Download Modern Physics – Fall 2016 Prof. Akhavan Sharif University of

Modern Physics – Fall 2016
Prof. Akhavan
Sharif University of Technology
Physics Department
Problems Set #6.
Due Date: 05th of November / 15th of Aban.
1 The Bohr Model.
(a) Show that the speed of an electron in the n th Bohr orbit of hydrogen is
αc
,where α is the fine
n
structure constant, equal to
(b) What would be the speed in a hydrogen-like atom with a nuclear charge of Ze?
(c) Calculate the energies and speeds of electrons in circular Bohr orbits in a hydrogen-like
atom using the relativistic expressions for kinetic energy and momentum.
(d) The electron-positron pair that was discussed in can form a hydrogen-like system called
positronium. Calculate the energies of the three lowest states and the wavelength of the
Lyman lines(Detection of those lines is a “signature” of positronium formation.).
2 The Bohr Model.
Consider the electron-nucleus system. The electron of mass m and the nucleus of mass M must
both move so that the center of mass(CM) remains fixed in space. Carry through the details of the
Bohr calculation for a nucleus of finite mass and:
(a) Obtain the following equation:
R=
mM
e4
1
1
1
, = R M Z2 ( 2 − 2 )
3
m+ M 4 π cħ λ
nf
ni
(b) Make a graph of Rydberg constant as a function of M/m. What is the maximum amount of
it? Speak over the limits.
(c) Calculate the speed of the proton in a grand state hydrogen atom.
3 The Bohr Model.
An alternative development of the Bohr theory begins by assuming that the stationary states are
those for which the circumference of the orbit is an integral number of de Broglie wavelengths.
(a) Show that this condition leads to standing de Broglie waves around the orbit, and gives the
angular momentum condition, Eq. 6.26, used in the Bohr theory (Required text book is
Modern Physics, Krane, 3rd edition.).
In the ground state of the hydrogen atom, according to Bohr's model, what are:
(b) the quantum number, (c) the orbit radius, (d) the angular momentum, (e) the linear
momentum, (f) the angular velocity, (g) the linear speed, (h) the force on the electron,
(i) the acceleration of the electron, (j) the kinetic energy, (k) the potential energy, (l) and
the total energy? How does the quantities (c) to (l) vary with the quantum number?
4 Line Spectra.
(a) The shortest wavelength of the hydrogen Lyman series is 91.13 nm. Find the three longest
wavelengths in this series.
(b) Use the Bohr theory to find the series wavelength limits of the Lyman and Paschen series
of hydrogen.
(c) The following wavelengths are found among the many radiations emitted by singly ionized
helium: 24.30 nm, 25.63 nm, 102.5 nm, 320.4 nm. If we group the transitions in helium as
we did in hydrogen by identifying the final state n0 and initial state n, to which series does
each transition belong?
5 Line Spectra.
Three isotopes of hydrogen occur in nature; ordinary hydrogen, deuterium, and tritium. Their
nuclei consist of, respectively, 1 proton, 1 proton and 1 neutron(deuteron), and 1 proton and 2
neutrons(triton)(Required text book is Modern Physics, Krane, 3rd edition.).
(a) Use following Equation to determine Rydberg constants for deuterium and tritium.
R=
μ k 2 e4
m k2 e4
1
1
=
(
) = R∞ (
)
3
3
1+ m/ M
4 πcħ
4 πcħ 1+m/ M
(b) Determine the wavelength difference between the Balmer a lines of deuterium and tritium
(c) Determine the wavelength difference between the Balmer a lines of hydrogen and tritium.
(d) Show that a small change in the reduced mass of the electron produces a small change in a
spectral line given by
Δλ Δμ
. Use this to calculate the difference Δλ in the Balmer red
=
λ
μ
line λ=656.3 nm between hydrogen and deuterium, which has a nucleus with twice the mass
of hydrogen.
6 The Bohr Model.
(a) The current i due to a charge q moving in a circle with frequency frev is q frev. Find the
current due to the electron in the first Bohr orbit.
(b) The magnetic moment of a current loop is iA, where A is the area of the loop. Find the
magnetic moment of the electron in the first Bohr orbit in units A-m2. This magnetic
moment is called a Bohr magneton.
(c) Derive a relation connecting the frequency of the electromagnetic radiation, emitted in a
transition between two states of a Bohr atom, and the orbital frequencies of the electron in
these states. Study this relation in the limit of large quantum numbers, and comment on its
correspondence with the predictions of classical physics.
7 The Bohr Model.
(a) Assume the angular momentum of the earth of mass 6.0 E24 kg due to its motion around
the sun at radius 1.5 E11 to be quantized according to Bohr's relation L = nh/2π.
What is the value of the quantum number n? Could such quantization be detected?
(b) If electric charge did not exist and electrons were bound to protons by the gravitational
force to form hydrogen, derive the corresponding expressions for a0 and En, and compute
the energy and frequency of the Hα line(which is the second line of the Balmer series
predicted by Bohr’s model. The Hα line is emitted in the transition from ni=4 to nf=2.) and
the limit of the Balmer series.
Compare these with the corresponding quantities for “real” hydrogen.
8 Questions.
(a) For the Bohr hydrogen atom orbits, the potential energy is negative and greater in
magnitude than the kinetic energy. What does this imply?
(b) If only lines in the absorption spectrum of the hydrogen need to be calculated, how would
you modify Eq. 6.33 to obtain them?
(c) On emitting a photon, the hydrogen atom recoils to conserve momentum. Explain the fact
that the energy of the emitted photon is less than the energy difference between the energy
levels involved in the emission process.
(e) What examples of degeneracy in classical physics, other than planetary motion, can you
think of?
(f) According to classical mechanism, an electron moving in an atom should be able to do so
with any angular momentum whatever. According to Bohr's theory of the hydrogen atom;
however, the angular momentum is quantized to L = nh/2π. Can the correspondence
principle reconcile these two statements?
9 General Problem.
When an atom emits a photon in a transition from a state of energy E1 to a state of energy E2,
the photon energy is not precisely equal to E1 – E2. Conservation of momentum requires that the
atom must recoil, and so some energy must go into recoil kinetic energy Krecoil.
Show that Krecoil ≃ (E1−E2)2 / 2Mc2 where M is the mass of the atom. Evaluate this recoil energy
for the n = 2 to n = 1 transition of hydrogen. Compare the wavelength of the light when the recoil
is taken into account to the wavelength without accounting for recoil.
10 General Problem.
In a muonic atom, the electron is replaced by a negatively charged particle called the muon.
The muon mass is 207 times the electron mass.
(a) What is the radius of the first Bohr orbit of a muonic lead(Z = 82) atom?
(b) Calculate the magnitude of the lowest energy state for this atom.
(c) Ignoring the correction for finite nuclear mass, what is the shortest wavelength of the
Lyman series in a muonic hydrogen atom? In what region of the electromagnetic spectrum
does this belong?
(d) How large is the correction for the finite nuclear mass in this case(Hint: See the discussion at
the beginning of Section 6.8 of Modern Physics, Krane, 3rd edition.)?