*Exploring Quantum Physics – Week 3, Lecture 7*

- Super conducting first observed by Heike Kamerlingh Onnes in 1913
- Bellow 4.2K, resistivity of Hg drops to
**zero** very quickly
*Magnetic flux expulsion* (a.k.a. Meissner effect) Magnetic fields “avoid” super conductor
- equivalent to the Anderson-Higgs mechanism

- High temperature super conductor (“High-Tc”), a.k.a. Cooper superconductor
- Theory of High-Tcs not yet discovered

## Particle braiding & spin-statistics theorem

- “Particle braiding” = particles switching positions
- What are the possible values of the quantum statistical phase, ?
- For bosons (integer spin),
- For fermions (half-integer spin),

## Pauli exclusion principle for fermions

- What are the consequences of ?
- An important one for is
- This means the probability of finding two identical fermions in the same point is zero
- More generally:
**two identical fermions can not occupy the same quantum state.**
- No such constraint for bosons

- Given a single particle quantum state , the possible occupation numbers for fermions are , while for bosons

## Ground state of many boson system: Bose-Einstein condensate (BEC)

What happens if we “pour” identical bosons into a prescribed landscape of quantum states (i.e. single-particle states, , with energies )?

A: At low temperatures, they form the lowest-energy state (ground state), which in the absence of interactions corresponds to putting all bosons into a single lowest-energy state

**Superfluidity**

- Closely related to BEC
- Condensate of bosons forms a quantum liquid which has zero viscosity and flows without resistance
- Discovered experimentally in Hellium by Pyotr Kapitsa 1937
- Mathematical theory was put together by Lev Landau

## Ground state of a many-fermion system: Fermi gas

Representation of a potential well with the inner energy levels filled. The outer electrons can change energy levels, but the inner cannot due to Pauli

- Fermi Temperature = threshold momentum, below which all the low energy states are occupied (up to the Fermi-level)
- = Fermi energy

## Two Particle Schrödinger

- Consider generic case of particles with masses and interacting via potential

- Change of variables:
- Schrödinger equation:

**Calculating **

- Focus on one (x-) component of the Laplacian with

and
- Change of variables in the derivatives:

- Change of variables in the kinetic energy (Laplacians):

Reduced mass,

**Putting it together**

*where is the Laplacian with respect to the center of mass (R)*

Left term looks like , can move it into the energy (E) on the right:

## The Cooper problem

**Isotope effect**

- only mass differs on isotope lattice
- electrons exchange “phonons” (waves running through the lattice) which leads to
*weak effective phonon mediated attraction* between electrons

**Spherical-cow model of the phonon-mediated attraction**

- Electrons interact by exchanging phonons. Energy and momentum must be conserved!
- The phonon energies available are pretty low compared to the Fermi energy. When converted to temperature, and
- The exact interaction is complicated, simplified model:

*where the first condition describes electrons on the Fermi shell*

**Cooper pairing problem**

- 2-particle Schrödinger:

- Momentum space version (apply Fourier transform):

- Result (same method from Lecture 5)