Lecture Notes for Physics 262

	(c) Copyright by Prof Robert Ehrlich
	George Mason University

Note:  Bring these notes to class with you.  They are incomplete, and it is intended that you will take notes directly on them.

	Chapter 17 Temperature & Heat

What are the main ideas of this chapter?

Temperature and thermometers

Intuitively, what is the meaning of temperature?  

What is meant by the concepts "thermal contact" and "thermal equilibrium?"

What is the "zeroth law of thermodynamics?"

Why is the zeroth law necessary for a formal definition of temperature?

What are some physical properties that depend on temperature that can be used to construct a thermometer?

What is the idea behind the constant volume gas thermometer?
How do different gases behave in this thermometer?

How is the zero point on the Kelvin (absolute) scale defined?
Why is it called absolute zero?  What happens to a gas at absolute zero temperature?

How are temperatures converted between the Celsius and Fahrenheit scales?  Between the Celsius and Kelvin scales

At what temperature do the Celsius and Fahrenheit scales read the same value?

How do the dimensions of most solids and liquids change when they are heated or cooled?

What equation describes thermal expansion?

What happens to the diameter of a hole in a washer when it is heated?

What happens to the volume of a solid or liquid when it is heated?

How is the coefficient of volume expansion related to the coefficient of linear expansion?  Why the factor of 3?

What is a bimetallic strip?  What is it used for

How do the thermal properties of water differ from most other materials?

Why do lakes freeze from the top down, rather than the bottom up?

What is the definition of the calorie -- a unit of thermal energy or heat?  How does it relate to the food calorie?  What is a Btu?

What was the point of Joule's experiment on the mechanical equivalent of heat?  How was the experiment done?

What was the result of Joule's experiment?

How much warmer would water be at the bottom of a 100 m high waterfall?

Would it take the same amount of heat to raise the temperature of different substances by 1 C?

What is meant by the specific heat of a substance?  What is the molar specific heat?

What substance has one of the highest specific heats?  What are the environmental consequences?

What is a calorimeter?  What can it be used to measure?  How is the measurement done?

What is the specific heat of a metal if a 50 gram piece of it heated to 200 C raises the temperature of 400 grams of water from 20 to 22.4 C?

Does a substance always have its temperature changed when heat is added or removed?  What would be an example

What is meant by the term "latent heat?"  How are the latent heats of fusion and vaporization defined?  What are their values in the case of water?

How would the temperature of a fixed mass of water change if heat is added or removed at a constant rate?

How much heat must be added to one gram of ice initially at -30 C to produce one gram of steam at 120 C?

What happens to the molecules of a substance when heat is added: (a) assuming no phase change occurs, (b) assuming a phase change does occur?

Why does it feel so cool when alcohol is rubbed on your arm?

Why is it so hot in a steam room compared to a dry sauna at the same temperature?

What are three ways to transfer heat between a system and its surroundings?

What is heat conduction, and what is actually happening on a microscopic scale?

What substances are good conductors of heat, and what substances are poor conductors (good insulators)?

What is necessary for heat conduction to occur" What does the rate of heat flow depend on?  

How can we understand the presence of each variable in the equation?  What does the value of the thermal conductivity k tell us?

Suppose we had N parallel slabs of different thicknesses and thermal conductivities in contact.  In the steady state situation, how is the heat flow through each slab related?

How can we derive the heat flow equation in this case?

What is meant by the engineering term "R-value?"

How is heat transferred by convection?  What would be some examples of natural and forced convection?

Why can water no longer be heated by convection below 4 C?

What is the difference in heating a fluid if the source of heat is on the top or the bottom?

What is the third way of transferring heat or thermal energy between a system and its surroundings?  What form is that radiant energy?

How much radiant energy does the Earth receive from the sun?  Why doesn't the Earth keep getting hotter?

Why is it warmer in summer than winter?

How does the rate at which an object radiates energy to its environment depend on its absolute temperature?  What does the emissivity e depend on?   How does a thermos bottle work?

How much energy would a star emit compared to the sun if the star had twice the sun's temperature, and twice its radius, assuming the emissivity was the same?

How do we account for the radiant energy objects absorb from the environment?

	Chapter 18 – Thermal Properties of Matter

What are the main ideas of this chapter?

What three quantities define the state of a gas in thermal equilibrium?  

What is meant by the "equation of state" of a gas?

What is an "ideal gas?"

How is one mole of a substance defined?  What is the basis of that definition?  What is Avogadro's number?

What is the equation of state of an ideal gas?  (The ideal gas law)  When does it apply?

What is the constant R?  What volume does one mole of any ideal gas occupy at STP?

What is an alternative way to write the ideal gas law?  What's the difference between n and N?

Suppose on a hot day the temperature of your car's tires rises from 20 to 40 C.  If the gauge pressure is initially 35 psi, what is the pressure after the tires heat up?  

Approximately how many molecules of air are inside each tire?

How does the behavior of real gases differ from ideal gases?  How is the “critical temperature” defined?

What is Van der Waals equation?  What do the constants a & b represent?

On a molecular basis, why does a gas exert a pressure on the walls of its container?

What are the assumptions of the molecular model of an ideal gas?

What average force does a single molecule exert on the walls of the container?  What do we use for the collision time dt?  Why?

How is the formula for the pressure of an ideal gas derived?

What is the relationship between the preceding result and the ideal gas law?  What conclusion can be drawn about the molecular basis of temperature?

What is the definition of the root-mean-square (rms) speed of molecules in a gas?  

How would you actually compute the rms speed given the individual speeds of some number of molecules?

What is the total kinetic energy of a gas containing N molecules?

In a mixture of two gases (say H and He) in thermal equilibrium, do the molecules have the same average KE or the same rms speed?

What is the ratio of the rms speeds of the two types of molecules here?

The Maxwell Distribution Law:  What's this all about?  How does it vary with temperature?  Why do chemical reaction rates vary drastically with temperature?

Kinetic theory simulation

How are the average (mean) speed and the most probable speed found for the Maxwell distribution?  How do they compare to the rms speed?

Example of a calculation using the Maxwell distribution:  Compare the relative number of molecules having speeds equal to 1,2,3, and 10 times vrms. 

How can we find the mean free path of a gas molecule between collisions?

What would the mean free path be for the air molecules in this room?

Specific heat(s) of a gas:

Does the heat needed to raise the temperature of a gas from Ti to Tf depend on the path taken?  How come?

For now we consider the case where the gas volume remains fixed.  How is the molar specific heat Cv?

How can we predict  Cv for a monatomic ideal gas?  How well does the predicted value agree with experiment

Equipartition of Energy:

What does this mean?  What are “degrees of freedom?”

What is the number of degrees of freedom for a monatomic gas

How many degrees of freedom does a diatomic gas have?

What does the equipartition theorem predict Cv should be in the case of a diatomic gas assuming (a) rotations, but no vibrational excitations, (b) rotations and vibrations

How does Cv for a diatomic gas vary with temperature.  What is the explanation?

