Download Document 8624085

2008 Exam. 2
1. Wave Particle Duality
a. Write the relationship for the kinetic energy and momentum for particle moving at speeds much slower
than the speed of light.
b. Find the wavelength of an electron in an x-ray machine having a kinetic energy 10 keV.
c. Write the relationship for the kinetic energy and momentum for a particle moving at speeds which are on
the order of the speed of light.
d. Write the relationship for the kinetic energy and momentum for a photon.
e. The maximum energy of an x-ray photon produced by a 10 keV electron is 10 keV. Find the wavelength of
such an x-ray photon.
2. Schroedinger’s Equation
A completely free beam of electrons is moving in the +x direction with a kinetic energy of 10 keV.
a. Write the Schroedinger equation for a particle moving in the x direction.
b. Show that the wave function in a. is a solution to the Schroedinger equation.
3. Schroedinger’s Equation
An electron is confined to move freely in a one dimensional box of length L=1.0 nm having infinite potential
walls.
a. Write the space part wave function for the ground state, and draw it in the upper left provided axes.
b. Write the space part probability density and draw it in the lower left provided axes.
c. Draw the wave function and probability density for the same situation but for the case where the height of
the potential walls is finite.
d. Which state, a. or c., has the lower energy. Explain in one sentence.
ψ
ψ
x
P
x
x
P
x
4. In momentum space (k-space) the separation of states is given by Δk x = Δk y = Δkz = π / L .
a. Find the number of states in a volume V=L3 with momentum less than k and kinetic energy less than E.
b. Find the Fermi energy for neutrons in a neutron star having 5 × 10 57 neutrons with radius 10 km.
c. Find the total zero-point kinetic energy of the neutrons at temperature T=0 K.
2007 Exam. 2
1. Relativity A star is emitting light in the positive x direction. The wavelength of the light is 400 nm. a. (5 pt) What is the period Δt in ns of one oscillation of light in the star’s fixed reference frame. Assuming the wave turns on at t=0 b. (5 pt) How far does it go in t =100 ns in the star’s fixed frame? c. (5 pt) Write the 4-­‐vector for the space-­‐time position after a time 100 ns. d. (5 pt) Obtain the space-­‐time invariant interval that the light travels in 100 ns. Suppose the star moves away from the earth in the positive x direction with a velocity 0.8c. e. (5 pt) What is the period Δt ′ in ns of one oscillation of light in the earth’s moving reference frame? f. (5 pt) How far does the light travel after one oscillation as seen by the earth. f. (5 pt) Write the 4-­‐vector for the space-­‐time position after a time t ′ corresponding to one oscillation as seen from the earth’s reference frame. g. (5 pt) Obtain the space-­‐time invariant interval in the earth’s frame that the light travels in 100 ns 2.) Bohr model. According to the Bohr model of the hydrogen atom, an electron in the ground state orbits at a radius of about 0.5 Ao. Suppose the electron is replaced by a muon ( mµc2=105 MeV) to form a muonic atom. a. (10 pt) What is the radius of orbit for the muonic atom in its ground state? b. (10 pt) What are the energies of the ground and first and first excited states? c. (10 pt) What is the wavelength corresponding to the transition between the first excited state and the ground state? 3.)Schroedinger equation. A simple harmonic oscillator (SHO) has a mass m and spring constant K. The potential energy is 1 / 2Kx 2 . a. (10 pt) Write the Schroedinger equation for the space part of the SHO. bx 2
b. (10 pt) The wave function for the ground state has the form Ae . By direct substitution show this is a solution, and thereby finding the constant b in terms of m and K c. (10 pt) Write the probability distribution for the ground state, and carefully graph it. d. (10 pt) Write an integral whichwould be used to obtain the normalizing constant A .You do not need to solve this integral) 4.)Schroedinger Eq. in 3 dimensions. Consider a three dimensional cubic potential well with rigid (infinite) walls, having sides of dimension Lx = Ly = Lz = L=0.1 nm. a. (5 pt) Write the Schroedinger equation for a particle within the well. b. (5 pt) Write the quantum conditions on kx , ky and kz. c. (5 pt) Obtain the quantum condition on the wave number k2. d. (5 pt) Obtain the quantum condition on the allowed energies E. e. (5 pt) Write the ground state solution Ψ(x, y, z) to the Schroedinger equation for a particle within the well. f. (5 pt) Write the probability density for a particle within the well in the ground state. g. (5 pt) Obtain the numerical result on the allowed energies E in units of eV. h. (5 pt) Obtain the number of electrons which can be accommodated at each of the lowest 3 energy levels. Take into account that different combinations of quantum numbers can have the same energy, and that two electrons, corresponding to spin up and down can fit into each combination of spatial quantum numbers. 2006 Exam. 2
1. A baby seal in the pacific ocean has a body temperature of 310 K. If the mean temperature of the water is 287 K at what rate will the seal lose energy by radiating photons? ( σ = 5.7 × 10
−8
-2
-4
