Homework
set-1 and study help

1.
(a)
Calculate the energy of photons, electrons, neutrons, in electron volts, when
their de Broglie wavelength is 10A.  Such a wavelength is useful for probing
the structure at the atomic scale.

You
can simplify the calculation if you use the resource from the link in the next
problem.

(b) The diffraction of
C60, the fullerine, or buckeyball,  has been observed in experiments. Read
the original paper in Nature, 401, 680 (1999),   ( Article ).
Summarize the paper using less than 300 words.

2.
When dealing with electrons, one often uses the
atomic units to describe physical quantities. In atomic units, you set m=e==1.
Common physical quantities like length, time,  energy etc are then expressed in atomic
units as well. Here is a good link that you may want to save

http://www.phys.ksu.edu/personal/cdlin/mainpage/atomic-units.mht

Use
the link above, do the following calculations. Express the quantitites in
atomic units.

Find the numbers from books or web resources.

(a)
The energy separation between the 1s and 2p
levels of  atomic hydrogen.

(b)
The fine
structure separation of the 2p1/2 and 2p3/2 levels of
atomic hydrogen.

(c)
The separation between 2s1/2 and 2p1/2
levels in atomic hydrogen. (This is called Lamb shift.)

You can find
accurate data for H from  http://physics.nist.gov/PhysRefData/HDEL/data.html

3.
Describe in words the Hilbert space and its
dimension, the basis state vectors, and some examples of operators and what can
be measured,  for the following
systems.

(a) An electron with spin 1/2 in a
magnetic field.

(b) A particle of mass m is
confined to an infinite square well potential.

(c) A particle of mass m is
confined in a two-dimensional square well potential.

(d)  Two noninteracting particles of mass m1
and m2 confined in an infinite square well potential.

Review and Preview –

Review part

1.
Quick review of Planck distribution law or black-body
radiation

See
http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html

Exercise: use the graph here  http://en.wikipedia.org/wiki/File:Firas_spectrum.jpg.  It shows the cosmic microwave background spectrum measured from COBE. Calculate the average isotropic
temperature of the universe based on the above graph. (The anisotropy is in the
order of 10-4  to 10-5
from modern meaurements).

1.
Quick review of Compton scattering

See  http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/compdat.html#c1

The
resulting formula

The derivation is straightforward, but a
quick review is good for you.

see http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/compeq.html#c1

Exercise: What is the Compton wavelength? In Angstroms.

Question: In what spectral region
(microwave up to gamma-ray) should you worry about   Compton scattering?

Here is a graph of the spectral range

http://en.wikipedia.org/wiki/File:Electromagnetic-Spectrum.png

While we are on this topic, you may want to
take a look at the transmittance or opacity
of our atmosphere from this graph

http://en.wikipedia.org/wiki/File:Atmospheric_electromagnetic_transmittance_or_opacity.jpg

2.
Double slit interference experiment with
single  electrons

A sketch of this experiment is described
here http://www.hitachi.com/rd/research/em/doubleslit.html

A video clip of the electron
“buildup” can be seen by clicking the link right below the figure
there. The original paper is published in Tonomura et al, Am. J. Physics 57,
117-120 (1989).

Double-slit electron diffraction was ranked
as the most beautiful physics experiment by physicists. Here is the top 10
list:

See
http://physics-animations.com/Physics/English/top10.htm

First evidence of electrons as waves was
carried out in the

Davisson-Germer experiment, see

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/davger2.html

Review what is the Bragg’s
diffraction law

Here
http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/bragg.html#c1

Preview:

In quantum mechanics, we first
have to deal with Hilbert space.
Hilbert space is generalized from the linear vector space.

Quick review of the properties of a Vector Space

See       http://en.wikipedia.org/wiki/Vector_space

In particular, the definition; axiom of additions; bases;
dimension; matrices;  eigenvalues
and eigenvectors; norm; inner product (scalar product)

Later: direct product and direct sum

Qucik preview of polarization of light

See
http://en.wikipedia.org/wiki/Photon_polarization

And look at : Jones vector;
linear polarization; circular polarization;

Energy, momentum and angular momentum of photons