Why the absence of vibrational excitations at room temperature?  What happens at higher temperatures?

Why do most solids have a molar specific heat of about 3R at high temperatures?   What happens as the temperature is reduced?

How does the behavior of real gases differ from ideal gases?  How is the “critical temperature” defined?

What does the phase diagram look like for water, and what is the “triple point” in such a diagram?

What are PVT surfaces?

Why does evaporation occur?  Why does the rate of evaporation increase with a liquid’s temperature?  

Why does evaporation cause cooling?

	Chapter 19 -- First Law of Thermodynamics

What are the main ideas in this chapter?

What is the thermal (or internal) energy of a system?  Under what circumstances does it change?

How does a system's thermal or internal energy differ from the term "heat?"

What is thermal energy on a microscopic scale?  What is the internal energy of an ideal gas?  How does heat differ from temperature?

Why must a system be in thermal equilibrium in order to define its macroscopic state?  What would be an example?  

What is meant by a "quasistatc" expansion of a gas?

How much work is done by a gas on a piston if the gas is allowed to expand quasistatically  What does the sign of the work tell us?

What must be specified before we can evaluate the integral?

Which of the three sample paths involves the greatest and least amount of work

If a gas is maintained at a fixed temperature during an expansion, does any heat (thermal energy) flow into it

Does the heat added to a gas, Q, depend on the path taken from state i to f?  Does the work W done by the gas also depend on the path  Does the quantity Q - W depend on the path?  How come?

What does the first law of thermodynamics say?  How does it relate to the law of conservation of energy?

What are the sign conventions on Q, W and dU

What would be two examples of processes for which dU = 0?  What does the first law say in each case?

What would be an example of a process for which W = 0?

What would be an example of a process for which Q = 0

Why do we say U is a state function?  Is Q or W?

What is the definition of an adiabatic process?

What must happen to the temperature of a gas if it expands or contracts adiabatically?  What would be some examples?

What is the process known as an adiabatic free expansion of a gas  How does the temperature of a gas change in such a process?  Why?

What is an isobaric process?  What is the work done by the gas in such a process?

What is an isovolumetric process?  What would be an example?

What is an isothermal process?  What path does a gas follow on a PV diagram

How much work is done by an ideal gas during an isothermal expansion?

How much work would be done by two moles of an ideal gas which doubles its volume while remaining at 0 C

How much heat is needed to boil water at 100 C if the volume of the water is 1 cc, and that of the resulting steam is 1671 cc.?  Why is the change in the internal energy of the water not equal to the heat added

What is the change in internal energy in this example?  What fraction of the heat added went into doing work during the expansion?

How can we find the difference between Cp and Cv for an ideal gas?  
Why is Cp bigger than Cv for an ideal gas?

What is the value of Cp/Cv for an ideal gas?

Why does the previous result, namely Cp - Cv  = R, hold for any type of ideal gas, not just monatomic?

Adiabatic processes:

What are they again?  What are two means of achieving an adiabatic expansion or compression of an ideal gas?

How can it be shown that during a reversible adiabatic process the quantity PV^gamma remains constant?  (Gamma is Cp/Cv)

What path does the relation PV^gamma = constant trace out on a PV diagram?

Suppose that during the compression stroke in a diesel engine the air-fuel mixture is adiabatically compressed to one tenth its original volume.  What is the final temperature and pressure if the air is initially at 20 C?  (Note: Air is mostly nitrogen.)

Suppose we considered the problem of adding heat to a solid object -- what fraction of the heat added would go into doing external work in that case?  What would the ratio Cp/Cv be for a solid?

	

Chapter 20 -- The Second Law of Thermodynamics

What are the main ideas on this chapter

What is the 2nd Law of Thermodynamics (Clausius version)?  What sorts of processes does it forbid?

What is a heat engine  What are some examples?

How do we schematically represent energy transfers for a heat engine?  Why is a temperature difference needed to run any engine?

What is an implication of the fact that all heat engines operate by taking some “working substance” through a cycle?  How is this represented on a PV diagram?

How is the thermal efficiency of a heat engine defined?

What is the Kelvin-Planck form of the Second Law of Thermodynamics?

What are reversible and irreversible processes?  Are any real processes reversible?  What two things must be true for a process to be reversible?

The Carnot Engine:

What are the four steps in a Carnot cycle?  Exactly what happens in each step?

What is the efficiency of a Carnot engine?  How is the result relevant to real engines?

In what two ways can increase the efficiency of a Carnot engine?  Which of these two ways is practical?

Suppose a steam engine has a boiler that operates at 500 K.  What is the maximum work it can do each cycle if the heat absorbed from the high temperature reservoir each cycle is 200 J?

What is Carnot's Theorem?  How can it be proven using the Second Law of Thermodynamics?

What is the Otto cycle, and what happens during each step?

What is the efficiency of an ideal Otto cycle?  How can this result be proven?

How well real gasoline engine efficiencies compare with the values computed for an ideal Otto cycle?

Why do efficient engines tend to have a high “compression ratio?”

Refrigerators, air conditioners and heat pumps:

What is a heat pump?  How are heat pumps (and air conditioners and refrigerators) all related to heat engines?

How is its coefficient of performance (COP) defined?

What is the maximum COP based on the Carnot cycle?

Under what conditions would the Carnot COP be greatest?  How high a COP can be achieved in practice for a heat pump or refrigerator?

Entropy:

How do we define the change in entropy of a system?  What is the basis for this definition?

Under what conditions is dS zero, positive or negative?

How much does the entropy of a block of ice of mass m change when it melts?

How is dS calculated when the temperature of a system is not constant?

What does it mean to say entropy is a “state function?”

What are the entropy changes during each step of a Carnot cycle?  What is the change in entropy for a complete cycle?  (This is true for any reversible process.)

Irreversible processes:

How are entropy changes computed in this case?  Why does that work?

What do we always find for the entropy change in a system PLUS its surroundings for any irreversible process?  

Example: What is the net entropy change that occurs when heat flows through the walls of your house? 

What would happen if heat could flow from cold objects to hot ones?

What is an adiabatic free expansion of a gas?  How do we compute the change in entropy during such an expansion?  What reversible process can we use to do the calculation?  How come?

What is the net entropy change in the universe when one mole of an ideal gas doubles its volume?

What is the net entropy change if one kg of water at 100 C is mixed with one kg of water at 0 C?

What is the general statement of the 2nd Law of Thermodynamics based on entropy?  Why has entropy sometimes been called “time’s arrow?”

What is another meaning of the entropy of a system?  How is the "amount of disorder" of a system measured

Given that entropy is a measure of disorder, why does the entropy increase when each of the following processes occur:

(a) an adiabatic free expansion of a gas?

(b) an ice cube melts

(c) two volumes of water at different temperatures are mixed?

Which of the preceding processes are reversible?

When systems spontaneously become more orderly – as in the case of evolution or the development of organisms -- doesn’t that violate the 2nd Law?

How can the amount of order or disorder of a system be defined using the laws of statistics?  What is an example using a set of coins?  What is an example using molecules of air in two halves of a box?

