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) 98^{3}, b) 98^{1/2}, c) 31^{1/5}. No calculators, please.

a) NASA Control?

b) Biff? 4. The tail of a spaceship traveling at 2.0´ l0

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 10^{5} light years across. The most energetic particles in it have energies around 10^{20}eV. Let us assume that a particle with the same mass as an electron has E=10^{20}eV. 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?

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 =1/2mv

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?

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.

radius: a=0.000276cm

density: r=0.9561g/cm

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.

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.

Planck: u(f,T)=8p hf

h= 6.626´ l0

k=1.38´ 10

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

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 10

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?

a) what is the angular frequency, w, as a function of k?

b) what is the ratio of the group velocity, v

c) under what circumstances is v

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, Dp

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 10

a) What is w?

b) Classically, E = kA

a) If E

b) What is E

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/c

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(

b) U=cos(pxy

c) U=exp(-ar

2. Calculate the Laplacian of y for

a) y =tan(kx)sin(ly)cos(mz)

b) y =cosh(kx)sinh(my ); where coshx=(e

c) y =exp(-ar

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".

E

where Z

Element | Ionization Energy | Z |
n_{max} |
Z_{eff} |

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)=(1/3a_{o})^{3/2} 4Ö
2/3 r/a_{o} [1-r/6a_{o}] exp(-r/3a_{o}).

Calculate the most probable radius. (The Bohr prediction is <r>=9a_{o}.)

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

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

E

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.

a=1/2a

a) Given that r=[x^{2}+y^{2}+z^{2}]^{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-x^{2}/r^{3}-ax^{2}/r^{2}]exp(-ar), show that the Laplacian of R is (a^{2}-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)= - ke^{2}/r.

You may set. Y_{1,0}(q
,j
)=1.

(For extra credit. you may show the result in part (b) that we took as given. )

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?