Chapter 27

Elements of Quantum Physics

• Black Body Radiation:

• Wien’s Displacement Law

• Planck’s Hypothesis

• The Dual Nature of Light and Matter:

• Photoelectric Effect

• Elements of Quantum Theory:

• Wave Function

• The Uncertainty Principle

Need for Quantum Physics

• At the end of the 19th century, Newtonian mechanics and Maxwell’s theory of

electromagnetism (we’ll study it next semester) seemed to provide an almost

complete perspective about natural order

• Moreover, at the beginning of 20th century, Einstein’s Theory of Relativity

subsumed the Newtonian Mechanics as a particular case: the science of motion of

macroscopic objects moving with speeds much smaller than the speed of light

• However, several only apparently minor problems remained unsolved within the

framework of the classical understanding of Physics:

• Blackbody radiation: the electromagnetic radiation emitted by a heated object

• Photoelectric effect: emission of electrons by an illuminated metal

• Spectral Lines: discrete emission of light by gas atoms in an electric discharge tube

• The anomalous phenomena were explained with the advent and booming

development of Quantum Mechanics (a.k.a. wave mechanics) between 1900-1930

• The new science was highly successful in explaining the behavior of atoms,

molecules, and nuclei

• While Max Planck introduced the initial basic ideas, its developments and

interpretations involved such people as Einstein, Bohr, Schrödinger, de Broglie,

Heisenberg, Born, Dirac

• In order to understand the initial scientific conundrums that led to the birth of

Quantum Mechanics, we have to first talk about the wave-like nature of light

• In the context of electromagnetic theory, light is seen as an electromagnetic wave: a

combination of oscillating transversal electric and magnetic fields propagating with

the speed of light, c ~ 3108 m/s2 (in vacuum)

• Visible light is just electromagnetic waves no different from radio waves of X-ray

waves, except that their wavelengths lay in a range (between 400-700 nm) which

makes them perceivable by the eye

• Even though not mechanical in nature, in many situations light waves behave much

like mechanical waves do. However, not always – one such unexplainable

phenomena was the Blackbody radiation:

Light as a Wave

• An object at any temperature emits electromagnetic

waves or radiation, (also called thermal radiation)

• The emission is throughout the entire spectrum, with a

maximum of intensity at a wavelength depending on the

temperature and properties of the object

• The experimental data was not explainable within the

classical theory

Blackbody Radiation

• Light producing thermal interactions inside

an object correspond to a range of energies,

such that the radiation emitted by an object

range across all wavelengths

• The intensity of various light waves versus

the wavelengths of a perfect radiator is called a

blackbody spectrum

• The larger the temperature, the more violent

thermal collisions there are, and the more

radiation is emitted at smaller wavelengths:

Object at

7000 K

Object at

6000 K

Object at

5000 K

Blue object

Red object

Green object

Ex: Black body spectra at high T

Produced

by violent

collisions

Produced

by mild

collisions

Wien’s Displacement Law: The peak of

the black body spectrum shifts towards

shorter wavelengths λmax when the

temperature increases, as given by

where the temperature T is measured in

Kelvins (the lowest temperature possible is

0 K, and the room temperature ~ 273 K)

3

max

2.89 10 m K~

T

Predominant

average

collisions

Planck’s Hypothesis

• The bell-shape of the black body spectrum cannot be understood in the context of

classical theories where the emitted radiation can carry any amount of energy, such

that the intensity should keep increasing with decreasing wavelength (so, no peak)

• Incidentally, this means that at extremely short wavelengths, the intensity should

tend to infinity: this paradox was called the ultraviolet catastrophe

• The blackbody spectrum was explained by Max Planck in 1901 by assuming that

the energy was emitted discretely by elementary oscillating sources (resonators)

• Each resonator had a specific frequency f, and was allowed to emit

only an integer number n of quanta of energy proportional to that

frequency, via Planck’s constant, h = 6.63×10-34 J∙s E hf

n

cE nhf nh

• Hence, the intensity of a certain wavelength is given by the

quantum state corresponding to the quantum number n of the

resonators with frequencies corresponding to that wavelengths:

