It is very important that you remember that time is the horizontal axis! A lot of people see the drawing above and think of it as two particles coming together at an angle. These two particles are in a head-on collision, not hitting at an angle.
Note that space and time axes have been left out; they are understood to be there. Also note that the arrow on the bottom is supposed to be backwards. We do that any time we have an antiparticle. Most people like to think of antiparticles as traveling backwards in time, and this is roughly explained by CPT symmetry.
Scattering Diagram: Here is the Feynman diagram for two electrons coming towards each other then repelling each other through the electromagnetic force (via exchange of a virtual photon). Note that the particles are always separated in space (vertical axis) so that they never touch. Hence they are scattering by exchanging virtual photons which cause them to repel. You can think of a virtual photon as existing for an instant of time. Therefore there is no movement in time (horizontal) axis.
Example
Question: For the following Feynman diagrams, describe in words the process that is occurring. For instance: (a) what type of interaction: annihilation or scattering (b) what are the incoming articles? (c) which kind of boson mediates the interaction? (d) which fundamental force is involved in the interaction? (e) what are the outgoing particles? Also, is the interaction "allowed"?
Answer: In this Feynman diagram, one of the down quark from a neutron splits into a upward quark, an electron, and a electron neutrino via a particle. Because this does not break any laws, this interaction is allowed. Infact, this is a interaction that we already know. This is decay.
Feynman Diagrams Problem Set
For the following Feynman diagrams, describe in words the process that is occurring. For instance: (a) what type of interaction: annihilation or scattering (b) what are the incoming articles? (c) which kind of boson mediates the interaction? (d) which fundamental force is involved in the interaction? (e) what are the outgoing particles?
Also, for each, decide if the interaction shown is allowed. An interaction is allowed if it does not violate any of the rules set out by the Standard Model of physics. If the interactions violate some rule, state which rule it violates. If they do not violate a rule, say that the interactions are allowed.
Hint: the best approach is to verify that the incoming and outgoing particles can interact with the boson (force particle) then to look at each vertex where more than one particle is coming together. Look immediately to the left of the vertex (before) and immediately to the right of the vertex (after). For instance, one rule states that the total electric charge before a vertex must equal the total electric charge after a vertex. Is that true? Check all the conserved quantities from the previous chapter in this way.
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In this case, the electron and positron are exchanging virtual electron/positron pairs.
Draw all of the possible Feynman diagrams for the annihilation of an electron and positron, followed by motion of an exchange particle, followed by the creation of a new electron and positron.
Draw the Feynman diagram for the collision of an up and anti-down quark followed by the production of a positron and electron neutrino.
Answers to Selected Problems
Allowed: an electron and anti-electron(positron) annihilate to a photon then become an electron and anti-electron(positron) again.
Not allowed: electrons don’t go backward though time, and charge is not conserved
Not allowed: lepton number is not conserved
a. Allowed: two electrons exchange a photon b. Not allowed: neutrinos do not have charge and therefore cannot exchange a photon.
a. Allowed: an electron and an up quark exchange a photon b. Not allowed: lepton number not conserved
Not allowed: quark number not conserved
Allowed: electron neutrino annihilates with a positron becomes a then splits to muon and muon neutrino.
Allowed: up quark annihilates with anti-up quark becomes , then becomes a strange quark and anti-strange quark
Not allowed: charge not conserved
Allowed: this is a very rare interaction
Not allowed: electrons don’t interact with gluons
Not allowed: neutrinos don’t interact with photons
Allowed: the electron and the positron are exchanging virtual electron/positron pairs
Allowed: this is beta decay, a down quark splits into an up quark an electron and an electron neutrino via a particle.
Allowed: a muon splits into an muon neutrino, an electron and an electron neutrino via a particle.
Chapter 28: Quantum Mechanics Version 2
The Big Idea
Quantum Mechanics, discovered early in the century, completely changed the way physicists think. Quantum Mechanics is the description of how the universe works on the very small scale. It turns out that we can't predict what will happen, but only the probabilities of certain outcomes. The uncertainty of quantum events is extremely important at the atomic level (and smaller levels) but not at the macroscopic level. In fact, there is a result called the correspondence principle that states that all results from quantum mechanics must agree with classical physics when quantum numbers are large -- that is, for objects with large mass. The foundation of quantum mechanics was developed on the observation of wave-particle duality.
Electromagnetic radiation is carried by particles, called photons, which interact with electrons. Depending on the experiment, photons can behave as particles or waves. The reverse is also true; electrons can also behave as particles or waves.