W ⋅ m ⋅ K ) 2. Wave particle duality. Compare the wavelength and frequency of a photon and electron, each with kinetic energy 10 KeV. 3. Bohr model. a. Use the Bohr model of the atom to estimate the energy levels of positronium, in which an electron orbits a positron. b. The ionization energy (binding energy) of an electron in hydrogen is 13.6 eV. What is the ionization energy of positronium? 4. Particle in a box. Approximate an atomic nucleus as an infinite cubical box of side L=2 fm, where 1 fm = 10-­‐15 m, in which the nucleons move freely. a. Obtain an expression for the wavelength of the ground, or lowest lying energy state. b. What is the kinetic energy of a neutron in the ground state of this atom. The rest energy of a neutron is mc2=939 MeV. 5. Simple harmonic oscillator. An approximate representation of the interaction between two atoms in a diatomic molecule is a spring like force F=-­Kx with oscillator frequency ω =
K / m . Take the force 3
constant to be 8 × 10 eV/nm2 = 1000 N/m, and the mass of each atom around to be 5 × 10
−27
kg 2
( mc = 4.69 GeV). The wave function for the ground state of a simple harmonic oscillator is ψ 0 (x) = ⎛⎜ mω ⎞⎟
⎝ π ⎠
1/4
− mω x2
2
e
. a. What is the energy of the ground state? b. Find the wave function in momentum space by performing a Fourier transformation. 6. Density of states and Fermi energy. a. Find the average energy of an electron in a white dwarf star of radius 10,000 km 57
containing 2 × 10 nucleons, half of which are protons. The density of states distribution is 3/2
dN
Vm
1/2
=
E . 2 3
dE
2π 
b. From the results in part a, comment on whether it is approporiate to use non-­‐relativistic kinematics. Other problems from previous exams: 1. A free electron has kinetic energy 1000 eV. It moves in the x-­y plane in a direction which makes an angle 30 deg. relative to the x axis. a. Find its momentum p , wavelength λ and wave number k . b. Write the wave function Ψ(x, y, z,t) in symbols (not numerical values) in Cartesian coordinates. c. Write the probability density P(x,y,z). d. What can you say about the uncertainty in the electron’s position. Approximate a nucleus consisting of free nucleons in a spherical rigid wall potential with radius R=4 fm. For the isotope 17O: a. What are the quantum numbers of each of the neutrons and protons? b. What are the energies of each of the neutrons and protons in the isotope 17O? 2. a. Write the wave function for a free particle moving in 3-­‐dimensional Cartesian coordinates. b. The relativistic version of the Schroedinger equation is called the Klein-­‐Gordon equation. Using E 2 = p 2 c 2 + m 2 c 4 , construct the Klein-­‐Gordon equation by expressing the energy and momentum in terms of differential operators. c. Show that the wave function in part a. is a solution to the Klein-­‐Gordon wave function that was constructed in part c. 3.) Consider an electron which moves freely in a 2 dimensional infinite square well of side a. a. Write the Schroedinger equation for this case. b. What are the allowed values of k x and k y c. What are the allowed energy levels? d. If a = 10 Angstroms, what is the lowest energy. e. Write the wave function for this state. 4.) The three primary terms which determine the binding energy of a nucleus are volume, surface and Coulomb, EV, ES, EC energies. a. What is the R and Z dependence of each, where R is the nuclear radius and Z the atomic number. Also indicate the sign of each. i EV ∝
ii ES ∝ iii EC ∝
b. What is the A and Z dependence of each, where A is the number of nucleons. Also indicate the sign of each. i EV / A ∝
ii ES / A ∝ iii EC / A ∝
c. Draw the magnitude of each as a function of A, as well as the sum of each. Be sure to clearly fill in the energy scale in the vertical axis and the number of nucleons in the horizontal axis at the position of the tic marks. 5.) In the blank spaces provided in the table, fill in the properties of the particle shown, as well as the energy scales and quark makeup where appropriate. particl
Charge Rest mass Units of Quark energy e energy Flavor content .93 p +1 GeV uud n
− .139 π
+
π e 0.511 ν γ
W 89 g
6.Draw a graph for the shape of the nucleon-­‐nucleon attractive potential energy, indicating the approximate range and depth. 238
7. a. 92 U captures a neutron, followed by asymmetric fission into 2 unbound neutrons and 92
140
238
38 Sr and 54 Xe . Obtain the difference in the binding energy between the initial 92 U 92
140
and the final 38 Sr and 54 Xe nuclides, and therefore the energy released. 92
b. Calculate the kinetic energy due to the electrostatic repulsion between the 38 Sr and 140
54 Xe when they are still touching, and show that it is the same order as your answer 1/3
with r0 ≈ 1.2 fm. ) in part a. above. (note: r = r0 A
8.Fill in the table below: 8
3 Li
16
8O
56
26 Fe
78
38 Sr
118
46 Pd
212
82 Pd
β− + ν
stable
9. a. The major source of energy production in the sun is the proton-­‐proton cycle. Trace the steps of the p-­p cycle as we discussed in class. b. If the final result is the fusion of 4 protons into 4He, calculate the total energy released in the cycle. 10. Draw a Feynman diagram for each of the following processes, and identify the exchanged quantum: +
a. e-­ +µ e-­ +µ
+
via the electromagnetic interaction. +
b. e-­ +µ e+ +µ-­ via the weak interaction. c. u + u → s + s via the strong interaction. 6. From the information on spin, baryon number and strangeness given in the table below, fill in the quark flavor content and decay interaction of each of the following hadrons. Decay
interact
ion
we
ak