• The bell-shaped blackbody spectrum is thus explained by the fact that high-n

quantum states are less likely to occur than low-n, while low-n will have low energy

• The idea that energy is quantized – that is, it can be exchanged only in discrete

packages or quanta – marked the birth of Quantum Mechanics, and was subsequently

extended as a fundamental characteristics of the microscopic world

Quiz:

1. The spectra emitted by stars is close to the ideal black body spectrum. Two of the brighter

stars observable in the northern hemisphere are in the Orion constellation: Betelgeuse is red

and Rigel is blue. Which one hotter?

2. A student states that the color of our bodies cannot be due to the light emitted by the body.

Is the student right? Why?

Problems:

1. Wien’s displacement law: The adjacent graph represents

the blackbody radiation spectra for several sources.

a) Which object has the largest temperature?

b) What is the temperature of the this hottest object?

2. Plank’s quantization: a) What is the frequency of the

quanta of energy carried by the dominating radiation

emitted by the hottest object in the adjacent graph? b) What

is the energy of those quanta?

A

B

C

D

E

Photoelectric Effect – Phenomenon

• From a classical perspective, if the incident radiation were regarded as a wave, the

photoelectrons should achieve higher and higher energy as the intensity of the

incident light is increased. However, the photoelectrons behave very differently:

1. if the frequency of incident light is under a cutoff value fc characteristic to the

material, no photoelectron is emitted, irrespective of the intensity of light

2. increasing the incident light intensity increases the number of emitted

electrons, but not their maximum kinetic energy

3. the maximum kinetic energy increases only if the frequency of the incident

wave is increased

4. the photoelectrons are emitted almost instantaneously, without the gradual

absorption of energy predicted by classical theories

• The idea of discreteness of energy was

subsequently used by Einstein in 1905 to explain

another physical phenomenon at odds with

classical theories: the photoelectric effect which

had been first observed by Hertz: if a metal is

irradiated with light, it emits electrons, as they

are provided with energy by the radiation

Photoelectric Effect – Quantum mechanical explanation

• Einstein explained the photoelectric effect by suggesting that the energy carried by

electromagnetic waves is quantized, such that the incident beam of light can be seen

as a collection of particle-like quanta – let’s call them photons – each carrying an

equal amount of energy E = hf, as prescribed by Planck’s theory

• Atomic electrons can be extracted from metals only if they receive at least an

amount of energy given by the work function, ϕ, characteristic to each material

• Electrons intercept photons and absorb their energy: if the

energy is larger than the work function, the difference is

converted into kinetic energy with a maximum value KEmax: maxKE hf

• Hence, the photoelectrons effect is immediately explainable:

1. if the frequency of incident photons is under a cutoff value fc, there are no

photoelectrons since they won’t have enough energy to bypass the work function

2. increasing the incident light intensity correspond to an increased number of

photons and so of photoelectrons, but with the same maximum kinetic energy

3. if the frequency of the incident wave is increased, the maximum kinetic energy

increases since the photons will have more energy to pass to the photoelectrons

4. the photoelectrons are emitted almost instantaneously, since the photoelectrons

absorb photons extremely fast

Problems:

3. Kinetic energy versus frequency in the photoelectric effect:

A metallic sample is illuminated by light with higher and higher

frequencies f, while the maximum kinetic energy KEmax of the

photoelectrons is monitored for each frequency. A linear KEmax vs

f graph is built, intersecting the frequency axis at a value f0 =

5.01014 Hz.

a) What is the significance of the frequency f0 in the context of

the photoelectric experiment?

b) Use f0 to find the work function of the respective material?

c) What is the maximum kinetic energy of the photoelectrons at

an incident frequency f = 3f0

KEmax

f0 Frequency, f

4. Compton effect: A photon of frequency f0 = 6.0001019 Hz experiences Compton

scattering with various scattering angles θ.

a) For which angle θ photons did not experience Compton scattering?

