The following is a collection of questions that the instructor has given as either exam questions or 'sample exam' questions in the past. They are collected here for two purposes: (a) to provide you with some idea of the sorts of questions he might ask in future exams, and (b) to provide him with an easy way to incorporate such questions into the homework. 
Table of Contents:
The space and time of relativity
Relativistic mechanics
Quantization of light
Quantization of atomic energy levels
Matter Waves
Schrödinger's equation: 1D
Schrödinger's equation: 3D
Selected Solutions 

1. Two clocks on opposite ends of a spaceship of length 100m (proper length) are synchronised in the rest frame. The spaceship passes an
observer(or vice versa) at 0.9c. We wish to determine the amount by which the two clocks differ in the observer's frame of reference.
a) What is the ship's length in the observer's frame of reference?
b) If light from a bolt of lightning starts at a point half-way between the clocks, how long would the observer say that it would take the light to reach the 'leading" clock?
c) How long would the light take to reach the 'leading" clock?
d) So, in conclusion, the observer would say that the ___________ clock leads the other clock by__________. (Fill in the blanks.)

2. Use the Binomial Approximation to approximate: a) 983, b) 981/2, c) 311/5. No calculators, please.

3. Two astronauts, Muffy and Biff, set off from Earth in identical, lkm long, spaceships on different assignments. The two return to Earth and pass each other from opposite directions. NASA Control in Houston measures each astronaut as traveling at 0.95c. What is the length of Muffy's spaceship as measured by...
a) NASA Control?
b) Biff?

4. The tail of a spaceship traveling at 2.0´ l08m/s passes you 5m sec after its front. What is the proper length of the ship?

1. When a neutron decays, it turns into a proton, an electron, and a massless (as far as we know) particle called an antineutrino. If the neutron began with no kinetic energy, the resulting particles will pick up some energy, because of conversion of mass to energy. The antineutrino being massless will pick up most at the energy. For argument's sake, let's say all of it.
a) Calculate the final energy of the antineutrino.
b) Calculate its momentum (Do NOT use p= g mu!!).
c)The electron and proton must counteract this momentum. If electron and proton have the same speed, and move in the opposite direction from the antineutrino, what is their speed? (Hint: it will be highly nonrelativistic.)

2. The Milky Way galaxy is about 105 light years across. The most energetic particles in it have energies around 1020eV. Let us assume that a particle with the same mass as an electron has E=1020eV. What is its time-dilation factor, g ? What is the shortest possible half life that this particle can have if it expects to cross the galaxy?

3. The electron in a hydrogen atom is bound by 13.6eV to a proton. By what percentage is the mass of the two separate particles decreased when they join together? How fast would the electron have to travel in order to relativistically make up for that mass difference?

4. A pion is created in the laboratory with a momentum of 268MeV/c and an energy of 300MeV.
a) What is its rest mass?
b) What is g?
c) What is the speed of the pion, as a fraction of c?

5. Calculate the lowest-order correction to the classical relation K =1/2mv2.

6. A proton ('uud', composed of two up quarks and one down quark) has a mass 938.28MeV/c2 and a neutron ('udd') has a mass of 939.57MeV. What should be the mass of the 'u' and the 'd' quark (if there were no force between them). Explain briefly why this answer is no good.

7. Two weightlifters, Hans and Franz, are each lifting one-ton (m=1000kg) weights aboard their separate spacecrafts. Hans says that Franz is lifting a three-ton weight and someone on the ground watching the two says that Franz is lifting a two-ton weight. What does the Earth-bound spectator measure for the weight that Hans is lifting?

General relativity
8. For the relativistic red shift, we showed that f' = [(1+v/c)/(1-v/c)]1/2 fo, where f' is the frequency of the radiation emitted by an object as observed by someone moving with a speed, v=?c, towards the source of the electromagnetic wave.
a) Use this expression to obtain an expression for the observed wavelength, ?, in terms of the wavelength emitted, l o.
b) How fast must an automobile be going in order to use the Doppler shift as an excuse for running through a red light? (Use
l =650nm for red light, l =525nm for green.)
c) What is the shift in wavelength reported by someone traveling at 55mph=24m/s?

1.In a Thomson q/m experiment, a physicist supplies a voltage of 2000V across a pair of parallel plates 10.0cm long separated by a 2.00cm gap. The particles are deflected by an angle of 0.20 radians.
a) Calculate the speed of the particles, given that a magnetic field of 0.045T cancels the deflection due to the electrostatic plates.
b) Calculate the charge--to--mass ratio of the ions in this experiment.

