seeing

how it implies that energy is quantized

wavefunction

A central principle of quantum mechanics is that the

wavefunc-tion contains all the dynamical informawavefunc-tion about the system it describes Here we concentrate on the information it carries

about the location of the particle

The interpretation of the wavefunction in terms of the location of the particle is based on a suggestion made by Max Born He made use of an analogy with the wave theory

of light, in which the square of the amplitude of an magnetic wave in a region is interpreted as its intensity and therefore (in quantum terms) as a measure of the probability

electro-of finding a photon present in the region The Born

interpre-tation of the wavefunction focuses on the square of the

wave-function (or the square modulus, |ψ|2 = ψ *ψ, if ψ is complex; see Mathematical background 3) For a one-dimensional sys-

tem (Fig 7B.1):

If the wavefunction of a particle has the value ψ at some point x, then the probability of finding the particle between x and x + dx is proportional to

|ψ|2dx.

Thus, |ψ|2 is the probability density, and to obtain the

prob-ability it must be multiplied by the length of the infinitesimal

region dx The wavefunction ψ itself is called the probability

amplitude For a particle free to move in three dimensions

(for example, an electron near a nucleus in an atom), the

Justification 7B.1 The plausibility of the Schrödinger

equation

The Schrödinger equation can be seen to be plausible by

not-ing that it implies the de Broglie relation (eqn 7A.14, p = h/λ)

for a freely moving particle After writing V(x) = V, we can

rearrange eqn 7B.1 into

treated in Mathematical background 4 at the end of Chapter 8;

we need only the simplest procedures in this Topic In this case a solution is

/

kx k 2m E V2

1 2



We now recognize that cos kx is a wave of wavelength λ = 2π/k,

as can be seen by comparing cos kx with the standard form

of a harmonic wave, cos(2πx/λ) (Foundations C) The tity E − V is equal to the kinetic energy of the particle, Ek, so

quan-k = (2mEk/2)1/2, which implies that Ek = k22/2m Because

Ek = p2/2m (Foundations B), it follows that p = k Therefore,

the linear momentum is related to the wavelength of the function by

wave-p=2λπ×2hπ=h λwhich is the de Broglie relation

Table 7B.1 The Schrödinger equation

2 2

Two dimensions

2 2

Alternative forms

Legendrian

φ θ θ θ θ θ

2 2

2 2

294 7 Introduction to quantum theory

wavefunction depends on the point r with coordinates x, y, and

z, and the interpretation of ψ(r) is as follows (Fig 7B.2):

If the wavefunction of a particle has the value ψ at some

point r, then the probability of finding the particle in an

infinitesimal volume dτ = dxdydz at that point is

proportional to |ψ|2dτ.

The Born interpretation does away with any worry about the

significance of a negative (and, in general, complex) value of ψ

because |ψ|2 is real and never negative There is no direct

sig-nificance in the negative (or complex) value of a wavefunction:

only the square modulus, a positive quantity, is directly

physi-cally significant, and both negative and positive regions of a

wavefunction may correspond to a high probability of finding a

particle in a region (Fig 7B.3) However, later we shall see that

the presence of positive and negative regions of a wavefunction

is of great indirect significance, because it gives rise to the

possi-bility of constructive and destructive interference between

dif-ferent wavefunctions

Example 7B.1 Interpreting a wavefunction

In Topic 9A it is shown that the wavefunction of an electron in the lowest energy state of a hydrogen atom is proportional to

e−r a/ 0, with a0 a constant and r the distance from the nucleus

Calculate the relative probabilities of finding the electron

inside a region of volume δV = 1.0 pm3, which is small even on the scale of the atom, located at (a) the nucleus, (b) a distance

a0 from the nucleus

Method The region of interest is so small on the scale of the

atom that we can ignore the variation of ψ within it and write the probability, P, as proportional to the probability density (ψ 2; note that ψ is real) evaluated at the point of interest mul- tiplied by the volume of interest, δV That is, P ∝ ψ 2δV, with

Therefore, the ratio of probabilities is 1.0/0.14 = 7.1 Note that

it is more probable (by a factor of 7) that the electron will be found at the nucleus than in a volume element of the same

size located at a distance a0 from the nucleus The negatively charged electron is attracted to the positively charged nucleus, and is likely to be found close to it

A note on good practice The square of a wavefunction is

a probability density, and (in three dimensions) has the dimensions of 1/length3 It becomes a (unitless) prob-ability when multiplied by a volume In general, we have

to take into account the variation of the amplitude of the wavefunction over the volume of interest, but here we are supposing that the volume is so small that the variation of

ψ in the region can be ignored.