What is the general definition of entropy as derived by Boltzmann?

What would this formula give for the change in entropy that occurs when a gas undergoes an adiabatic free expansion to twice its original volume?
	Chapter 33: The Nature and Propagation of Light	

The Ray Approximation in Geometrical Optics:  Light, unlike sound, to a good approximation travels in straight lines or "rays."  How might you demonstrate this behavior of light?  

How can you easily show that sound does not travel in straight lines – i.e., no “sound rays?”  Why the difference between light and sound?

What does the field of geometrical optics encompass?  What are its limitations?   (In this chapter and the next, we assume that the ray approximation holds true for light.)

Measurements of the Speed of Light:  Galileo's first (unsuccessful) attempt.  Why didn't it work?

First true measurement of the speed of light by Ole Roemer, based on eclipses of one of Jupiter's moons.  Basic idea of experiment?

Another approach to measure c:  Michelson’s Experiment -- basic idea?

Polarization of EM waves.  EM waves are transverse -- meaning?  Whether they are polarized depends on how they are produced.  What is meant by a plane (linearly) polarized wave?  By an unpolarized wave?  Do waves have to be transverse in order to be able to be polarized?

Polarization simulation

Consider an EM wave produced by a vertical antenna.  What are the directions of the E and B vectors at large distances?

This is a plane polarized wave.  The plane of polarization (for E) depends entirely on the antenna direction.  In an unpolarized wave we have a mixture of all different polarizations - all equally weighted.  

Light waves are emitted from individual atoms.  Do you suppose light is usually produced in the form of polarized or unpolarized waves?

Polaroid filters - useful for creating a polarized wave from an unpolarized one.  How does a polaroid filter work?

"Crossed" polaroids (two have polarization axis at right angles -- what fraction of the light will be transmitted through both?

General case: Suppose a polaroid makes an angle θ with the polarization plane of an EM wave.  What fraction of the wave intensity will be transmitted in the case?  (Malus Law)

Three polaroid case:  What happens if a third polaroid is inserted between two crossed polaroids?

Polarization by Reflection: Suppose an unpolarized beam of light is incident from air on a mirror of index n.  The reflected beam will be 100 % polarized for a certain angle of incidence called "Brewster's Angle." 

How is Brewster's Angle related to n?

Why do Polaroid sunglasses reduce glare?

Polarization by Scattering: Scattering is the absorption and subsequent reradiation of light incident on a medium consisting of many small particles, e.g. atoms or dust particles.  The scattered light is completely polarized when the scattering angle is 90 degrees. Demonstrate.

Reflection and Refraction of Light: Consider a light ray incident on a boundary between to media, such as air and glass.  What three things can happen to the light ray?

What are the two types of reflection of light from a surface?  examples?
Why does diffuse reflection occur?

Law of Reflection: θ1 = θ1'

Example: the "corner reflector": how does it work?  What are some uses?

Law of Refraction:			n1 sin θ1 = n2sin θ2 
(also known as Snell's Law)

Note: angles are measured with respect to the normal to the boundary.

What is n?  How is it related to the speed of light in a medium?  

Based on the drawing which medium has the higher index of refraction n, air or glass?

What happens to the speed of light after it enters glass, and then reemerges into the air?  How can that be?

What is the value of n for vacuum?

What are some values of n for typical transparent substances?

Snell’s Law simulation

What happens to the frequency, velocity and wavelength of light when it passes from one medium to another?

How could you use the following diagram to find the index of refraction of the medium?

Why does a ray of light emerge from a slab of glass travelling along its original direction?  (A slab, by definition has parallel faces.)

Does the value of n depend on wavelength?  How?  Why do rainbows form?

Total internal Reflection:  Suppose that a light ray starts from inside a medium with the higher n.  Show that there is a maximum angle of incidence, the critical angle, for the light ray to enter the lower index medium -- demonstrate.

Total internal reflection simulation

What happens to light rays incident on the boundary at angles larger than the critical angle?

What are several examples of total internal reflection?

What must n be for total internal reflection to occur for a 45 degree prism?

Why do diamonds sparkle more brightly than chips of glass?

How do optical fibers work?

Optical fiber simulation

	Chapter 34: Geometrical Optics 

Plane (flat) Mirrors:  Where is the image you see when looking at an object in a plane mirror?

Show why this occurs by drawing a ray diagram:

In equation form, we relate the image and object distances: s'=-s

Why is the image from a plane mirror said to be "virtual," as opposed to "real"?

What is the smallest vertical size mirror that would allow you to see your full height?  Does the result depend on your distance to the mirror?

Define the lateral magnification of a lens or mirror: M = ratio of image to object heights = h'/h.  What is the magnification of a plane mirror?

Spherical Mirrors:  What are the two types?
Which type is on the right side of your car?

Convex Mirrors (bulge out) &  Concave Mirrors (sink in -- think "cave")

Diagram showing the formation of a real image in a concave mirror:
(Why do we say the image is “real?”)

Based on similar triangles: M = h'/h = -s'/s.  (Why minus?)

Derivation of the Mirror Equation:

Definition of Focal Length of a Mirror:  Let s become infinite
What is “spherical aberration?”

Concave Mirror							Convex Mirror
(f > 0)								(f < 0)

How is the focal length, f, related to its radius of curvature?

What is the relation between the object distance, s, the image distance, s', and the focal length, f?  (The “mirror equation.”)

What are the sign conventions for s, s', f, and M?

What properties would an image have if its magnification is positive? negative?

Example:   An object is placed 10 cm in front of a mirror, and its image is found to be 20 cm in front of the mirror (on a screen placed there). (a) What is the radius of the mirror?  (b) For what range of object distances would a real image be produced?  (c) Where could the object be placed for the magnification to be -2?

Ray Tracing and Mirrors:

Show three rays whose paths can easily be traced.  (Note that only two rays are needed to locate an image point.)

Concave Mirror:							Convex Mirror:

Prove the result of part (c) of the previous example, using a ray tracing diagram.

Mirror simulation

Thin Lenses: What does “thin” mean?  How is focal length defined?  How is the “power” of a lens defined?

Converging lens (thicker in middle) -- causes rays to converge:

Diverging lens (thinner in middle) -- causes rays to diverge:

Ray Tracing with Lenses: Show the three rays whose paths can be easily traced:

Converging lens:							Diverging lens:

How are s, s', and f related for a lens?  (The “lens equation”)
Does it apply to both types of lenses?

On which side of lens are real and virtual images?  Which type of lens can only form virtual images? Why?

How is lateral magnification defined?  Are virtual images erect or inverted?

Lens simulation

Example: Consider a lens with focal length 20 cm.  An object is placed 40 cm in front of the lens  (a) locate the image, (b) what is its magnification?, (c) what are the properties of the image (real/virtual, erect/inverted, enlarged/reduced)?, (d) draw a ray diagram.

How do you find the image when a combination of several lenses is used?

What is the "Lensmaker's Equation"?

What is the sign convention for s, s', f, R1 and R2?