2. Consider an oil droplet from one of Millikan's early experiments:
radius: a=0.000276cm
density: r=0.9561g/cm3
rise dist.: d=10.21mm
rise time: t=80.708see
a) Calculate the mass and momentum of this droplet.
b) If the position of the drop is precise to the last significant digit in the radius quoted above, calculate the relative uncertainty in the momentum.

3. Alpha particles are being fired at a thin copper foil. An alpha-particle detector is rotated at a constant distance of 10m from the foil, and alpha particles are counted at various angles for ls. Since there are many more particles arriving at the detector for small angles than for larger angles, one needs to compensate by bringing the detector in closer to the foil or by taking a longer measurement. In order to get the same number of counts at 10° and at 90° ,
a) How much more time must one collect data at 90° ?
b) How close must the detector be at 90° if it is 10m away for 10° ? You can assume that the window of the detector is infinitely narrow.

1. An astronomer on Glorp, a tiny planet in a galaxy fax far away, happens to tune into WSLU, Canton, NY, at about 90MHz on the FM dial. Not realizing that this is an isolated FM signal, and imagining that this is the peak radiation coming from a stellar object, what will the Glorpian calculate for the temperature of Terra?

2. A hypothetical spaceship, looking at the moons of planets in our Solar System, tries to measure the surface temperature of one of those moons by the following method: It measures the radiated energy density (proportional to the energy density, u(f,T)) at two different frequencies, f=l012Hz and f'=2´ l012Hz. From the ratio of these two values, determine the temperature. Since the frequencies are rather small for all but the lowest of temperatures, you should take the appropriate limit of Planck's, law for low frequency. The ratio is equal to 2.00.
Planck: u(f,T)=8p hf3/c3[exp(hf/kT)-1]-1
h= 6.626´ l034J- s
k=1.38´ 10-22J/K

3. Assume that all the planets absorb 100% of the sunlight falling on them.
a) Calculate symbolically the fraction of the Sun's radiant energy, E, that a planet of radius r, located a distance R from the Sun receives.
b) Equating the total absorbed energy to the total emitted energy, estimate how the planet's average temperature varies with its distance from the Sun, and its radius.
c) Pluto is about 24% the size (diameter) of Earth, and is about 39 times further away. Estimate its surface temperature.

4. How much power is needed to keep a 900° C furnace in equilibrium with its surroundings. (23° C) if the furnace is perfectly insulated but has a peephole in its door that is l cm ´ lcm square?
(s =5.67x10-8W/m2/K4 )

5. The Sun has a radius of 700Mm, and a surface temperature of about 5500K. What is its power output? Why is it the surface and not the interior temperature that I asked for?

6. Light of wavelength 400nm falls on a surface and produces photoelectrons. A stopping potential of 0.50V is needed to stop them. What is the work function of the metal? What would be the stopping potential if the wavelength were 200nm?

7. A metal has a work function of 2.03eV. What is the minimum frequency needed to excite photoelectrons from the metal? What is the maximum wavelength? What will be the maximum kinetic energy of the photoelectrons for l=200nm?

Bragg scattering
8. In trying to do optics with X-rays, scientists run into the problem that there is no such thing as an X-ray lens. However, they make do by bending X-rays with the aid of "glancing" (low-angle) reflections off of metals. If we have 100keV X-rays bouncing off of a gold film (lattice constant of 2.02A, what is the lowest angle for which we get reflection? (Bragg's law: nl=2dsinq)

Bohr's Atom
1. It takes 79eV to ionize helium, that is, to remove both electrons. Use the Bohr model to approximate this known experimental value. Start by calculating the binding energy of one electron to the He++ ion and then, treating the nucleus plus the first electron as if it were a nucleus of charge Z=1, calculate the binding energy of the second electron.

2. A hydrogen discharge tube contains atoms in a large number of energy states, En. Now consider the highly-excited states, for which n>>l.
a) Calculate the amount of energy needed to excite an atom in state n to state n+l (Keep only the highest orders in "n" in numerator and denominator).
b) If the tube has a radius of lmm, what is the largest n could be for an electron to circle its hydrogen nucleus without colliding into the walls of the tube?

3. The pair of spectral lines in the sodium spectrum ("the sodium doublet") results from transitions from two energy levels that are very close in energy, resulting in lines at 588.995nm and 589.592nm. Transitions between the two states are forbidden, but if they were not, what would be the wavelength of a photon corresponding to such an electronic transition?

4. The Milky Way galaxy is about 105 light years across. What is the quantum number, n, that an electron belonging to a hydrogen atom would have if the nucleus were at the center of the galaxy and the electron were somewhere on the edge?