Figure 7B.1 The wavefunction ψ is a probability amplitude in

the sense that its square modulus (ψ*ψ or |ψ|2) is a probability

density The probability of finding a particle in the region

dx located at x is proportional to | ψ|2dx We represent

the probability density by the density of shading in the

superimposed band

dx dy dz

z

r

Figure 7B.2 The Born interpretation of the wavefunction

in three-dimensional space implies that the probability of

finding the particle in the volume element dτ = dxdydz at some

location r is proportional to the product of d τ and the value of

|ψ|2 at that location

Figure 7B.3 The sign of a wavefunction has no direct physical significance: the positive and negative regions of this wavefunction both correspond to the same probability distribution (as given by the square modulus of ψ and depicted

by the density of the shading)

7B Dynamics of microscopic systems  295

(a) Normalization

A mathematical feature of the Schrödinger equation is that if

ψ is a solution, then so is Nψ, where N is any constant This

feature is confirmed by noting that ψ occurs in every term in

eqn 7B.1, so any constant factor can be cancelled This

free-dom to vary the wavefunction by a constant factor means that

it is always possible to find a normalization constant, N, such

that the proportionality of the Born interpretation becomes an

equality

We find the normalization constant by noting that, for a

nor-malized wavefunction Nψ, the probability that a particle is in the

region dx is equal to (Nψ *)(Nψ)dx (we are taking N to be real)

Furthermore, the sum over all space of these individual

prob-abilities must be 1 (the probability of the particle being

some-where is 1) Expressed mathematically, the latter requirement is

N2 ψ ψ* d =x 1

−∞

∞

Wavefunctions for which the integral in eqn 7B.2 exists (in the

sense of having a finite value) are said to be ‘square-integrable’

Therefore, by evaluating the integral, we can find the value of N

and hence ‘normalize’ the wavefunction From now on, unless

we state otherwise, we always use wavefunctions that have been

normalized to 1; that is, from now on we assume that ψ already

includes a factor which ensures that (in one dimension)

where dτ = dxdydz and the limits of this definite integral are

not written explicitly: in all such integrals, the integration is over all the space accessible to the particle For systems with spherical symmetry it is best to work in spherical polar coor-

dinates (The chemist’s toolkit 7B.1), so the explicit form of eqn

7B.4c is

ψ ψ* r r2 sinθ θ φ

0

2 0

Self-test 7B.1 The wavefunction for the electron in its lowest

energy state in the ion He+ is proportional to e−2r a/ 0 Repeat the

calculation for this ion Any comment?

Answer: 55; more compact wavefunction

The chemist’s toolkit 7B.1 Spherical polar coordinatesFor systems with spherical symmetry it is best to work in

spherical polar coordinates r, θ, and ϕ (Sketch 1)

x r= sin cos ,θ φ y r= sin sinθ φ,z r= cos θ

where:

r, the radius, ranges from 0 to ∞

θ, the colatitude, ranges from 0 to π

ϕ, the azimuth, ranges from 0 to 2πThat these ranges cover space is illustrated in Sketch 2 Standard manipulations then yield

Sketch 2 The surface of a sphere is covered by allowing θ

to range from 0 to π, and then sweeping that arc around a complete circle by allowing ϕ to range from 0 to 2π

spherical polar coordinates

296 7 Introduction to quantum theory

(b) Constraints on the wavefunction

The Born interpretation puts severe restrictions on the

accept-ability of wavefunctions The principal constraint is that ψ must

not be infinite over a finite region If it were, it would not be square-integrable, and the normalization constant would be zero The normalized function would then be zero everywhere, except where it is infinite, which would be unacceptable (the particle must be somewhere) Note that infinitely sharp spikes are acceptable provided they have zero width

The requirement that ψ is finite everywhere rules out many

possible solutions of the Schrödinger equation, because many mathematically acceptable solutions rise to infinity and are therefore physically unacceptable We could imagine a solu-tion of the Schrödinger equation that gives rise to more than

one value of |ψ|2 at a single point The Born interpretation implies that such solutions are unacceptable, because it would

be absurd to have more than one probability that a particle is at the same point This restriction is expressed by saying that the

wavefunction must be single-valued; that is, have only one value

at each point of space

The Schrödinger equation itself also implies some ematical restrictions on the type of functions that can occur Because it is a second-order differential equation, the second

math-derivative of ψ must be well-defined if the equation is to be

applicable everywhere We can take the second derivative of a function only if it is continuous (so there are no sharp steps in

it, Fig 7B.4) and if its first derivative, its slope, is continuous (so there are no kinks in the wavefunction)

