Danny

Consider an atomic gas in which all atoms emit monochromatic light at the same wavelength lambda_0. However, because of the Doppler effect, a stationary observer detects the light emitted by a particular atom at some time as being slightly shifted in wavelength. Since the instantaneous velocities of the atoms are randomly distributed, a stationary observer detects the light emitted by the gas as being broadened in wavelength.

(a) What is the line shape function for this situation? Ignore relativistic effects.

(b) Find an expression for the fwhm.

(c) Evaluate the fwhm for a laser for with lambda_0 = 633 nm at room temperature. The light is emitted by the neon atoms.

To solve this problem you require the velocity distribution of atoms in a gas. This Maxwell distribution can be found in almost any text books on statistical physics.

An exercise related to particle kinematics. The notes on relativistic kinematics may be

useful for this exercise. A neutral meson Q decays via the reaction

 Q --> K+ plus K-

(a) Work out the momentum (in units of MeV/c) that each kaon would carry off if the meson decayed in its rest frame.

(b) Suppose that the kaons are emitted parallel to the y-axes in the Q rest frame. If the meson is travelling in the z-direction in the laboratory frame with energy 2 GeV, calculate the angle between the flight directions of the two kaons in this frame.

(a) How much energy is contained in the electromagnetic radiation lling a cubic meter of space

at the sun's surface?

1. A water molecule can vibrate in various ways, but the easiest type of vibration to excite is the "flexing" mode in which the hydrogen atoms move toward and away from each other but the HO

bonds do not stretch. The oscillations of this mode are approximately harmonic, with a frequency

of 4.8 x 10^13 Hz. As for any quantum harmonic oscillator, the energy levels are (1/2)hf, (3/2)hf, (5/2)hf, and so on. None of these levels are degenerate.

(a) Use the Boltzmann distribution to calculate the probability of a water molecule being in its flexing ground state and in each of the first two excited states, assuming that it is in equilibrium with a reservoir (say, the atmosphere) at 300 K. (Hint: You can estimate Z by adding up the first few Boltzmann factors until the rest are negligible, or do it numerically on a computer.)

(b) Repeat the calculation for a water molecule in equilibrium with a reservoir at 700 K (perhaps

in a steam turbine).

For plane waves the wavenumber k and the frequency ω are related by

k=nω⁄c

(a) The group velocity v_g of a wave is defined as v_g = dω/dk. In optics one often deals with the inverse group velocity k(1), defined as k(1) = dk/dω. If we define the group index n_g through k(1) = n_g/c, then show that n_g = n + ω (dn/dω).

(b) Consider a transparent material. Do you think n_g > n or n_g < n? Give reasoning.

Intelligent beings on a distant planet try to communicate with the Earth by sending

powerful radio waves swept in frequency from 10 to 50 MHz every minute. The linearly

polarized emissions must pass through a radiation belt plasma in such a way that E and

k are perpendicular to B0. It is found that during solar flares (on their sun), frequencies

between 24.25 and 28 MHz do not get through their radiation belt. From this deduce the

plasma density and magnetic field there. (Hint: Do not round off numbers too early.)

(a) The hydrogen nucleus has a spin 1/2 but the deuterium nucleus has spin 1. Why is the hydrogen atom a suitable candidate for Bose-Einstein condensation, but the deuterium atom is not? Justify your answer.

(b) Why is there only a singlet state corresponding to the ground state electron

conguration of HELIUM (1s2)? Justify your answer.

Plasmas used to produce nuclear fusion are heated to a temperature of approximately

100 million degrees Celsius. The vessel that contains this plasma, such as a Tokamak, is

made from steel. In less than half a page give reasons (qualitative and/or numerical) as

to why these vessels do not melt. Note the plasma in these vessels can (and often does)

make contact with the walls of the vessel.

For an electron in a 3p state

(a) What is the value of the quantum number l?

(b) What is the magnitude of orbital angular momentum |L|? Justify your answer.

(c) What are the possible values for Lz?

(d) Draw a vector model diagram for this electron state with appropriate labels to

indicate scale and units.

Consider a diatomic molecule with a moment of inertia about its centre of mass of I

and a natural vibration frequency of v0.

(a) Write down expressions for the:

(i) rotational energy, and

(ii) vibrational energy

of the molecule, indicating the possible values for relevant quantum numbers.

