**Classical Harmonic Oscillator**

- Hook’s force:
- Newton’s 2nd law:
- Rewrite as: , where

- Solving for x:

**Phase portrait of a harmonic oscillator**

- Choose initial condition s.t. ,
- Velocity:
- Energy:
- Expanding
*x*and*v*:

- Expanding

**Going quantum**

- In most situations we can approximate the lowest energy potential as a quadratic, since

- Classical energy:
- Usual transformation:

**Quantum Hamiltonian**

Making the Hamiltonian dimensionless:

Using :

*where *

Let be the **creation operator**

and be the **annihilation operator**.

**Summary:**

**Commutation relations**

**Generating the spectrum**

, so

latex \hat{a}^\dagger \hat{a} ( \hat{a}^\dagger | n \rangle) = \hat{a}^\dagger (1 + \hat{a}^\dagger \hat{a}) |n\rangle = \hat{a}^\dagger ( n + 1) | n \rangle = (n + 1)(hat{a}^\dagger | n \rangle)$

Annihilation:

Ground state:

**Deriving ground-state wave-function**

For the ground state,

Expanding operators:

Let

**Result**

Where *C* is the normalizing coefficient:

**Excited states**

- Hamiltonian

- Energy spectrum

- Relation between (normalized) eigenfunctions

- Ground-state wave function:

- Excited-state wave functions:

, etc.

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

- 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

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

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

- 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:

**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)

Single-particle quantum mechanics,

- Wavelength “quantization” in a guitar string: String anchored at end nodes, must have multiples of a half-wave between them

- Wavelength quantization in an infinite quantum well: potential “walls” anchor the wave function similar to guitar string

- For
- For
- Find the bound state(s) (with ) satisfying continuity constraints (problem definition):

*Where +0 means to the right of a point, and -0 means to the left*

**Using the symmetry**

- If the Hamiltonian commutes with an operator, , then solutions to the Schrödinger equation can be chosen to have definite
*a*and*E*,

- Potential is then inversion symmetric, . Eigenvalues of are
- General solution of :

- Can choose solutions with definite parity, i.e.

**Using the constraints at infinities**

- General solution of :

- E.g. for
*x > a/2*we must request that as . Otherwise probability would not be finite (does not make sense). - So drop
*B*term and the solution is

**Using matching conditions at transition points**

- $latex \displaystyle \psi_+(x) = \left\{ \begin{array}{ll}

A e^{\gamma x} & x < -a/2 \\

C \cos(kx) & |x| a/2

\end{array} \right. $ - Match solutions at
*x = a/2*

Divide Eq. 2 / Eq. 1:

**Making the self-consistency equation dimensionless**

- The non-linear self-consistency equation is not solvable analytically:

with - Start by introducing dimensionless parameters, and

- 2 limiting cases:
- Deep well:
- Shallow well:

**Solving**

**Fourier transforms**

- Can solve for shallow potential well, using Fourier transform:

- Specifically for Dirac -function:

- Use these identities to solve stationary Schrödinger equation for (with )

**Shallow level in the delta-well**

*What is E?*

Take the integral of :

…

**Final result:**

In 2 dimensions: , where is the angle of the k-vector

In D dimensions:

*where from the integral over the angle *

**Critical dimension (=2)**

- Main question: is there a bound state in a weak potential ?
- Can we make arbitrarily large by choosing
*E*?- Equivalent to asking whether diverges
- For D=2,

- If , it diverges (there is a bound state), otherwise it is finite (no bound state).

**Very shallow level in 2D quantum well**

**Result:**

As , this result has no Taylor expansion.

- Gaussian integral:
- There is no closed analytical expression for . But if there is a small parameter in the exponential, , one can simplify things.
- The max of occurs at the min of
*f(x)* - As approaches Gaussian
- E.x:

*where f_min is the minimum value of the function (stationary point), so there is no first derivative, and the 2nd order term of the Taylor expansion becomes the sqrt in the above*

- Quantum effects hinge on interference phenomena. Classical limit implies suppressing them, which happens if

or, formally,

Can think of this as increasing the scale, or shrinking the Planck scale. - Feynman path integral in the classical limit, , is “collected” from the trajectories for which the action is minimal.
- Consequence of destructive interference when , since will be out of phase.