What is the structure of the human eye? What is its normal “near point” and “far point?”

What are the most common defects of vision?

How do we determine the power of eyeglasses needed to correct for a near-sighted and far-sighted eye?

Suppose a nearsighted eye cannot see anything clearly beyond 40 cm, what power lens would be needed in this case?  

How about an eye that cannot see anything clearly if it is closer than 1 meter?

Eye glass simulation

How do we find the magnifying power of a magnifying glass?

What ray diagram shows how a refracting telescope works?  What is the magnification for this instrument?

How does a compound microscope work?  What is its magnification?

What are the most common aberrations of lenses and mirrors?  How can they be corrected?

Lens combination simulation

	

Chapter 35: Interference

In the last two chapters when considering ray optics we made no use of the wave nature of light.  Here we discuss a property that light shares with all other waves – what is that property?

Huygen's Principle: all points on a wave front serve as new wave sources.  These "wavelets" recombine (and interfere) to form the wave front at a later time.

What does Huygen's Principle give for a wave travelling in a single medium?

Plane waves						Circular waves

Now, consider a plane wave incident on an opening (slit):
Why would Huygen's Principle predict it should spread out (diffract)?

How can Huygen's Principle explain the Law of Refraction?

Interference: What is meant by this term?  

"constructive" interference:
(wave crests aligned)
What is resultant wave amplitude?

"destructive" interference:
(crests aligned with troughs)
What is resultant wave amplitude?

Can we get an intermediate result?  How?

Interference simulation

For interference pattern to be observed, the sources must be coherent.  Meaning?  What are some examples of coherent and incoherent sources of light and sound?

Consider two (incoherent) light sources.  Why is no interference pattern observable in this case?

Must two sources be monochromatic (single wavelength) to observe interference?

Why is it much easier to observe interference of sound or water waves than light waves?

Diffraction: The spreading out of waves when they encounter an opening or an obstacle.  

Young's Double Slit Experiment: The first convincing evidence of the wave nature of light (1801) -- also allowed Young to measure the wave length.    

At what points on the screen will the interference be constructive (a bright "fringe" -- or maximum intensity)?

At what points on the screen will the interference be destructive (a dark "fringe" -- or minimum intensity)?

Double slit simulation

What is the path difference between waves from each slit to reach a given point on the screen?

Along what directions are minima and maxima found?

maxima:						minima:
(constr.)						(destr.)

m = "order number" (m = 0,1,2,... are zeroth, first, second orders.)

Consider the case where the small angle approximation holds.  We can then find the location on the screen for bright and dark fringes:

Example: Adjacent maxima on a screen are observed to be 2.5 mm apart.  If the slit separation is 0.12 mm, and the screen is 55 cm distant, what is the wavelength of the light?

Nowadays, we often do Young's Experiment using a laser beam to illuminate the slits.  What would you expect to see on the screen in this case?

Intensity Pattern for Two Sources using “phasors”:
How to find a general expression for the intensity at any position on the screen -- along any angular direction?

How to write the electric field representing the waves from each slit? Why is there a phase difference φ between them?

How does the phase difference between the waves depends on the path difference?

What should this formula reduce to when the path difference is one wavelength?

What is the resulting electric field when the waves from the two slits are combined?

How to find the resulting intensity, I, from the two waves?

What does the resulting intensity versus position on the screen y look like?

How to show that the previous conditions for maximum and minimum intensity give the expected result for the intensity at a given point on the screen?

maximum								minimum

What would we find when combining two incoherent sources?

Change of Phase Due to Reflection: When an EM wave is reflected from the boundary of a medium having higher index of refraction n than where it starts it undergoes a 180 degree phase change on reflection.  Examples where phase change occurs and doesn't occur?  

Description of the Lloyd's mirror experiment:

How does the pattern of bright and dark fringes compare with that for the two slit experiment?

Interference in Thin Films:  Due to interference of light from the front and back surfaces of the film.  Why do you see pretty colors when looking at an oil film on a wet pavement?

Suppose light were reflected off the film at normal incidence.  What would you expect to be the condition on d, the film thickness, in order to get constructive interference from reflections from front and back of the film?  Note: we need to take into account: 

(1) the dependence of wavelength on n (index of refraction),  

(2) 180 degree phase change when waves reflect off medium with higher n.  

Thus, taking both effects into account, we can write down the conditions for maxima and minima for reflection off thin films:

case 1: no 180 degree phase changes

maxima:							minima:

case 2: one 180 degree phase change

maxima:							minima:

How about if both waves undergo a 180 degree phase change?

Which of the following apply to case 1, and which to case 2:
soap bubble film? air film between two sheets of glass?
camera lens coated with substance whose n is less than glass?

Example: A thin film is illuminated by white light.  A first maximum is observed at 1702 nm wavelength.  What other maxima would be found if (a) the film has a higher index than the surrounding media, (b) a lower index than the surrounding media, (c) a higher index than the medium on one side, and a lower index than the medium on the other side.

Thin film simulation

Varying Thickness Thin Films: creates a pattern of "fringes" of constant thickness.  What causes these fringes?

Which has a greater separation: red or blue fringes?  Why?

Suppose we had a thin film in the shape of a wedge of angle θ.  What is the distance between bright fringes?  Does this result depend on whether there are any 180 degree phase changes?

What would you expect to see when looking at a soap film on a vertical wire loop?  (Assume film is thicker at bottom due to gravity, and has near-zero thickness at the top.) Demonstrate

What is the Michelson interferometer?  How does it work and what is it used for?
	Chapter 36: Diffraction 

Diffraction: what is it?  For what type of waves does it occur?

How does the amount of spreading depend on the wavelength and the size of the opening?  

Diffraction also applies in the case of waves encountering an obstacle.  Consider the case of a penny blocking a light wave.  What does the diffraction pattern look like in this case?  What role did this observation play in the history of physics?

What is meant by "Fraunhofer diffraction?" 

Diffraction from a Single Slit: How do we apply Huygen's Principle to this situation?  

Will the intensity pattern must have a maximum or a minimum in the forward direction (θ = 0)? -- why?

How can we locate the first minima in the intensity pattern?  

What is the path difference to the screen between a wavelet starting from the slit center and one starting at the top of the slit?

What is the condition for these two wavelets to interfere destructively on reaching the screen?

Why do all the wavelets cancel out for the entire slit in this case?

What then is the angle θ for the first minimum in the intensity pattern?

To find the second minimum in the intensity pattern, imagine the slit to be divided into four equal zones, rather than two.  What is the condition for the wavelets originating from the top of each zone to destructively interfere?  What does this give for the angle of the second minimum?

What about the mth minimum?

Make a "guess" for the shape of the intensity curve, based on the locations of the minima, and the maximum at 0 degrees:

Diffraction simulation

Example: A slit of width a is illuminated by light of wavelength 650 nm, and the first minimum falls at 15 degrees.  (a) What is the slit width? (b) How many minima will the pattern have?  (c) What wavelength light will produce its first maximum at the first minimum of the first wavelength?  (Assume that the positions of maxima are midway between minima -- only approximately true.)