There are cases, and we shall meet them, where acceptable wavefunctions have kinks These cases arise when the poten-tial energy has peculiar properties, such as rising abruptly to infinity When the potential energy is smoothly well-behaved and finite, the slope of the wavefunction must be continuous;

if the potential energy becomes infinite, then the slope of the wavefunction need not be continuous There are only two cases

Example 7B.2 Normalizing a wavefunction

Normalize the wavefunction used for the hydrogen atom in

Example 7B.1.

Method We need to find the factor N that guarantees that the

integral in eqn 7B.4c is equal to 1 Because the system is

spheri-cal, it is most convenient to use spherical coordinates (The

chemist’s toolkit 7B.1) and to carry out the integrations

speci-fied in eqn 7B.4d Relevant integrals are found in the Resource

Note that because a0 is a length, the dimensions of ψ are

1/length3/2 and therefore those of ψ2 are 1/length3 (for

instance, 1/m3) as is appropriate for a probability density

If Example 7B.1 is now repeated, we can obtain the actual

probabilities of finding the electron in the volume element at

each location, not just their relative values Given (from inside

the front cover) that a0 = 52.9 pm, the results are (a) 2.2 × 10−6,

corresponding to 1 chance in about 500 000 inspections of

finding the electron in the test volume, and (b) 2.9 × 10−7,

cor-responding to 1 chance in 3.4 million

Self-test 7B.2 Normalize the wavefunction given in Self-test

7B.1

Answer: N=( 8/ πa0) 1 2 /

In these coordinates, the integral of a function f(r,θ,φ) over all

space takes the form

where the limits on the first integral sign refer to r, those on

the second to θ, and those on the third to φ.

7B Dynamics of microscopic systems  297

of this behaviour in elementary quantum mechanics, and the

peculiarity will be mentioned when we meet them

At this stage we see that ψ:

• must not be infinite over a non-infinitesimal

The restrictions just noted are so severe that acceptable

solu-tions of the Schrödinger equation do not in general exist for

arbitrary values of the energy E In other words, a particle may

possess only certain energies, for otherwise its wavefunction

would be physically unacceptable That is, as a consequence of

the restrictions on its wavefunction, the energy of a particle is

quantized We can find the acceptable energies by solving the

Schrödinger equation for motion of various kinds, and

select-ing the solutions that conform to the restrictions listed above

That task is taken forward in Chapter 8

Once we have obtained the normalized wavefunction, we can

then proceed to determine the probability density As an

exam-ple, consider a particle of mass m free to move parallel to the

x-axis with zero potential energy The Schrödinger equation is

obtained from eqn 7B.1 by setting V = 0, and is

−2m2 d2dψ x( )2x =E x ψ( ) (7B.5)

As shown in the following Justification, the solutions of this

equation have the form

ψ ( ) x A= eikx+Be− ikx E k m= 2 2

where A and B are constants (See Mathematical background 3

at the end of this chapter for more on complex numbers.)

We see in Topic 8A what determines the values of A and B;

here we can treat them as arbitrary constants that we can vary

at will Suppose that B = 0 in eqn 7B.6, then the wavefunction is

This probability density is independent of x; so, wherever we

look in a region of fixed length located anywhere along the

x-axis, there is an equal probability of finding the particle (Fig

7B.5a) In other words, if the wavefunction of the particle is given by eqn 7B.7, then we cannot predict where we will find

it The same would be true if the wavefunction in eqn 7B.6 had

A = 0; then the probability density would be |B|2, a constant

Now suppose that in the wavefunction A = B Then, because

coskx=1 eikx+e− ikx

2( ) (Mathematical background 3), eqn 7B.6

becomes

The probability density now has the form

| ( )| (ψ x 2= 2 cosA kx)*(2 cosA kx)=4A2cos2kx (7B.10)

Justification 7B.2 The wavefunction of a free particle in

one dimension

To verify that ψ(x) in eqn 7B.6 is a solution of eqn 7B.5, we

simply substitute it into the left-hand side of the equation and

show that E = k22/2m To begin, we write

−2 2 = −  + −

2

2 2 2