- I.e. When the Feynman path integral reduces to the principle of stationary action.

- To determine a point where a function is at a minimum:

*where , so drop first-order term* - To determine a trajectory where functional-action is minimal:

*and* - So set the first variation of the action to 0:

- Action:
- Calculate first variation:

The integrand in the last equation is , which is 0 (by the principle of least action), therefore ,

- Simplistic Drude model:

- With “friction” force

- In equilibrium,

*where is the expected time between collisions w/ imperfections*

- Probability to propagate

- Typical action,
- Quantum (interference) terms

- A “loop” in the path (a.k.a. “weak localization”) tries to bring electron back to where it came from
- Loop can be traversed in either direction, giving 2 paths of equal length, so the quantum interference is 0

- If many particles experience a random walk, their (average) density satisfies the diffusion equation:

*where is a function* - Density spread from a point (e.g. the origin)

*where d is the dimensionality of the space*

- Probability (density) to return to a close vicinity of the origin in a time
*t*

- Remember probability of electron to diffuse from :

**Conclusion:**

At “high” temperatures, the phase is disrupted and the particle behaves as described by the Drude model, but at low temperatures the quantum effects matter.

Developed in Feynman’s 1948 paper: Space-Time Approach to Non-Relativistic Quantum Mechanics

For a particle traveling from a coordinate

The probability of going to is the sum of the probability (or weight) of taking a particular path, over all possible paths:

- Consider a particle localized at at time
*t*=0, that is - Evolve according to Schrödinger:

- What “part” will propagate to ?

i.e. what is the overlap with , - How will it get there? (In Q.M. it follows all possible trajectories)

*Where is the evolution operator*

Plug into Schrödinger:

i.e. does not depend on the initial condition of

Verify:

*Asside: define the exponential of an operator, with it’s Taylor series:
*

a.k.a. the transition amplitude

Exponential of an operator is messy (infinite sum, see above) so simplify using:

Combining the above:

Reapplying this *N* times gives:

Let becomes small, and using a first order approximation:

**Propagator becomes a product:**

*Asside: difference between *

– Represents an “abstract” state of the system, coordinate-independent

– Represents in a coordinate system.

*Asside: the Dirac delta-function*

- Extension to a “continuum” basis

*where C normalizes:*

- Fourier transform:

*because is an eigen state of the coordinate system*

*where is the Lagrangian*

**Born Rule**: is the probability of finding the quantum particle (described by ) in the volume at time *t*.

Integrate over volume:

Recognizing Gauss’ theorem in the integral, we get:

From Born’s Rule, calculate the probability flux:

*Remember*

Apply to Schrödinger:

Expanding the Hamiltonian and canceling the potential gives:

*Remember K.E. is an operator:*

is the probability current

This connects the change in the probability of finding a particle in *V* with the flux of the probability current flowing through the surface.

**Classical properties (momentum, location, etc.) become operators acting on the wave function**

- momentum:
- location: (multiplication operator)
- kinetic energy:

E.g. Angular momentum becomes

Consider measuring particle position…

*x* becomes a probability function *F(x)*

*Notation:*

This generalizes to:

For a property *X* with operator :

Recall the time-*dependent* Schrödinger equation:

Often, is time independent, so we can separate the variables:

Canceling the exponential gives us the **time-independent Schrödinger equation**:

Notice that this looks like an eigenvalue equation, where *E* is the eigen value and is the eigen vector.

For any operator, , acting in a space of functions , one can define its **Hermitian-adjoint operator** with:

And is “Hermitian” if

Eigenvalue properties:

- eigenvalues (
*a*) are real if is Hermitian - eigenvectors, , form a basis

- Physical observables in quantum mechanics are described by Hermitian (a.k.a. self-adjoint, though there are subtle differences we will ignore) operators,
- Eigenvalues of a physical operator determine possible values of the observable that actually can be measured in an experiment.

- Eigenvectors form a basis in the sense that a wave-function can be expressed as their linear combination.