Single Slit Diffraction: Here we use the phasor technique and Huyghen's Principle to find the intensity pattern.  Imagine the slit opening, of width a, to be divided into N zones.  The zones have a width dy, so N.dy = a.  If the screen is very distant, what is the path difference and the phase difference between wavelets originating from one strip and the next one? 

How do the little phasors representing waves from all N strips combine geometrically, when we consider waves going in the forward direction (θ = 0)?  How about some other angle?

Forward direction:					General case:

See derivation in the book for the equation for the intensity as a function of angle:
I = Im [(sin β/2)/β/2]2,  where β = (2πa/λ)sin θ.

What does the formula give for the intensity in the forward direction?

Show that the preceding equation has minima at the previously identified angles: sin θ = mλ/a,  m=1,2,3,...

Why is the amount of spreading out of a wave passing through an opening proportional to λ/a?

Use the preceding formula for the intensity I to find the height of the secondary maxima of the intensity curve, assuming they lie midway between the minima.

Diffraction in the double slit experiment.  What did we find earlier for the intensity pattern for the double slit case?  

How does this result need to be corrected to take into account the effect of diffraction?  

How does the number of interference fringes in the central diffraction peak depend on the slit width and slit separation?

Resolving Power: The ability to clearly separate two adjacent point images.  The greater the resolving power, the closer two images can get and still be resolved as two rather than one.  What is the connection between resolving power and diffraction?

Rayleigh's Criterion: For two images to be just resolvable, the first minimum of one diffraction pattern coincides with the central maximum of the other.  (See figs on p1081.)  Draw examples of two point sources that are: (a) clearly resolved, (b) barely resolved (based on Rayleigh's Criterion), and (b) not resolved:

Rayleigh criterion simulation

Based on Rayleigh's Criterion, what is the minimum angular separation for two images (or objects) to be resolved?  What is the criterion when the wavelength is small compared to the slit diameter?

Diffraction from a Circular Hole: The analysis is similar to what was done for a slit, but the math is more complicated.  The result is a central maxima surrounded by concentric rings.  The middle of each dark ring is a minimum intensity.  For the first minimum, the result is slightly different than for a slit:

You can observe this type of diffraction yourself by looking at bright point source through a tiny pinhole in a piece of aluminum foil -- demonstrate.

The stars are so far away that we do not see a disk -- yet we don't see actual points either.  How come?

What does Rayleigh's Criterion for the limit of resolution become in the case of a circular aperture?

Why do large diameter telescopes give better resolution than small diameter telescopes?

Why must radio telescopes often have huge diameters?

The Diffraction Grating: Many closely spaced slits engraved on a glass plate.  (Typically 5,000 lines per cm.)

Maxima in the pattern occur when wavelets from all slits arrive in phase at screen -- draw diagram.  What is the condition for a maximum?

The condition for a maximum is the same as in the case of two slits.  The big difference between N slits and 2 slits is in the sharpness of the maxima -- Show the two cases:

Demonstrate intensity patterns for the two cases using a laser which produces monochromatic light.  

Can also look at the case of a source with many wavelengths present.  In this case, we get maxima in the intensity pattern for each wavelength: "spectral lines."  The lines are very narrow because of the effect just discussed.  What practical use can be made of this? Demonstrate

Resolving Power and the Diffraction Grating: With a diffraction grating we often want to be able to resolve as distinct two very nearby wavelengths.  The definition of resolving power in this case is given by:

Why is this a reasonable definition?

It can be shown that the resolving power of a diffraction grating can be expressed as: R = Nm, where N = the number of slits, and m = the order.  Why should the resolving power increase with both N and m?

	Chapter 37: Relativity 

Where did physics stand at the end of the 19th century?  Why is relativity needed?  Is Newtonian mechanics "wrong"?

The Theory of Relativity is actually two separate theories: special relativity (1905), and general relativity (1915).  We will here be primarily concerned with special relativity, which rests on two basic postulates (assumptions):

Postulate 1. The principle of relativity: The laws of physics are the same for observers in all inertial reference frames.  (There is no "preferred" reference frame.)  What is an inertial reference frame?  How can you tell if you are in one?

Are we in an inertial reference frame right now?

Another way to express the principle of relativity: There is no absolute motion and absolute rest.  For two observers in relative motion, each can consider himself at rest.  Does this postulate seem plausible to you?

How did Maxwell’s discovery of the laws of electromagnetism challenge the relativity pinciple?

How did the physicists of that era respond to that challenge?

What was the Michelson-Morely experiment, and why was it done?

What were the results of the experiment, and why were they so puzzling?

Back to Einstein’s theory of relativity:

Postulate 2. Constancy of the speed of light: The speed of light in vacuum is the same in all inertial reference frames (the same for all inertial observers) -- independent of motion of light source or observer.  Does this postulate explain the M-M result?  Does the postulate seem plausible to you?

Suppose you are in a rocket ship going past earth at half the speed of light, and you shine a flashlight, which emits light at speed c relative to the ship.  

What would you normally expect the light speed to be according to an earth observer?

What does the second postulate predict?

Is there an empirical check on the correctness of Einstein’s "crazy" 2nd postulate? 

Still another interpretation of the 2nd postulate:  the speed c is the "limiting speed in the universe" -- you can approach it but you cannot exceed it.  

What would happen if you kept accelerating an object indefinitely?

An "event": Anything that happens at a particular point in space, and at a certain moment in time.  How many coordinates to specify an event?  Give an example of an event.

How can an observer in a certain reference frame find the space and time coordinates of an event?

Why is it tricky to create an array of synchronized clocks to record the time of events at different points in space?

How can light be used for this purpose?

Relativity of Simultaneity:  Suppose a pair of events occur which one observer judges to have occurred at the same time (simultaneously).  Will another observer judge the events to be simultaneous?

Consider a specific example: two passing trains (or rocket ships).  Observer B is at the center of "moving" ship, and observer A is at the center of the "stationary" ship.  Why the quotes?

Observer A sees lightning strikes hit the two ends of the ships just as they pass.  How did A know the lightening strikes were simultaneous?

What will observer B conclude about the two lightning strikes?

Which observer's judgement should be trusted as to whether the lightning strikes were actually simultaneous?

What do we conclude from this example?

How specifically did the conclusion follow from the two postulates of relativity?

Relativity of time (time dilation): In pre-relativity physics, the rate of passage of time was considered to be universal -- the same for all observers.  (Not considering psychological time here, which can depend on circumstances.)

In relativity, there is no universal time.  The time between a pair of events -- say the ticks of a clock -- depends on the observer's motion. 

What is the formula?  v is the speed relative to what?

How can we use the two postulates of relativity to derive this relationship?

The "bouncing light pulse clock":  How does it work?

What is the time t0 for a round trip of a light pulse between the mirrors for an observer who is at rest relative to the clock?

What is the time t for a round trip of a light pulse between the mirrors for an observer who watches the clock move past?

How are t and t0 related?  Which is the longer time?