5. The threshold energy for creating matter-antimatter pairs is 1.022MeV, twice the rest mass of an electron. Conceivably, that much energy might be created by ejecting the most tightly-bound electron (n=l) from a heavy atom, like uranium (Z=92). Assuming that we can still use the hydrogenic atom model to get a rough estimate of the binding energies of the innermost electrons, calculate what Z would have to be in order for the binding energy of the atom would be greater than 1.022MeV.

1. Use DeBroglie's argument that the wavelength of the ground-state-hydrogen electron can be wrapped around the circumference of a circle of radius ao (the Bohr radius) to determine the speed of the electron. (Avoid MKS units if possible.)

2. Professor deBroglie strolls into class (at lm/s) through a lm doorway. He is 0.2m wide and has a mass of 80kg.
a) What is his wavelength?
b) By how much will his wave packet have spread when he reaches the far end of the room, 3m from the doorway?

Wave packets
3. If the phase velocity of a wave is of the form vp=akn, that is, a plane wave of a single wavenumber, k, travels with a speed vp, then
a) what is the angular frequency, w, as a function of k?
b) what is the ratio of the group velocity, vg to vp?
c) under what circumstances is vg>vp?

4. Professor X applies for NSF (National Science Foundation) money to measure the relation between position and speed for an object approximately 10-6m big and 10-6kg heavy. This object is vibrating at the end of a thin fiber with a maximum displacement of 10-5m at a frequency of 1000Hz, so that its speed is about 10-2m/s. Should she be denied funds because of the quantum 'fuzziness' of the problem?

5. A po meson, which plays an important role in nuclear physics, has a mass [according to the text] of 135.0MeV/c2. It has a lifetime of 0.83´ l0-16sec. What is the inherent uncertainty of the mass?

6. A blink of the eye lasts (if I remember correctly) 1/40 of a second. How precisely can one measure the rest mess of a particle that is still there after you have blinked? (Please DO NOT answer in MKS.)

7. Use Heisenberg's Uncertainty Principle to determine an lower limit for the lifetime of the first excited state of hydrogen.

8. If a runner can cover a mile (1620m) in four minutes, what is her
a) average speed (in MKS)?
b) wavelength. if m=50kg?
c) minimum uncertainty in speed if the uncertainty in her position is equal to her wavelength?

9. A particle traveling at 14% of the speed of light experiences a length contraction or time dilation of only l%. We can consider this particle to be "marginally relativistic".
a)What is the momentum of an electron traveling at 0.14c?
b) What is its wavelength?
c) What is the minimum percent uncertainty in momentum, Dp/p, if Dx=l

10. We showed in class that the kinetic energy of the electron in the ground state of the hydrogen atom was equal to the absolute value of the total ground state energy.
a) Use this fact to solve for the momentum of this electron.
b) An electron circling a nucleus may have a fixed magnitude of momentum, but the direction is undecided. Set the uncertainty, Dpx, equal to the momentum in part (a) and solve for Dx.
c) Compare the answer in part (b) to the Bohr radius.

11. Two deuterons (a neutron and a proton stuck together) can fuse to create helium and a lot of excess energy if they come close enough together. Since a deuterium atom (a deuteron with an electron circling it) is about the same size as a normal hydrogen atom, calculate the uncertainty in the momentum of one deuteron that would be sufficient to place it next to the other deuteron. Let p=Dp, and calculate the kinetic energy of this deuteron. Since the kinetic energy equals (3/2)kT, which equals (1/40)eV at room temperature (300K), calculate the temperature at which you would expect deuterons to fuse.

12. Scientists can create light pulses of less than 100 femtosecond duration.
a) How long is such a pulse, distance-wise?
b) If the light is approximately 500nm wavelength, how many wavelengths fit in this pulse?
c) What is the relative uncertainty of the wavelength?

13. The mass of the Universe is believed to be about 1053kg, give or take a factor of 100. How long a lifetime would a particle have whose mass were uncertain by this mount? (Another way of looking at the problem is that particles as massive as the Universe can zip in and out of existence, but only for time periods smaller than the one you will have calculated.)

Simple Harmonic Oscillator
1. For the Simple Harmonic Oscillator, Eo = 1/2(hbar) w. Treat an electron circling a hydrogen nucleus as though it were a SHO with Eo = 13.6eV.
a) What is w?
b) Classically, E = kA2/2 = mw2A2/2, where A is the classical amplitude. Solve for A for this problem.