Sulfur has a density of 2.1×10^3 kg.m^-3, is number 16 in the periodic table and has an atomic mass

of 32. Sulfur has a dielectric constant of 4. The internal electric field in a piece of sulfur is 10^7 V.m^-1. Assuming all electrons to be displaced by the same distance d with respect to the nucleus, find d. How does d compare to the size of an atom? Comment on this result. Given:

e = 1.60×10^-19 C

Avagadro's no. = 6.02×10^23 particles.mol^-1

ε0 = 8.85×10^-12 C^2.N.m^-2

(1) Consider a single sulfur atom. Draw a diagram showing the electron distribution.

(2) The internal electric field in a piece of sulfur is 10^7 V.m^-1. Draw a diagram showing the

electron distribution and the electric field.

(3) On the diagram show the electric dipole moment of a sulfur atom (vector), the charge that

has been separated and the separation distance d and the equation for the electric dipole

moment of a single sulfur atom.

(4) What does the internal electric field do to the piece of sulfur?

(5) Determine the electric susceptibility of the sulfur.

(6) Determine the number density of the sulfur.

(7) Determine the electric dipole moment of a sulfur atom.

(8) Determine the charge separation distance d.

(9) Comment on the magnitude of d.

(10) You can also approach the problem from an algebraic point of view.

Express the separation distance d in terms of the following quantities:

εr, εo, E, M (molar mass), Z (atomic number), ρ, NA

where NA is Avagadros number.

A parallel plate capacitor has an area A = 20.0 cm^2 and a plate separation of d = 4.00 mm.

(a) Find the capacitance in air and the maximum voltage and charge the capacitor can hold.

(b) A Teflon sheet is slid between the plates, filling the entire volume, Find the new capacitance

and maximum charge.

(c) Before the insertion of the Teflon, the plates are connected to a 24.0 V battery and then

disconnected. What are the energies in the capacitor before and after the Teflon is inserted?

(d) Was work done in inserting the Teflon?

Suppose now, that the Teflon sheet that is inserted between the plates is only 2 mm thick and

fills only half the volume. Before the Teflon is inserted, the disconnected capacitor carries a

charge of 1.00 nC.

(e) Find the electric field everywhere.

(f) Find the new capacitance.

Given:

Dielectric strength (air) = 3.00×10^6 V.m^-1

Dielectric strength (Teflon) = 6.00×10^7 V.m^-1

Dielectric constant (Teflon) = 2.1

When I first push the power button to my pc, the monitor reads "No signal detected". However about 40 seconds later, when windows has started, the Monitor starts working, reading "DVI blabla" or whatever it says. I want to access my bios settings but my screen is black throughout the entire boot! My RAM is installed fine. Does anyone know what might be the issue? I have updated my video card and monitor drivers as well. 10 points to best answer as soon as this is fixed!

An electron is trapped in an infinitely deep potential well of width 0.500 nm.

1. If the electron is in its ground state, draw a careful diagram to show the probability function for the electron. (Would be great if I could just get a general description of what this function would look like in terms of shape and maybe a few values to plot).

2. Explain clearly, with aid of your diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.125. Note: Your method of estimation does not need to be accurate, but you must show clearly how you arrive at the final value.

3. Consider now that the electron is in the 99th energy state above the ground state. Explain clearly again, with the aid of a new diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.25. Again you must show clearly how you arrive at your value.

A proton and deutron (a particle of same charge as proton, but twice the mass) attempt to penetrate a rectangular potential barrier of height 10 MeV and thickness 10^(-14) m. Both particles have total energies of 3 MeV.

1. Use qualitative arguments to predict which particle has the highest probability of succeeding.

2. Estimate quantitatively the probability of success for both particles.

A rough criterion for the applicability of ordinary Newtonian mechanics is that the de Broglie wavelength delta, be much less than some characteristic linear dimension 'L' of the system. For L <= delta, wave mechanics is needed. Apply these criteria to the following systems.

1. The valence electrons in a large organic molecule. Take the length of the molecule to be 20 nm, and the typical ground energy to be a few eV. Do you need to take quantum mechanics into account when describing such a particle?

2. Electrons in a TV set. The electrons in a typical television receiver have kinetic energies of the order of 10 keV. Do the designers of the image-forming systems in television sets have to use quantum physics in their calculations?