Chapter 40

Introduction to

Quantum Physics

Need for Quantum Physics Problems remained from classical mechanics

that relativity didn’t explain Attempts to apply the laws of classical physics

to explain the behavior of matter on the atomic scale were consistently unsuccessful

Problems included: blackbody radiation

The electromagnetic radiation emitted by a heated object photoelectric effect

Emission of electrons by an illuminated metal

Quantum Mechanics Revolution Between 1900 and 1930, another revolution

took place in physics A new theory called quantum mechanics was

successful in explaining the behavior of particles of microscopic size

The first explanation using quantum mechanics was introduced by Max Planck Many other physicists were involved in other

subsequent developments

Blackbody Radiation An object at any temperature is known

to emit thermal radiation Characteristics depend on the temperature

and surface properties The thermal radiation consists of a

continuous distribution of wavelengths from all portions of the em spectrum

Blackbody Radiation, cont. At room temperature, the wavelengths of the

thermal radiation are mainly in the infrared region

As the surface temperature increases, the wavelength changes It will glow red and eventually white

The basic problem was in understanding the observed distribution in the radiation emitted by a black body Classical physics didn’t adequately describe the

observed distribution

Blackbody Radiation, final A black body is an ideal system that

absorbs all radiation incident on it The electromagnetic radiation emitted

by a black body is called blackbody radiation

Blackbody Approximation A good approximation of a

black body is a small hole leading to the inside of a hollow object

The hole acts as a perfect absorber

The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity

Blackbody Experiment Results The total power of the emitted radiation

increases with temperature Stefan’s law (from Chapter 20):

P = AeT4

The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases Wien’s displacement law maxT = 2.898 x 10-3 m.K

Stefan’s Law – Details P = AeT4

P is the power is the Stefan-Boltzmann constant

= 5.670 x 10-8 W / m2 . K4

Stefan’s law can be written in terms of intensity I = P/A = T4

For a blackbody, where e = 1

Wien’s Displacement Law

maxT = 2.898 x 10-3 m.K max is the wavelength at which the curve

peaks T is the absolute temperature

The wavelength is inversely proportional to the absolute temperature As the temperature increases, the peak is

“displaced” to shorter wavelengths

Intensity of Blackbody Radiation, Summary

The intensity increases with increasing temperature

The amount of radiation emitted increases with increasing temperature The area under the

curve The peak wavelength

decreases with increasing temperature

Active Figure 40.3

(SLIDESHOW MODE ONLY)

Rayleigh-Jeans Law An early classical attempt to explain

blackbody radiation was the Rayleigh-Jeans law

At long wavelengths, the law matched experimental results fairly well

4

2I , Bπck T

λ Tλ

Rayleigh-Jeans Law, cont. At short wavelengths,

there was a major disagreement between the Rayleigh-Jeans law and experiment

This mismatch became known as the ultraviolet catastrophe

You would have infinite energy as the wavelength approaches zero

Max Planck Introduced the

concept of “quantum of action”

In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

Planck’s Theory of Blackbody Radiation In 1900 Planck developed a theory of

blackbody radiation that leads to an equation for the intensity of the radiation

This equation is in complete agreement with experimental observations

He assumed the cavity radiation came from atomic oscillations in the cavity walls

Planck made two assumptions about the nature of the oscillators in the cavity walls

Planck’s Assumption, 1 The energy of an oscillator can have only

certain discrete values En

En = nhƒ n is a positive integer called the quantum number h is Planck’s constant ƒ is the frequency of oscillation

This says the energy is quantized Each discrete energy value corresponds to

a different quantum state

Planck’s Assumption, 2 The oscillators emit or absorb energy

when making a transition from one quantum state to another The entire energy difference between the

initial and final states in the transition is emitted or absorbed as a single quantum of radiation

An oscillator emits or absorbs energy only when it changes quantum states

Energy-Level Diagram An energy-level diagram

shows the quantized energy levels and allowed transitions

Energy is on the vertical axis

Horizontal lines represent the allowed energy levels

The double-headed arrows indicate allowed transitions

More About Planck’s Model The average energy of a wave is the average

energy difference between levels of the oscillator, weighted according to the probability of the wave being emitted

This weighting is described by the Boltzmann distribution law and gives the probability of a state being occupied as being proportional to

where E is the energy of the state BE k Te

Planck’s Model, Graphs

Active Figure 40.7

(SLIDESHOW MODE ONLY)

Planck’s Wavelength Distribution Function Planck generated a theoretical

expression for the wavelength distribution

h = 6.626 x 10-34 J.s h is a fundamental constant of nature

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Planck’s Wavelength Distribution Function, cont. At long wavelengths, Planck’s equation

reduces to the Rayleigh-Jeans expression

At short wavelengths, it predicts an exponential decrease in intensity with decreasing wavelength This is in agreement with experimental

results

Photoelectric Effect The photoelectric effect occurs when

light incident on certain metallic surfaces causes electrons to be emitted from those surfaces The emitted electrons are called

photoelectrons

Photoelectric Effect Apparatus When the tube is kept in the

dark, the ammeter reads zero

When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter

The current arises from photoelectrons emitted from the negative plate and collected at the positive plate