Why is t0 called the "proper time"?

Time dilation simulation

Suppose observers A and B each have their own clocks, and they move past one another at a fraction of the speed of light.  What would A observe about B's clock?

What would B observe about A's clock?

Is time dilation some kind of illusion, or an it be actually observed?

The twin paradox: Suppose one of two twins (A) takes a space ship journey to a distant star and returns.  Which twin is older on A's return?

The paradox: If motion is relative, why can't we say that the earth twin (B) was the one who took the journey, and he should be the younger one?

Could this really happen?  Has it been done?  What is one resolution of the paradox?

Examples: (a) At what speed would a clock run five times slower than when it is at rest?  

(b) At what speed would a clock run slower by 1 s after one year?

Length Contraction: Relativity predicts that an observer's motion affects length measurements as well as time measurements, at least for lengths along the direction of motion.  

How could you measure the length of an object as it moves past?  Why couldn't you just move a ruler along with it?

How can we understand length contraction as a consequence of the relativity of simultaneity?

What do we mean by the "proper length" of an object?  Is that its "true" length?

What is the length of an object according to a relatively moving observer?

How can we understand length contraction as a consequence of time dilation?

What is the length of a meter stick according to someone watching it move past at 98% the speed of light?

Suppose the meter stick moves perpendicular to its length?

The transformation equations (The Lorentz Transformation): Answers the question: How are two relatively moving observer's measurements of the same event related?

Specifically: Assume observer S finds that an event has coordinates (x,y,z,t), and another observer S' moving with velocity v relative to S along the x-axis finds the coordinates of the same event to be (x',y',z',t').

According to pre-relativity physics (common sense), what would the relation be?

The Galilean Transformation:

But, the preceding relation is now known to only be true at "low" speeds.  What do we mean by low?

What is the correct transformation that holds at any speed?

The Lorentz Transformation:

What about the "inverse" transformation?

All the previously discussed relativistic effects can be derived from the Lorentz Transformation equations:

1. Simultaneity.  Suppose two events occur at locations separated by a distance d according to observer S who judges them to be simultaneous.  What will observer S' find?

2. Time Dilation. Suppose two ticks of a clock are separated in time by t according to observer S moving with the clock.  What does observer S' watching the clock move past at speed v find?

3. Length Contraction. Suppose an object's length is Δx according to an observer watching it go past.  (He measures its length by simultaneously marking its end points.)  What is the object's length according to someone moving with the object?

4. Addition of Velocities (this one not treated earlier)
Observer S' moves at velocity u relative to observer S.  Both observers measure the speed of the same object.  S says its speed is v = Δx/Δt, and S' says its speed is v' = Δx'/Δt'.  How are the two speeds related?  What is the connection to the classical ("common sense") relation?

Example. Suppose a fire cracker explodes on earth at x=0, t=0, and a second firecracker explodes at a point 100 meters to the west 20 ns later.  These two explosions are observed by a UFO moving to the west at half the speed of light.  (a) for UFO occupants find the spatial and time separations between the two events, (comment on the sign of Δt,

(b) for what speed UFO would the two explosions be simultaneous?, 

(c) what would be your answer to (b) if the difference in time between the two explosions were 20 μs instead of 20 ns?,

(e) under what conditions can the time order of a pair of events get reversed?

A new look at momentum: p = mv

What does the Law of Conservation of Momentum say about a collision?

Suppose we observe the collision in a different reference frame.  Do we find the same pf and pi?  What do we find using the classical Galilean Transformation?

So, why is there a problem?

What are three ways to resolve the problem?

How must momentum be redefined?

Why is the formula obviously right at low speeds?

What does it predict for speeds approaching the speed of light?

What does the formula say about how a particle's mass varies with its speed?

How can this behavior "explain" why c is the limiting speed in the universe?  Why quotes?

A new look at energy: The kinetic energy of an object also had to be redefined for particle's moving at relativistic speeds.  Derived (on p.939), based on the work done to accelerate a particle from rest:

Using the Taylor Series expansion, show that the relativistic formula reduces to the non-relativistic formula if the velocity of the object is low:

In relativity, we define the total energy of a particle:
What is the meaning of the term: mc2?

Can the total rest mass of the particles change during a reaction?  What does not change?

During what kinds of reactions is energy "liberated"? During what kinds of reactions is it "consumed"?

Why do we normally not notice mass changes in chemical reactions?

Why is so much more energy liberated in nuclear reactions compared to chemical reactions?

Momentum and energy: What is the relativistically correct relation between momentum and energy?

Example: Up to what velocity will the non-relativistic formula for the kinetic energy of an electron give an answer that is correct to 10%?

Example: If a particle had zero rest mass, show that its speed could never be anything but c.

Example: Two electrons collide head-on with equal and opposite momenta.  What is the threshold momentum for the electrons to create two additional particles having the mass of the electrons.  Could this actually happen?

	Chapter 38: Photons, Electrons & Atoms

Beginnings of Quantum Theory:  What physics problem at the early part of the 20th century set the stage for the quantum revolution?

Why is it called blackbody radiation?  What would be an example of an ideal black body?

What is the shape of the spectrum?  What does the shape of the spectrum depend on?  

Blackbody simulation

How does the peak of the distribution depend on absolute temperature experimentally?  How does the total amount of radiation depend on T?

What did classical physics predict for the shape of the spectrum?
What was the "ultraviolet Catastrophe?"

What formula did Max Planck propose in order to fit the spectrum shape?  How did he get the value of h?

What assumption did Planck make in order to derive his formula to fit the blackbody spectrum?  On what basis was this assumption made?

The Nature of Light: Here we use the term "light" to include all forms of EM radiation, not just visible light.  In previous chapters we have seen convincing evidence for the wave nature of light.  What was this evidence again?

Now, we shall see that convincing evidence also exists that light behaves like a stream of particles!  Later on, we shall see how to reconcile these seemingly contradictory ideas.

The photon: In 1905 Einstein extended the ideas of Planck.  What proposal did Einstein make, and why was it such a radical idea?

What does the fact that h is such an incredibly tiny number imply?

How does this relation give us a new way to look at the electromagnetic spectrum?

If we think of the photon as a particle, what is its rest mass? Why?

Where did Einstein get the idea for his photon hypothesis?

The Photoelectric Effect: The photon hypothesis was made to explain this effect.  Light shining on a metal surface causes electrons to be emitted.  The ejected electrons constitute an electric current that flows to a "collector" electrode.  The current can be reduced to zero by applying a large enough stopping potential.  

What property of the emitted electrons do we measure by observing the voltage that is required to shut the current off?

What quantities are actually observed in this experiment?

A number of features of the experiment are inexplicable according to the wave theory, but they make perfect sense under the photon model of light:

Observation				Wave Theory		Photon Model

1. Current shuts off at the same
stopping potential for all
light intensities.

2. Stopping potential to shut
off the current is proportional
to the frequency of light

3. A "cutoff" frequency exists
below which no electrons are
emitted.

4. Electrons are emitted as
soon as light is turned on

What equation describes the photoelectric effect?  What is the basis of this equation?