b) Find an expression for the change in frequency in terms of Compton wavelength and θ

c) What fraction of energy is lost by photons scattered under angle θ = 30°

The Dual Nature of Light and Matter

• So, like any electromagnetic radiation, light has a dual nature: it exhibits both wave

and particle characteristics

• The particle-like behavior becoming easier to observe at higher frequencies (since

each photon carries more energy), and is demonstrated by phenomena such as the

photoelectric effect and Compton scattering

• On the other hand, phenomena such as interference or diffraction (bending of light

by small obstacles) offer evidence of the wave nature of light

• The theory of Quantum Mechanics extends this duality: in 1924, Louis de Broglie

postulated that all forms of matter can be seen both as particle-like or wave-like

• Furthermore, extrapolating the quantum aspects for the photon and Einstein’s mass-

energy equivalence, the wave-like (f, λ, c) and particle-like (mass m, momentum p)

characteristics associated with any particle of matter can be related:

2

E hf h c hcpc

E mc pc

h

vp

m

h

• The wave-like nature of particles was demonstrated in many experiments, starting

with the Davisson-Germer experiment which proved that beams of electrons diffract

like waves

• The dual nature of matter have been demonstrated experimentally

not only by showing that previously considered pure waves – such

as EM radiation – was also corpuscular, but also by showing that

previously considered pure particles – such as electrons, protons or

even molecules – behave like waves in interference and diffraction

experiments

• For instance, if electrons are fired at a screen in a double-slit

interference experiment, the hits will arrange to form fringes of

interference – a behavior associated with interacting waves –

corresponding to the De Broglie wavelength of electrons

• Notice on the figure that, if the electrons are fired one by one, they

will hit the screen like particles forming the fringes in the manner of

a pointillist painting: even one electron will interfere with itself, like

going through both slits then getting localized in a point on the

screen corresponding to a wave-like distribution of probability

• The wave-like nature of electrons (and other particles) is ordinarily

used in microscopy: for instance, neutrons are scattered by the

crystalline lattice of solids and form patters of interference offering

information about the lattice geometry and nature

The Dual Nature of Light and Matter – Electrons as waves

Ex: Electron-wave

interference fringes

The Wave Function and the Uncertainty Principle

• The function describing the state of a particle associated with a wave is called a

wave function typically denoted Ψ

• In 1926, Erwin Schrödinger discovered the wave equation describing the space and

time dependency of the wave function

• The solution of the wave equation adapted to the any given conditions is

the particle wave function. Then, the probability per unit volume that the

particle occupies a certain region of space is proportional to the square of

the wave function at the respective location:

2

• A bizarre consequence of quantum mechanics is that certain pairs of physical

quantities – such as the position x and the momentum p of a particle – cannot be

simultaneously known with arbitrarily small uncertainties, as stated by

Heisenberg’s Uncertainty Principle: If the position of a particle is determined

with a precision Δx, then the precision Δp of a simultaneous measurement of

momentum will necessarily have to satisfy 4x p h

• This principle is completely at odds with the percepts of classical mechanics where

the measurement of physical quantities was assumed to be limited only by the

precision and accuracy of the apparatus

A similar expression holds for energy and time: 4E t h

Quiz:

1. De Broglie particle-wave duality and uncertainty principle: We usually see a tennis ball

as a particle, not a wave. Why? How is this related to the fact that throughout this semester

we’ve considered many times the instantaneous position and momentum of particles

accurately known at the same time, in spite of Heisenberg’s uncertainty principle.

2. Uncertainty principle: A popular misconception states that at a temperature of absolute

zero all motion stops. Why this cannot be true?

Problems:

5. De Broglie particle-wave duality: A proton and an electron (which is about 1800 times

lighter than a proton) move with the same speed. Which one has a longer wavelength when

seen as de Broglie waves?

6. Uncertainty principle: Say that you want to

measure the position of an electron. In order to do

that, you use a microscope that sends lights of

wavelength λ on the electron: consider a photon

of that light beam intercepting and colliding the

electron so it bounces back to your eye indicating

its position. What is the uncertainty in the

momentum of the electron?