Assignment in lieu of Quiz III
From systems of identical particles to relativistic quantum mechanics
1. Possible configurations of identical bosons: Three identical bosons with spin s = 1 are in the same
orbital states described by the wavefunction ψ(r).
(a) Write down the normalized spin functions for the total system.
(b) How many independent states are possible?
(c) What are the possible values of the total spin of the system?
(a) Calculate the radius of a neutron star with the mass of the sun.
(b) Also calculate the neutron Fermi energy, and compare it with the rest energy of a neutron. Isit reasonable to treat such a star non-relativistically?
3. Partial waves and phase shifts: Recall that, using the method of partial waves, we had obtained
the scattering amplitude to be
,
where Pl(x) denotes the Legendre polynomials and al was called the l-th partial wave amplitude.
(a) Using the above result, arrive at the corresponding expression for the differential cross-section and show that total cross-section can be written as
.
(b) Focusing on a particular l, show that the partial wave amplitude al can be expressed in terms of the phase shift δl as follows:
.
(c) Use this form of al to arrive an expression for the total cross-section in terms of the phase shifts δl.
4. Scattering amplitude and cross-section in the Born approximation: Using the Born approximation,
find the scattering amplitude as well as the total scattering cross-section of a particle in the following two central potentials: V (r) = αe−µr and V (r) = α/r2.
5. Simultaneous diaganolization of pˆ, Hˆ and Σˆ describing a Dirac particle: Recall that the Hamilto-
nian Hˆ of a Dirac particle is given by
Hˆ = α · pˆ+ β mc2,
where pˆ is the momentum operator. The quantities αi with i = (1,2,3) and β are the Dirac matrices defined as
,
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where σi and I are the 2 × 2 Pauli matrices and the unit matrix given by
.
Also, note that the spin operator describing the Dirac particles can be expressed in terms of the Pauli matrices σ as follows:
Sˆ ,
where
.
Show that the commutators [pˆ,Hˆ], [pˆ,pˆ· Σˆ] and [H,ˆ pˆ· Σˆ] vanish.
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