What is the meaning of φ?

How does this equation explain why a cutoff frequency must exist?

Could there be a metal for which red light produces a photoelectric effect, but blue light does not?  Why?

Example: When light of wavelength 600 nm is shone on a metal surface the maximum energy of ejected electrons is 1.8 eV.  (a) What is the work function of the metal?, (b) What is the cutoff frequency?, (c) What would the maximum energy electrons be if the wavelength were 500 nm?

A second consequence of the photon model of light:

The Compton Effect: The scattering of light by electrons.    Based on the photon model, it is clear that light should carry momentum as well as energy.  How could we figure out what happens to a photon that collides with an electron?

What is actually observed in the Compton Effect?  

Why does the existence of two intensity peaks (i.e., the "Compton Shift") hard to understand if light is a wave?

How did Compton explain the wavelength shift?

What did he predict for the magnitude of the wavelength shift based on conservation of momentum and energy? (See derivation in book – p955-956.)

Based on this formula, how should the magnitude of the wavelength shift depend on the incoming photon wavelength?

For what angle of the scattered photon is the shift predicted to be greatest?  How can this be easily understood?

Why does the experiment show two intensity peaks -- one shifted, and one not?

Example: For what incoming energy would photons scattered by 90 degrees experience a 10 percent shift in the wavelength?

Pair production.  What is an additional interaction of light with matter that is consistent with the photon picture?  What is the minimum energy (the “threshold”) for the process to occur?  What kind of photon is needed?

The dual nature of Light: In the beginning of the "Quantum Revolution" Einstein suggested in 1905 that light, previously considered to consist of waves, sometimes behaved like a stream of particles (photons).  What experiments support this idea again?  

So, is light a wave or a particle?  Under what conditions is the wave nature of light most evident? the particle nature?  What is the “principle of complimentarity?”

Atomic Structure: What were some early models of the atom, after the electron had been discovered?

What was Rutherford’s experiment about, and how did he discover the atomic nucleus?

Natural model that follows from Rutherford’s discovery is the planetary model – what are its fatal defects?

One of the main clues at to the structure of the atom is the light they emit – their spectrum.   Why is it particularly useful to look at the light emitted by a heated gas as against a solid?

What does the spectrum look like if a gas is heated to the point where it glows?  How does it differ from that of black body radiation?  How does the spectrum differ from element to element?

What element would be the most promising one to study to get clues regarding atomic structure?

Before the days of quantum theory much data had been accumulated on the spectrum of hydrogen and other elements.  It was found that the wavelengths of lines in the hydrogen spectrum could be described by some empirical formulas whose basis was not understood.  One such formula was discovered in 1885 by a Swiss school teacher:

Balmer Series:

Niels Bohr and his model of the Hydrogen Atom (1913):

1. Electron orbits the nucleus (a proton) in a circular orbit, and is attracted by the electrostatic force -- nothing new here.

2. Only certain orbits are stable.  Only for these stable orbits does the electron not radiate energy ("stationary states")

3. "Quantum jumps:"  When the atom emits or absorbs radiation it goes from one stationary state to another.

4. Angular momentum quantization.  The condition that determines the size of the stable orbits is the quantization of angular momentum – meaning?

What does conservation of energy predict for the frequency of light emitted by the hydrogen atom?  Why does this equation predict the presence of a discrete (line) spectrum?

Using Bohr's four assumptions, how can we calculate the radii of the allowed orbits, and also their energies?

What is the meaning of the minus sign?  How can energy be negative?

What is the energy of the three lowest energy levels?  How much energy is required to strip an electron from a hydrogen atom initially in its lowest energy level (ground state)?  This is called the ionization energy.

What is the wavelength of light emitted (or absorbed) during a transition from one level to another?

When does emission occur, and when does absorption occur?

How does Bohr's model explain the Balmer formula?

Example: Consider a hydrogen atom in its n=3 state.  (a) How many transitions can occur in which a photon is absorbed?, (b) How many transitions can occur in which a photon is emitted?, (c) What are the energies and wavelength for all these transitions?

Bohr model simulation

How could Bohr's formulas be extended to deal with other one electron systems, e.g., the ions He+, or Li++?

Atomic Transitions and Lasers.  Under what conditions does an atom absorb light?

Distinction between spontaneous and stimulated emission of light by an atom?

Three conditions necessary to achieve "lasing" action

1. Atoms must be in a "population inversion" -- meaning?  Why is this unusual?  How can it be created?

2. Excited state must be "metastable" -- meaning? Why necessary?

3. Stimulated photons must stay in system long enough.  How achieved? 

Reason for the name LASER?
Laser simulation

What are some uses of lasers?

In what ways does laser light differ from ordinary light?

	Chapter 39: The Wave Nature of Particles

Bohr's model of the atom – what were its successful features, and what questions could it not answer?  

The dual nature of matter:  In 1923 Louis deBroglie made an even more bizarre suggestion that particles, such as electrons, sometimes behave like waves!  What did deBroglie propose for the wavelength associated with a particle?  Where did he get the idea?

Example: What is the deBroglie wavelength of an electron accelerated from rest through 120 Volts?  What is the wavelength of a baseball of mass 0.15 kg travelling at 35 m/s?

What is a consequence of the fact that ordinary mass objects -- like baseballs -- have such a tiny wavelength?

Testing deBroglie's Hypothesis by Experiment: How could we check whether electrons do in fact have a wave nature?

Why would a crystal make a good diffraction grating to use for electrons?

The Davisson-Germer Experiment -- what was the basic method?  How did the results confirm the wave nature of the electron?

Bohr's postulate of quantization of angular momentum was earlier pulled out of thin air.  How can we can understand where this condition comes from, based on deBroglie's wavelength for an electron?  Think of standing waves on a string:

How about particles other than electrons?  Has their wave nature been observed?

What practical applications have been made of the wave nature of electrons and other material particles?

What is the more complete theory of atomic physics developed (around 1930) and who were its developers?  

What is the relationship between quantum mechanics and the classical physics applicable to the macroscopic world?

Introduction to Quantum Mechanics: The "wave function": What is it?  Is ψ itself a measurable quantity like say the electric field E?

If ψ itself is not a measurable quantity how is it connected to something that is?  What is the basis of this definition?

Why is the volume element dV needed in the definition here?

For a one-dimensional system (meaning?), we write:

				P(x) = |ψ(x)|2 dx

The Double Slit Experiment Revisited: If electrons have a wave nature, they should show an interference pattern in a double slit experiment (just as light does).  How can we can set up the experiment so as to show both the particle and wave nature of the electron?  

Double slit simulation

Is the interference being observed here between one electron and another?  How can an electron “interfere with itself?”

What is observed in the experiment if one slit is covered up half the time?  Can we say that the electron actually goes through one slit, but we just don’t know which one?

Heisenberg's Uncertainty Principle:  In classical physics, we can make exact predictions of the future state of a system.  What would be some examples?

