.. _lecture22:

Lecture 22: Quantum gases and condensates
++++++++++++++++++++++++++++++++++++++++++

.. note::

    *Bose-Einstein condensation is one of the most intriguing phenomena predicted by quantum statistical mechanics.* -- Wolfgang Ketterle
    
.. warning::

   This lecture corresponds to Chapter 30 of the textbook. 
   

Summary
-------
.. attention::

   This lecture uses results from :ref:`lecture21` to study the
   thermal properties of gases of quantum particles obeying either the
   Fermi-Dirac or Bose-Einstein distributions.

   First, we study the :index:`Fermi gas`: In a Fermi gas (*i.e.*, a gas of
   fermions), fermions fill states up to :math:`E_F` at absolute
   zero. The Pauli exclusion principle ensures that fermions only
   singly occupy states. The results for a Fermi gas can be applied to
   the electrons in a metal. At non-zero temperature, electrons with
   energies within :math:`k_B T` of :math:`E_F` are important in
   determining the properties. We find that the heat capacity in such
   a system scales linearly with temperature, in strong contrast with
   the physics of phonons we saw in :ref:`lecture16` where we saw that
   heat capacity due to phonons scales as the third power of
   temperature. We conclude that heat capacity at room temperature is
   dominated by phonons.

   Second, we study a Bosonic gas and realize that the treatment of
   the partition function using an integral rather than a sum leads to
   difficulties in understanding the behavior of the system at very
   low temperature. It so happens that at high density and low
   temperature, a new phase of matter develops: it is called the
   Bose-Einstein condensate where there is a diverging occupation for
   the lowest energy state. This is allowed by quantum mechanics since
   more than one bosons can occupy the same state.

   
   
Key Definitions
---------------

.. note:: 

   :index:`Fugacity`:
      Fugacity :math:`z= e ^{\beta \mu}` is... At low density, we have: :math:`z= \ll 1`. For bosons treated in a non-interacting scheme, we have that :math:`0<z<1`. The larger :math:`z`, the more evident are the quantum effects. 

   :index:`Fermi level`:
	  Fermi level is the highest occupied Fermionic state at :math:`T=0`.

   :index:`Fermi surface:`
	  Set of points in :math:`k`-space whose energy is equal to the chemical potential 

   :index:`Degeneracy`:
	  Degeneracy describes the fact that two different states (*i.e.*, different wave-functions) correspond to the same eigenvalue, *e.g.* energy. 

   :index:`Polylogarithm function`:
	  The polylogarithm function is defined as:
	  
	     .. math::
		Li _{n}(z)=\sum_{k=1}^{\infty} \frac{z^{k}}{k^{n}}.

   :index:`Zeta function`:
       The Riemann Zeta function is defined as:
       
	     .. math::
		\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}.

   :index:`Gamma function`:
       The Gamma function is defined as:
       
	     .. math::
		\Gamma(n)=\int_{0}^{\infty} x^{n-1} e ^{-x} d x.


   
   A full list of terms, including the ones provided here, can be
   found in the :ref:`genindex`. 
	  
	  
Learning Material
-----------------

Copy of Slides
~~~~~~~~~~~~~~

The slides for Lecture 22 are available in pdf format here:  :download:`pdf <_pdfs/slides/lecture22.pdf>`


.. raw:: latex
	 
	 The slides for Lecture 22 are available in pdf format \href{https://homepages.rpi.edu/~meuniv/TSM/_downloads/7ae5c9bdc7186171617cf1dbe42ce403/lecture12.pdf}{here}.
   

	    
Screencast
~~~~~~~~~~


.. raw:: html

	 <iframe width="560" height="315" src="https://www.youtube.com/embed/42cSmQr3FY4" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

.. raw:: latex

	 This lecture is available as a YouTube recording at :  \href{https://youtu.be/42cSmQr3FY4}{chapter 30}. 
 
   
Test your knowledge
-------------------




.. hint::      
   Find the answer keys on this page: :ref:`answerkeys`. Don't cheat! Try solving the problems on your own first!

Homework Assignment
-------------------

Solve the following problems from the textbook: