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Chapter 28

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Chapter 28

Quantum Theory

April 8th, 2013

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

- 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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

- 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

- 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

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

- 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

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

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

- 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 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

- 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!

- 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

- 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

- 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)!

- 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

- 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

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.