In quantum physics there are specific limits to our ability to make such predictions.  Heisenberg's Principle of Uncertainty says there is a correlation between the minimum uncertainty in position and momentum 

How might we measure the position of an electron?  What is the "price" of measuring the position of an electron very accurately?  Example.

What is the "price" of measuring its momentum very accurately? Why?

What is the usual way to write the Uncertainty Principle as an equation?

Particle in a box: Uncertainty in position equals the size of the box.  What happens to its uncertainty in momentum as the box is shrunk?  How can that be?

Second version of the Uncertainty Principle:

What is one implication of this second version of the Uncertainty Principle?

Does the Uncertainty Principle apply to ordinary size objects like baseballs?  Why are its restrictions negligible in such a case?

What is the wave function ψ(x) for a free particle not subject to any forces?  (Hint: a free particle has a fixed momentum p and hence a fixed wavelength)

The Schrodinger Equation: How we find ψ(x) in any given situation, where a particle is acted on by known forces (or in a potential U(x))?
Special case of a) non-relativistic, b) time-independent, c) one-dimensional Schrodinger equation:

How can we show that this equation gives the right answer in the case of a free particle?  Can the Schrodinger equation be derived?  Why do we believe that it is true?

	Chapter 40:  Quantum Mechanics

A classical digression involving standing waves on a string.  What are standing waves?

How could we draw the four longest possible wavelengths?  What is the wavelength in each case, and the frequency?

What is the general formula for the allowed frequencies for a standing wave on a string of length L?

Matter waves and electrons: How could we set up a standing matter wave in the case of electrons?

Particle in a box

Consider a pair of rigid walls separated by a distance L, between which electrons are free to move.  We call this a "one-dimensional potential well."  By analogy with a wave on a string, what would be the mathematical expression for the wave function Ψ2(x)?  (assume one end is x=0, and the other x=L)

If we require that a particle must be located somewhere inside the box, what mathematical condition would this impose on the wave function?
"normalization condition":

What would Ψ2(x) look like for the three longest wavelength states?

What would Ψ2(x) look like for the 100th state? (n = 100)

What are the allowed wavelengths for an electron in a one-dimensional well?

Based on deBroglie's hypothesis, what are the allowed energy levels for the electron?

What is the "ground state" (zero point) energy for an electron in a one-dimensional box?

Example: An electron is confined to an atomic size well of width L = 120 pm.  What are the first three energy levels?  Why is the quantization of energy not noticed for objects whose mass is much greater than electrons?

How does the situation for a well or box with finite height walls differ from the case of infinite walls?

"Barrier Tunneling": A surprising consequence  of the wave nature of matter.  Suppose you throw a pebble at your window -- what happens?

Now suppose an electron approaches a barrier with less energy than needed to make it over (or through) the barrier -- what happens in this case?

What does quantum theory predict for an electron to tunnel through a barrier?  How would the probability change if the barrier width were to double?

How come electrons can tunnel through barriers, but pebbles cannot tunnel through windows?

What are three practical applications of quantum tunnelling?

Can tunnelling be observed using photons, as well as electrons?  How?

Chapter 41: Atomic Structure

As already noted, Bohr's model of the hydrogen atom has been superseded by quantum mechanics (~1930).  What is the essential difference between the two approaches?

What is the ground state of the hydrogen atom in the two descriptions?

Hydrogen wave function simulation

What did quantum mechanics predict for the energy levels of the hydrogen atom?  (Note: We only quote results here.  They follow from solving the Schrodinger Equation for hydrogen -- too messy for here.)

In quantum mechanics, allowed states (wave functions) of the hydrogen atom are specified not only by an integer n (as in the Bohr model), but two other integers or “quantum numbers,” l and ml.  

Each of the three quantum numbers is only allowed to assume certain values (physical significance considered later)
							Allowed values
n= ”principal quantum number”
l= ”orbital quantum number”
ml=”magnetic quantum number”

All quantum states with a given value of n are said to form a "shell."  Those corresponding to a given n and l form a "subshell."  How many states are there in the n = 2 shell?  What are they? How do their energies compare in hydrogen?

The spin quantum number: Based on observation of spectra, it was found that the number of quantum states for each n is actually twice what we just said.  What was actually observed?

The fourth quantum number of "electron spin" ms was added to the three others, so as to double the number of quantum states -- two values of ms for each n, l and ml.  Meaning of ms?  Allowed values for three quantum numbers?

What about the spin quantum number is inconsistent with classical physics?

Wave functions for the hydrogen atom:  These are obtained by solving the Schrodinger Equation.  What do the wave functions tell us about the atom?

What is the ground state (n = 1) wave function?

What is the probability of finding an electron a distance r from the nucleus?  Why do we need to include the dr?

What is the most likely distance between the electron and the nucleus?  Why is that result to be expected?

Example:  If the radius of the nucleus of the atom is 10-5 times the Bohr radius, what is the probability of an electron being found inside the nucleus?  (Important result in nuclear physics)

The Quantum Numbers:  Here we consider the physical significance of the quantum numbers.

1. The Orbital Quantum Number, l, specifies the angular momentum of the quantum state.  But, remember no orbiting electron in quantum mechanics.  How is the angular momentum related to l?

2. The Magnetic Orbital Quantum Number, ml specifies the component of angular momentum along the direction of a magnetic field (if any), based on:

What if there is no magnetic field?  Why is the term "space quantization" used here?

What are the possible orientations of the angular momentum vector L with respect to the direction of magnetic field, in general?

Example: How many possible orientations are there for a particular value of L? Find all possible orientations for the l = 2 case.

3. Electron Spin Quantum Number, ms specifies the spin direction of the electron.  The Stern Gerlach Experiment (1921) proved that electron spin was space quantized.  How was the experiment done, and what about the result verified space quantization?

What about the experimental result indicated a value for the spin quantum number of 1/2?

How is the spin angular momentum related to the spin quantum number, s?

What are the allowed values of the z-component of the spin angular momentum for an electron?

What are the two allowed orientations for the angular momentum of a spinning electron with respect to a magnetic field?  Which one has higher energy?

Many electron atoms

The Pauli Exclusion Principle:  Pauli (1925) applied the same set of four quantum numbers seen for the hydrogen atom to atoms with many electrons:  

"No two electrons in an atom can be in the same quantum state, i.e., can have the same set of four quantum numbers."

What would happen in a universe in which this principle were not true?

What determines the order in which the shells and subshells of an atom are filled by electrons?

How many electrons are in the nth shell?  Why are the inert gases (helium, argon, krypton, xenon) so stable?

Notation for electron configuration in an atom: use s,p,d,f for l = 0,1,2,3, numbers for n value and superscripts for numbers of electrons.  What are the quantum numbers and electron configurations for the first four elements?  (Regularities in electron configurations explain the periodic table of elements.)

 X-ray spectra: How are X-rays produced? 

What is a typical spectrum for X-rays?  What is the cause of each component of the spectrum?

How can we estimate the wavelengths of characteristic X-rays emitted by a given element of atomic number Z?