Chapter 28
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Chapter 28. Quantum Theory April 8 th , 2013. Quantum Theory. Two things are really different at the atomic and sub-atomic scale: Wave-particle duality Particles are waves, waves are particles Energies are not continuous but discrete. They vary by discrete increments called quanta

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Chapter 28

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Chapter 28

Chapter 28

Quantum Theory

April 8th, 2013


Quantum theory

Quantum Theory

Two things are really different at the atomic and sub-atomic scale:

  • Wave-particle duality

    • Particles are waves, waves are particles

  • Energies are not continuous but discrete. They vary by discrete increments called quanta

    • e. g. electron binding energies


Demos

demos

  • Young’s double-slit experiment demonstrates the wave-nature of light

  • Wave-particle duality: electron are waves, they diffract like x-ray photons (1956 electron diffraction instrument)


Quantum regime

Quantum Regime

  • Macroscopic-world explanations fail at the atomic-scale

    • Newtonian mechanics

    • Maxwell’s equations describing electromagnetism

  • The atomic-scale world is referred to as the quantum regime

  • Quantum refers to a very small increment, or parcel, or packet of energy

  • The discovery and development of quantum theory began in the late 1800s and continued during the early 1900s


Waves vs particles

Waves vs. Particles

  • In the world of Newton and Maxwell, energy can be carried by particles and waves

  • Waves produce an interference pattern when passed through a double slit

  • Classical particles (bullets) will pass through one of the slits and no interference pattern will be formed


Particles and waves classical

Particles and Waves, Classical

  • Waves exhibit interference; particles do not

  • Particles often deliver their energy in discrete amounts

  • The energy carried or delivered by a wave is not discrete

    • The energy carried by a wave is described by its intensity

    • The amount of energy absorbed depends on the intensity and the absorption time


Interference with electrons

Interference with Electrons

  • The separation between waves and particles is not found in the quantum regime

  • Electrons are used in a double slit experiment

  • The blue lines show the probability of the electrons striking particular locations


Interference with electrons cont

Interference with Electrons, cont.

  • The probability curve of the electrons has the same form as the variation of light intensity in the double-slit interference experiment

  • The experiment shows that electrons undergo constructive interference at certain locations on the screen

  • At other locations, the electrons undergo destructive interference

    • The probability for an electron to reach those location is very small or zero

  • The experiment also shows aspects of particle-like behavior since the electrons arrive one at a time at the screen


Particles waves quanta

Particles, Waves, Quanta

  • All objects, including light and electrons, can exhibit interference

  • All objects, including light and electrons, carry energy in discrete amounts

    • These discrete parcels of energy are called quanta


Work function

Work function

  • In the 1880s, Hertz discovered the work function and the photoelectric effect

  • If V is the electric potential at which electrons begin to jump across the vacuum gap, the work function is Wc = eV

  • The work function, Wc is the minimum energy required to remove a single electron from a piece of metal

  • This energy can be delivered either as electric potential or by shining light on the metal

Wc = eV


Work function cont

Work Function, cont.

  • A metal contains electrons that are free to move around within the metal

  • The electrons are still bound to the metal and need energy to be removed from the metal

    • This energy is the work function

  • The value of the work function is different for different metals


Work functions w c of several metals

Work Functions Wc of several metals


Photoelectric effect

Photoelectric Effect

  • Another way to extract electrons from a metal is by shining light onto it

  • Light striking a metal is absorbed by the electrons

  • If an electron absorbs an amount of light energy greater than Wc, it is ejected off the metal

  • This is called the photoelectric effect


Photoelectric effect cont

Photoelectric Effect, cont.

  • No electrons are emitted unless the light’s frequency is greater than a critical value ƒc, the intensity of light does not matter

  • When the frequency is above ƒc, the kinetic energy of the emitted electrons varies linearly with the frequency, not the intensity of light

  • These results could not be explained with the classical wave theory of light


Photoelectric effect problems

Photoelectric Effect, Problems

  • Trying to explain the photoelectric effect with the classical wave theory of light presented two difficulties:

    • Experiments showed that the critical frequency is independent of the intensity of the light

      • Classically, the energy is proportional to the intensity

      • It should always be possible to eject electrons by increasing the intensity to a sufficiently high value

      • Yet it is observable that below the critical frequency, there are no ejected electrons no matter how great the light intensity

    • The kinetic energy of an ejected electron is independent of the light intensity

      • Classical theory predicts that increasing the intensity will cause the ejected electrons to have a greater kinetic energy

      • Yet experiments show that the electron kinetic energy depends on the frequency of light, not at all on its intensity


Photoelectric effect solution photons

Photoelectric Effect, solution: Photons

  • Einstein proposed that light carries energy in discrete quanta, now called photons