2. A simple harmonic oscillator exists for which Eo = 1J. What is w? What is E29?

3. The energies with which a carbon monoxide molecule can vibrate can be calculated by treating it as a simple harmonic oscillator with a spring constant of 187N/m. Calculate the energy of the three lowest energy states. (Use a mass of 6.9au or 6.9 proton masses for m.)

4. A classical pendulum that. consists of a 10g bob at the end of a 1m string is released from the point at which it is 1cm higher than it is at its lowest position. What is the total mechanical energy of the pendulum? If w=2p(l/g)1/2 then calculate the angular frequency of the pendulum. What is the quantum number of this system?

Particle in a box
5. Consider an electron in an infinite square well for which E1=1J.
a) If E1=h2/8mL2, find L, the width of the well.
b) What is E29?

6. For an infinite square well, E1= p2(hbar)2/2mL2.
If the electron in a hydrogen atom were confined to a square well of L=1A, what would the lowest energy level be?
What would be the energy of a photon emitted when an electron in the next higher energy state decayed into the ground state?

7. Consider the particle-in-a-box that I brought into class when we started Chapter 5 -- a metal ball (let's say m=l0g) inside a l0cm ´ l0cm box. Let's approximate this by a one-dimensional infinite box of l0cm size. If the velocity of the ball is lnm/yr, calculate
a) the energy of the "particle"
b) the quantum number, n, of this energy state.

8. An ¡ (upsilon) particle consists of a combination of a bottom and an anti-bottom quark (not to be confused with a top quark). Each of these particles has a mass of 4.3GeV/c2 and the energy between the lowest and next lowest energy states is 0.558GeV. Treating the system as an infinite square well in which one particle is trapped in the well created by the other, estimate the size of an ¡ particle (the maximum distance between the quarks).

9. If y (x) = cos2x is a solution to Schrödinger's equation, show that, if E = 2(hbar)2/m, then U = (hbar)2/my .

10. Show that Co exp(-ax2) is a solution to Schrodinger's equation. Express a as a function of m, E, and (hbar), if U µ x2.

11. For = y (x.)=exp(-ax4), show that, U(x)-E = Ax6+Bx2.Find A and B.

12. Consider the following wavefunction: y (x)=A(L-|x|)1/2, for -L<x<L, and y (x)=0 otherwise. Find the constant A. Calculate <x>, <x2> and Dx.

13. A wave function has the following form: y (x)=A(1-Bx2), for -L<x<L, and y (x)=0 otherwise.
Solve for the potential function U(x) for |x|<L. Your solution may contain the energy, E, but not A or B.

1. Calculate the gradient of U for
a) U=cos(kx)sin(ly)e-bz
b) U=cos(pxy2z)
c) U=exp(-ar2). where r2=x2+y2+z2
2. Calculate the Laplacian of y for
a) y =tan(kx)sin(ly)cos(mz)
b) y =cosh(kx)sinh(my ); where coshx=(ex+e-x)/2, sinhx=(ex-e-x)/2
c) y =exp(-ar2), where r2=x2+y2+z2

Particle in a box
3. The lowest energy state of a particle in a 3-dimensional box is (3/8)eV. What are the possible energy states lower than 8eV? Which of these states are non-degenerate?

4. The lowest energy state of a particle in a 3-dimensional box is leV. What are the possible energy states lower than 6.5eV? What is the degeneracy of each?

5. We've discussed the most probable radii of the hydrogen atoms.
a) For a particle in a 3-dimensional box, describe the most probable regions for a particle in the box. Consider a box for which the ground-state energy is 3 eV
b) What are the quantum numbers for each of the 6eV degenerate states?
c) For each of these states, define each of the most probable "regions".

Hydrogenic atoms
6. Given the periodic table on the attached sheet, and assuming that the energy of any electron state for any atom is determined by n and l alone, and does not depend on Z, sketch the energy level diagram for the elements. (That is, rank the (n,l) subshells in order of increasing energy. You will, of course, need to know the degeneracies of the subshells to figure out which one is partly filled at each element.)