Because the electron has a wavelength, its position and momentum can never be precisely established. This is called the uncertainty principle. (What has been said above about the electron is true for protons or any other particle, but, experimentally, the effects become undetectable with increasing mass.)
The Key Concepts
The energy of a photon is the product of its frequency and Planck’s Constant. This is the exact amount of energy an electron will have if it absorbs a photon.
A photon, which has neither mass nor volume, carries energy and momentum; the quantity of either energy or momentum in a photon depends on its frequency. The photon travels at the speed of light.
The five conservation laws hold true at the quantum level. Energy, momentum, angular momentum, charge and CPT are all conserved from the particle level to the astrophysics level.
If an electron loses energy the photon emitted will have its frequency (and wavelength) determined by the difference in the electron’s energy. This obeys the conservation of energy, one of the five conservation laws.
An electron, which has mass (but probably no volume) has energy and momentum determined by its speed, which is always less than that of light. The electron has a wavelength determined by its momentum.
If a photon strikes some photoelectric material its energy must first go into releasing the electron from the material (This is called the work function of the material.) The remaining energy, if any, goes into kinetic energy of the electron and the voltage of an electric circuit can be calculated from this. The current comes from the number of electrons/second and that corresponds exactly to the number of photons/second.
Increasing the number of photons will not change the amount of energy an electron will have, but will increase the number of electrons emitted.
The momentum of photons is equal to Planck’s constant divided by the wavelength.
The wavelength of electrons is equal to Planck’s constant divided by the electron’s momentum. If an electron is traveling at about this wavelength is then not much smaller than the size of an atom.
The size of the electron’s wavelength determines the possible energy levels in an atom. These are negative energies since the electron is said to have zero potential energy when it is ionized. The lowest energy level (ground state) for hydrogen is . The second level is. Atoms with multiple electrons have multiple sets of en
ergy levels. (And energy levels are different for partially ionized atoms.)
When an electron absorbs a photon it moves to higher energy level, depending on the energy of the photon. If a photon hits a hydrogen atom it ionizes that atom. If a photon strikes hydrogen the electron is moved to the next level.
Atomic spectra are unique to each element. They are seen when electrons drop from a higher energy level to a lower one. For example when an electron drops from to in the Hydrogen atom a photon is emitted. The spectra can be in infra-red, visible light, ultra-violet and even rays. (The photon is ultra-violet.)
The wave nature of electrons makes it impossible to determine exactly both its momentum and position. The product of the two uncertainties is on the order of Plank’s Constant. (Uncertainty in the electron’s energy and time are likewise related.)
The Key Equations
Relates energy of a photon to its frequency.
Relates the momentum of a photon to its wavelength.
The Debroglie wavelength of an electron.
This is the Heisenberg Uncertainty Principle, (HUP) which relates the uncertainty in the momentum and position of a particle.
Relates the uncertainty in measuring the energy of a particle and the time it takes to do the measurement.
Planck’s constant.
The most convenient unit of energy at the atomic scale is the electron volt, defined as the potential energy of the charge of an electron across a potential difference of volt.
A photon of energy of has a wavelength of and vice versa. This is a convenient shortcut for determining the wavelengths of photons emitted when electrons change energy levels, or for calculations involving the photoelectric effect.
Problems Set: Quantum Mechanics
Calculate the energy and momentum of photons with the following frequency: From an station at
Infrared radiation at
From an station at
Find the energy and momentum of photons with a wavelength: red light at
ultraviolet light at
gamma rays at
Given the energy of the following particles find the wavelength of: X-ray photons at
Gamma ray photons from sodium at
A electron
The momentum of an electron is measured to an accuracy of . What is the corresponding uncertainty in the position of the electron?
The four lowest energy levels in electron-volts in a hypothetical atom are respectively: . Find the wavelength of the photon that can ionize this atom.
Is this visible light? Why?
If an electron is excited to the fourth level what are the wavelengths of all possible transitions? Which are visible?
Light with a wavelength of strikes a photoelectric surface with a work function of . What is the stopping potential for the electron?
For the same surface in the previous problem but different frequency light, a stopping potential of is observed. What is the wavelength of the light?
An electron is accelerated through . It collides with a positron of the same energy. All energy goes to produce a gamma ray. What is the wavelength of the gamma ray ignoring the rest mass of the electron and positron?
Now calculate the contribution to the wavelength of the gamma ray of the masses of the particles? Recalculate the wavelength.
Was it safe to ignore their masses? Why or why not?