  • Each photon carries a quantum of energy Ephoton = hƒ

    • h is a constant of nature called Planck’s constant

    • h = 6.626 x 10-34 J ∙ s

  • A beam of light should be thought of as a collection of photons

    • Each photon has an energy dependent on its frequency

  • If the intensity of monochromatic light is increased, the number of photons increases, but the energycarried by each photon does not change

  • Vending machine analogy


Photoelectric effect solution photons1

Photoelectric Effect, solution: Photons

  • The introduction of photons accounts for all the problems with the classical explanation

    • The absorption of light by an electron is just like a collision between two particles, a photon and an electron

      • The photon carries an energy that is absorbed by the electron

      • If this energy is less than the work function, the electron is not able to escape from the metal

      • The energy of a single photon depends on frequency but not on the light intensity


Photoelectric effect solution photons2

Photoelectric Effect, solution: Photons

  • The kinetic energy of the ejected electrons depends on light frequency but not intensity

    • The critical frequency corresponds to photons whose energy is equal to the work function

      h ƒc = Wc

    • This electron is ejected with 0 kinetic energy

    • If the photon has a higher frequency, the difference goes into kinetic energy of the ejected electron

      KEelectron = h ƒ - h ƒc = h ƒ - Wc

    • This linear relationship is what was observed experimentally


Chapter 28

Photoelectric EffectNobel prizeAlbert Einstein 1921

(his discovery was in 1905)

KEelectron = h ƒ - h ƒc = h ƒ– Wc

With his explanation of the photoelectric effect, Einstein introduced the idea that

light is made of particles, now called photons,and that their energies are quantized.

We now say that:

a photon is a quantum of light


Momentum of a photon

Momentum of a Photon

  • A light wave with energy E also carries a certain momentum

  • Particles of light called photonscarry a discrete amount of both energy and momentum

  • Photons have two properties that are different from classical particles

    • Photons do not have any mass

    • Photons exhibit interference effects


Blackbody radiation

Blackbody Radiation

  • Blackbody radiation is emitted over a range of wavelengths

  • To the eye, the color of the cavity is determined by the wavelength at which the radiation intensity is largest


Blackbody radiation classical

Blackbody Radiation, Classical

  • The blackbody intensity curve has the same shape for a wide variety of objects

  • Electromagnetic waves form standing waves as they reflect back and forth inside the oven’s cavity

  • The frequencies of the standing waves follow the pattern ƒn = n ƒ where n = 1, 2, 3, …

  • There is no limit to the value of n, so the frequency can be infinitely large

  • But as the frequency increases, so does the energy

  • Classical theory predicts that the blackbody intensity should become infinite as the frequency approaches infinity. This is nonsense!


Blackbody radiation and quanta

Blackbody Radiation and Quanta

  • The disagreement between the classical predictions and experimental observations was called the “ultraviolet catastrophe”

  • Planck proposed solving the problem by assuming the energy in a blackbody cavity must come in discrete quanta

  • Each parcel would have energy E = h ƒn

  • His theory fit the experimental results, but gave no reason why it worked

  • Planck’s work is generally considered to be the beginning of quantum theory


Particle wave nature of light

Particle-Wave Nature of Light

  • Some phenomena can only be understood in terms of the particle nature of light

    • Photoelectric effect

    • Blackbody radiation

  • Light also has wave properties at the same time

    • Interference

  • Light has both wave-like and particle-like properties


Wave like duality

Wave-like Duality

  • The notion that the properties of both classical waves and classical particles are present at the same time is also called wave-particle duality and it is essential for understanding the micro-scale world

  • All particles at all scales are capable of wave-like properties, as first proposed by Louis de Broglie

  • De Broglie suggested that if a particle has a momentum p, its wavelength is

  • Even baseballs, although those would have a wavelength, albeit a very small one (e.g. 10-34m)!


Electrons are waves

Electrons are waves!

  • To test de Broglie’s hypothesis, an experiment was designed by Davisson-Germer to observe interference of electrons

  • The experiment showed conclusively that electrons have wavelike properties: the diffraction pattern they form is identical to that obtained by x-ray photons.

  • The calculated wavelength was in good agreement with de Broglie’s theory


Wavelengths of macroscopic particles

Wavelengths of Macroscopic Particles

  • From de Broglie’s equation and using the classical expression for kinetic energy

  • As the mass of the particle (object) increases, its wavelength decreases

  • In principle, you could observe interference with baseballs

    • Has not yet been observed


Problem 28 36

Problem 28.36

An electron and a neutron have the same wavelength. What is the ration of (a) their kinetic energies and (b) their momenta? Assume the speeds are low enough that you can ignore relativity.


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