Active Figure 40.9

(SLIDESHOW MODE ONLY)

Photoelectric Effect, Results At large values of V, the

current reaches a maximum value

All the electrons emitted at E are collected at C

The maximum current increases as the intensity of the incident light increases

When V is negative, the current drops

When V is equal to or more negative than Vs, the current is zero

Photoelectric Effect Feature 1 Dependence of photoelectron kinetic energy

on light intensity Classical Prediction

Electrons should absorb energy continually from the electromagnetic waves

As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy

Experimental Result The maximum kinetic energy is independent of light

intensity The maximum kinetic energy is proportional to the

stopping potential (Vs)

Photoelectric Effect Feature 2 Time interval between incidence of light and

ejection of photoelectrons Classical Prediction

At low light intensities, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal

This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal

Experimental Result Electrons are emitted almost instantaneously, even at

very low light intensities

Photoelectric Effect Feature 3 Dependence of ejection of electrons on light

frequency Classical Prediction

Electrons should be ejected at any frequency as long as the light intensity is high enough

Experimental Result No electrons are emitted if the incident light falls below

some cutoff frequency, ƒc

The cutoff frequency is characteristic of the material being illuminated

No electrons are ejected below the cutoff frequency regardless of intensity

Photoelectric Effect Feature 4 Dependence of photoelectron kinetic energy

on light frequency Classical Prediction

There should be no relationship between the frequency of the light and the electric kinetic energy

The kinetic energy should be related to the intensity of the light

Experimental Result The maximum kinetic energy of the photoelectrons

increases with increasing light frequency

Photoelectric Effect Features, Summary The experimental results contradict all four

classical predictions Einstein extended Planck’s concept of

quantization to electromagnetic waves All electromagnetic radiation can be

considered a stream of quanta, now called photons

A photon of incident light gives all its energy hƒ to a single electron in the metal

Photoelectric Effect, Work Function Electrons ejected from the surface of the

metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax

Kmax = hƒ – φ φ is called the work function The work function represents the minimum energy

with which an electron is bound in the metal

Some Work Function Values

Photon Model Explanation of the Photoelectric Effect Dependence of photoelectron kinetic energy

on light intensity Kmax is independent of light intensity K depends on the light frequency and the work

function Time interval between incidence of light and

ejection of the photoelectron Each photon can have enough energy to eject an

electron immediately

Photon Model Explanation of the Photoelectric Effect, cont. Dependence of ejection of electrons on

light frequency There is a failure to observe photoelectric

effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron

Without enough energy, an electron cannot be ejected, regardless of the light intensity

Photon Model Explanation of the Photoelectric Effect, final Dependence of photoelectron kinetic

energy on light frequency Since Kmax = hƒ – φ As the frequency increases, the kinetic

energy will increase Once the energy of the work function is

exceeded There is a linear relationship between the

kinetic energy and the frequency

Cutoff Frequency The lines show the

linear relationship between K and ƒ

The slope of each line is h

The x-intercept is the cutoff frequency This is the frequency

below which no photoelectrons are emitted

Cutoff Frequency and Wavelength The cutoff frequency is related to the work

function through ƒc = φ / h The cutoff frequency corresponds to a cutoff

wavelength

Wavelengths greater than c incident on a material having a work function φ do not result in the emission of photoelectrons

ƒcc

c hcλ

φ

Arthur Holly Compton 1892 - 1962 Director of the lab at

the University of Chicago

Discovered the Compton Effect

Shared the Nobel Prize in 1927

The Compton Effect, Introduction Compton and Debye dealt with

Einstein’s idea of photon momentum The two groups of experimenters

accumulated evidence of the inadequacy of the classical wave theory

The classical wave theory of light failed to explain the scattering of x-rays from electrons

Compton Effect, Classical Predictions

According to the classical theory, em waves incident on electrons should: have radiation pressure

that should cause the electrons to accelerate

set the electrons oscillating

Compton Effect, Observations Compton’s

experiments showed that, at any given angle, only one frequency of radiation is observed

Compton Effect, Explanation The results could be explained by

treating the photons as point-like particles having energy hƒ

Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved

This scattering phenomena is known as the Compton effect

Compton Shift Equation The graphs show the

scattered x-ray for various angles

The shifted peak, λ’ is caused by the scattering of free electrons

This is called the Compton shift equation

1' cosoe

hλ λ θ

m c

Compton Wavelength

The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms

The Compton wavelength is

0002 43 nm.e

h

m c

Photons and Waves Revisited Some experiments are best explained by the

photon model Some are best explained by the wave model We must accept both models and admit that

the true nature of light is not describable in terms of any single classical model