7. One can use an 'effective field' model to account for the screening of part of the nuclear charge (as seen by the outermost electron) by the inner electrons in complex atoms. In this model,
where Zeff is the effective charge. Now consider the alkali metals, for which one lone electron is in the outermost n-level, typically having a much larger average radius than the other electronic orbits. Given n and the ionization energy, you can calculate Zeff for each atom. Fill in the following chart, estimating Zeff. for francium from the Zeff's that precede it (The Handbook of Chemistry and Physics gives '4eV' for its ionization energy.):

Element Ionization Energy Z nmax Zeff
H 13.595eV 1 1  
Li 5.39 3 2  
Na 5.138 11 3  
K 4.339 19 4  
Rb 4.176 37 5  
Cs 3.893 55 6  
Fr   87 7  

8. a) Show that for any 3-dimensional system, the most probable radius of a particle is equivalent to the solution to the following:
The partial derivative of |g(r)| with respect to r equals zero.
b) For the hydrogen atom, for n=3 and l=1, R(r)=(1/3ao)3/2 4Ö 2/3 r/ao [1-r/6ao] exp(-r/3ao).
Calculate the most probable radius. (The Bohr prediction is <r>=9ao.)

9. Imagine a particle which has spin = 2.
a) What are the possible values for the magnitude of the spin angular momentum component parallel to some magnetic field? (Warning: get those units right.)
b) What are the possible angles between the spin vector and the magnetic field?
c) If the magnitude of m is 10-22A .m2 and B is 2T. then what are the possible values of the interaction energy? (You may want to use the results of part b.)

10. Given charts of Y(q ,j) and R(r),
a) Determine y (r, q ,j) for the 3p electronic state with the largest m, for hydrogen,
b) Find. the most probable radii, remembering that minimizing P(r) is equivalent to minimizing its square root.
c) There are three least probable radii, where P(r)=0. One is r=¥ , where are the others?

11. One can use the quantum defect model mentioned in Chapter 8 (although we didn't talk about it in class) to model the electronic energy levels in a complex, non-hydrogenic atom. In this model, the energy is given as

where n and l are the quantum numbers and D(l) is a number that depends on l, but not on n. Given that the sodium yellow lines are approximately 589nm, corresponding to the 3p to 3s transition, and given that it takes 5.138eV to ionize the ground state of the outermost electron of sodium (the 3s state), calculate D(0) and D(1) for sodium, and estimate the amount of energy needed to ionize the 2s electron.

12. Given that the pseudowavefunction g(r) is equal to g(r)= r2 exp(-ar), where a= 1/2ao, and given that l=1, calculate the potential energy function and the energy in terms of a, given that the limit of U(r) is 0 as r approaches ¥ .

13. For the hydrogen atom, for n=1, the function R(r) is proportional to exp(-ar), where
a=1/2ao= mke2/(hbar)2

a) Given that r=[x2+y2+z2]1/2 show that the partial derivative of r with respect to x is x/r.
b) Give that the second partial derivative of R(r) with respect to x is -a[1/r-x2/r3-ax2/r2]exp(-ar), show that the Laplacian of R is (a2-2a/r)exp(-ar)
c) Using the Cartesian-coordinate form of Schrödinger's Equation in three dimensions, show that the appropriate wavefunction y (x,y,z) is a proper wavefunction for U(r)= - ke2/r.
You may set. Y1,0(q ,j )=1.
(For extra credit. you may show the result in part (b) that we took as given. )

14. Let us imagine that we want to do NMR on the cheap. To save on materials, we will use water, for which the magnetic moment of the proton is m = 2.79mn. To save money on electromagnetic radiation, let's use a 60Hz signal, easily obtained from Niagara Mohawk. What is the size of the magnetic field we would need to induce resonance absorption by the protons? (Note: the Earth's magnetic field is, on average, about 1/2G, and this 60Hz effect has been suggested as the cause of the alleged physiological effects of a.c. electromagnetic radiation.)

15. The Legendre polynomial of l=3 and m=1 is given by Y3,1= -(1/8)[21/p]1/2 sinq (5cos2q-1)eij .
a) Show that the most probable angles at whch a particle may be found occur when the partial of Y with respect to theta is 0.
b) Solve for the most probable angles, theta. (Hint: convert all cosines into sines first.)

1. Consider two step barriers, one with a height of Uo and a width of w, the other with a height of Uo/2 and a width of 2w. For an incident particle with energy E<<Uo/2, which barrier will provide a better probability of successful tunneling? Why? You may assume that this is a 'wide' barrier, such that T<<l.

2. Consider three metal cylinders aligned on the same axis. The middle cylinder, 10mm long, is held at -10V and the others are held at 0V. An electron along the axis thus sees a potential energy barrier of 10eV, 10mm long. Calculate the tunneling probability for the electron.

3. "The little train that could":
A little (15000kg) train almost makes it to the top of a hill, but ends up lm vertical distance from the top. If it could tunnel (quantum-mechanically, of course) 20m through the hill, it could continue on its way. What. is the probability of the train making it through on its first try?