An photon of strikes an electron. What is the increase in speed of the electron assuming all the photon’s momentum goes to the electron?
A ray in the -direction strikes an electron initially at rest. This time a ray is observed moving in the direction after collision. What is the magnitude and direction of the velocity of the electron after collision?
The highly radioactive isotope Polonium has a half-life of and emits a gamma ray upon decay. The isotopic mass is . How much time would it take for of this substance to decay?
Suppose you had of how much energy would the emitted gamma rays give off while decayed?
What is the power generated in kilowatts?
What is the wavelength of the gamma ray?
Ultra-violet light of strikes a photoelectric surface and requires a stopping potential of volts. What is the work function of the surface?
Students doing an experiment to determine the value of Planck’s constant shined light from a variety of lasers on a photoelectric surface with an unknown work function and measured the stopping voltage. Their data is summarized below: Construct a graph of energy vs. frequency of emitted electrons.
Use the graph to determine the experimental value of Planck’s constant
Use the graph to determine the work function of the surface
Use the graph to determine what wavelength of light would require a stopping potential.
Use the graph to determine the stopping potential required if light were shined on the surface.
Laser Wavelength Voltage
Helium-Neon
Krypton-Flouride
Argon
Europium
Gallium arsenide
An element has the following six lowest energy (in ) levels for its outermost electron: Construct a diagram showing the energy levels for this situation.
Show all possible transitions; how many are there?
Calculate the wavelengths for transitions to the level
Arrange these to predict which would be seen by infrared, visible and ultraviolet spectroscopes
A different element has black absorption lines at when white light is shined upon it. Use this information to construct an energy level diagram.
An electron is accelerated through and is beamed through a diffraction grating, which has lines per . Calculate the speed of the electron
Calculate the wavelength of the electron
Calculate the angle in which the first order maximum makes with the diffraction grating
If the screen is away from the diffraction grating what is the separation distance of the central maximum to the first order?
A light source of is used to power a photovoltaic cell with a work function of . The cell is struck by photons per second. What voltage is produced by the cell?
What current is produced by the cell?
What is the cell’s internal resistance?
A ray moving in the positive direction strikes an electron, which is at rest. After the collision an ray of is observed to move degrees from the positive axis. What is the initial momentum of the incident ray?
What are the and components of the secondary ray?
What must be the and components of the electron after collision?
Give the magnitude and direction of the electrons’ final velocity.
Curium has an isotopic mass of and decays by alpha emission; the alpha particle has a mass of and has a kinetic energy of . What is the momentum of the alpha particle?
What is its wavelength?
Write a balanced nuclear equation for the reaction.
Calculate the isotopic mass of the product.
If the alpha particle is placed in a magnetic field of what is the radius of curvature? (The alpha particle has a double positive charge.)
f. If the alpha particle is moving in the direction and the field is in the direction find the direction of the magnetic force. g. Calculate the magnitude and direction of the electric field necessary to make the alpha particle move in a straight line.
A student lab group has a laser of unknown wavelength, a laser of known wavelength, a photoelectric cell of unknown work function, a voltmeter and test leads, and access to a supply of resistors. Design an experiment to measure the work function of the cell, and the wavelength of the unknown laser. Give a complete procedure and draw an appropriate circuit diagram. Give sample equations and graphs if necessary.
Under what circumstances would it be impossible to measure the wavelength of the unknown laser?
How could one using this apparatus also measure the intensity of the laser (number of photons emitted/second)?
; The momentum of an electron is measured to an accuracy of . What is the corresponding uncertainty in the position of the same electron at the same moment? Express your answer in Angstroms (, about the size of a typical atom).
Thor, a baseball player, passes on a pitch clocked at a speed of . The umpire calls a strike, but Thor claims that the uncertainty in the position of the baseball was so high that Heisenberg’s uncertainty principle dictates the ball could have been out of the strike zone. What is the uncertainty in position for this baseball? A typical baseball has a mass of . Should the umpire rethink his decision?
Consider a box of empty space (vacuum) that contains nothing, and has total energy . Suddenly, in seeming violation of the law of conservation of energy, an electron and a positron (the anti-particle of the electron) burst into existence. Both the electron and positron have the same mass, . Use Einstein’s formula to determine how much energy must be used to create these two particles out of nothing.
You don’t get to violate the law of conservation of energy forever – you can only do so as long as the violation is “hidden” within the HUP. Use the HUP to calculate how long (in seconds) the two particles can exist before they wink out of existence.
CK-12 People's Physics Book Version 2 Page 25