Also, the particle model and the wave model of light complement each other

Louis de Broglie 1892 – 1987 Originally studied

history Was awarded the

Nobel Prize in 1929 for his prediction of the wave nature of electrons

Wave Properties of Particles Louis de Broglie postulated that

because photons have both wave and particle characteristics, perhaps all forms of matter have both properties

The de Broglie wavelength of a particle is

h hλ

p mv

Frequency of a Particle In an analogy with photons, de Broglie

postulated that a particle would also have a frequency associated with it

These equations present the dual nature of matter particle nature, m and v wave nature, λ and ƒ

ƒE

h

Davisson-Germer Experiment If particles have a wave nature, then

under the correct conditions, they should exhibit diffraction effects

Davisson and Germer measured the wavelength of electrons

This provided experimental confirmation of the matter waves proposed by de Broglie

Complementarity The principle of complementarity

states that the wave and particle models of either matter or radiation complement each other

Neither model can be used exclusively to describe matter or radiation adequately

Electron Microscope The electron microscope

depends on the wave characteristics of electrons

The electron microscope has a high resolving power because it has a very short wavelength

Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light

Quantum Particle The quantum particle is a new model

that is a result of the recognition of the dual nature

Entities have both particle and wave characteristics

We must choose one appropriate behavior in order to understand a particular phenomenon

Active Figure 40.20

(SLIDESHOW MODE ONLY)

Ideal Particle vs. Ideal Wave An ideal particle has zero size

Therefore, it is localized in space An ideal wave has a single frequency

and is infinitely long Therefore,it is unlocalized in space

A localized entity can be built from infinitely long waves

Particle as a Wave Packet Multiple waves are superimposed so that one

of its crests is at x = 0 The result is that all the waves add

constructively at x = 0 There is destructive interference at every

point except x = 0 The small region of constructive interference

is called a wave packet The wave packet can be identified as a particle

Wave Envelope

The blue line represents the envelope function

This envelope can travel through space with a different speed than the individual waves

Active Figure 40.21

(SLIDESHOW MODE ONLY)

Speeds Associated with Wave Packet

The phase speed of a wave in a wave packet is given by

This is the rate of advance of a crest on a single wave

The group speed is given by

This is the speed of the wave packet itself

phaseωv k

gdωv dk

Speeds, cont. The group speed can also be expressed

in terms of energy and momentum

This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent

2 1

22 2g

dE d pv p u

dp dp m m

Electron Diffraction, Set-Up

Electron Diffraction, Experiment Parallel beams of mono-energetic

electrons that are incident on a double slit

The slit widths are small compared to the electron wavelength

An electron detector is positioned far from the slits at a distance much greater than the slit separation

Electron Diffraction, cont. If the detector collects electrons

for a long enough time, a typical wave interference pattern is produced

This is distinct evidence that electrons are interfering, a wave-like behavior

The interference pattern becomes clearer as the number of electrons reaching the screen increases

Active Figure 40.23

(SLIDESHOW MODE ONLY)

Electron Diffraction, Equations A minimum occurs when

This shows the dual nature of the electron The electrons are detected as particles at a

localized spot at some instant of time The probability of arrival at that spot is determined

by finding the intensity of two interfering waves

sin or sin 2 2

λ hd θ θ θ

pd

Electron Diffraction, Closed Slits If one slit is closed, the

maximum is centered around the opening

Closing the other slit produces another maximum centered around that opening

The total effect is the blue line It is completely different from

the interference pattern (red curve)

Electron Diffraction Explained An electron interacts with both slits

simultaneously If an attempt is made to determine

experimentally which slit the electron goes through, the act of measuring destroys the interference pattern It is impossible to determine which slit the electron

goes through In effect, the electron goes through both slits

The wave components of the electron are present at both slits at the same time

Werner Heisenberg 1901 – 1976 Developed matrix

mechanics Many contributions

include: Uncertainty principle

Rec’d Nobel Prize in 1932 Prediction of two forms

of molecular hydrogen Theoretical models of

the nucleus

The Uncertainty Principle, Introduction In classical mechanics, it is possible, in

principle, to make measurements with arbitrarily small uncertainty

Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy

Heisenberg Uncertainty Principle, Statement The Heisenberg uncertainty principle

states: if a measurement of the position of a particle is made with uncertainty x and a simultaneous measurement of its x component of momentum is made with uncertainty p, the product of the two uncertainties can never be smaller than

2xx p

Heisenberg Uncertainty Principle, Explained It is physically impossible to measure

simultaneously the exact position and exact momentum of a particle

The inescapable uncertainties do not arise from imperfections in practical measuring instruments

The uncertainties arise from the quantum structure of matter

Heisenberg Uncertainty Principle, Another Form Another form of the uncertainty principle

can be expressed in terms of energy and time

This suggests that energy conservation can appear to be violated by an amount E as long as it is only for a short time interval t

2E t