- If and are solutions to the Schrödinger equation,

,

Then is also a solution. - This motivates the notion of a Hilbert space – a linear vector space where quantum states live.
- The wave function, , is a specific representation of a quantum state (much like coordinates of a vector).
**Dirac notation**for “vectors” of quantum states: and- For a basis {}:

- Physical observables in quantum mechanics are associated with linear Hermitian operators.
- For a generic operator, , the eigenvalue problem defines eigenvectors that form a basis in the Hilbert space.
- is the wave-function in the
*a*-representation. - Standard choices are:
- coordinate representation:
- momentum representation:

**Photo electric effect**: beam of light hitting charged plates with a potential across them creates current.*Expected*: amount of current is independent of frequency of the light, dependent on the intensity.*Actual*: current depends on frequency of light (no current below frequency threshold), independent of intensity.- Einstein proposed photons (nobel prize)

**Electron diffraction**: (Davisson-Germer experiment)- Crystalline structure acts as a diffraction grating, electron beam demonstrates wave interference like light.
- Experiment supports DeBroglie hypothesis:

, - For a derivation of the DeBroglie wave length, see

http://www.chip-architect.com/physics/deBroglie.pdf

Wave equation:

*Where k is the wave number*

*Where c is the wave velocity*

We can represent particles as a wave (but not the other way around) via the **Fourier transform**:

*Where is the “fourier harmonics” (amplitudes)*

Represent a particle as a Gaussian wave with a sharp peak:

*Where is the wave function, and is the constant of proportionality, ignore for now.*

Since and , can write as:

Taking the derivative:

(or for a momentum vector: )

For a free particle (not in a potential field), , so:

Giving the **free schrödinger equation**:

Using the Hamiltonian for energy gives the **fundamental equation of quantum physics**:

Take a Gaussian wave-packet

Solve schrödinger with this gives

*where delocalization time*

Recognize the **Gaussian integral**:

Gives wave-function in momentum space:

Notice the uncertainty , but . This is a manifestation of the **Heisenberg uncertainty principle**:

A fairly accessible article which opens with a review of evidence (as of 2007) in support of inflation, and then goes on to speculate about its implications.

Since 2007, we only have more evidence in support of inflation. The Planck results fit the model of inflation with (figure 1. in the paper) more closely. And very recently, the BICEP2 results are very strong evidence of inflation.

**Supporting evidence**

- Scale of the universe
- Hubble expansion
- Homogeneity and isotropy
- (approximate) Flatness of the universe
- Absence of magnetic monopoles
- Anisotropy of the cosmic background radiation

**“Standard” FRW cosmology**– FRW stands for “Friedman, Robertson, Walker” (and sometimes FLRW with Lemaitre), describes a*homogenous and isotropic*universe. I think “standard” implies not expanding (contracting) in this case.**type 1a supernovae**– A type of “standard candle” which lets helps to measure distances or occlusions.**Anisotropy**– “non uniform” (opposite of isotropic)- – scalar field measuring the… rate of expansion? I’m fuzzy on this one, see http://physics.stackexchange.com/questions/105145/what-is-phi-as-refered-to-in-guth-2007
**De Sitter space equation of motion**– I don’t understand this. #unresolved**Anthhropic reasoning**– The idea that the parameters of our universe are the way they are because we couldn’t exist anywhere else, so we happen to have evolved (existed) in a universe which matches those parameters.

**Magnetic monopole**– A particle predicted by all grand unified theories which is extremely massive and carrying a net magnetic charge.**Youngness paradox**– The rate of new expansion is so high that at any given time almost all the existing universes are very very young. Guth uses this to argue that we’re the only (first) “advanced” species in our universe (but he then states, “I find it more plausible that it is merely a symptom that the synchronous gauge probability distribution is not the right one”).

- Inflation implies eternal inflation
- Eternal inflation implies an infinite multiverse
*(did I understand this correctly?)* - Eternal inflation (and thus infinite multiverses) is a boon for
*anthropic reasoning*(see above) - Probability is not well-defined in an infinite and eternally expanding multiverse, as anything and everything is infinitely probable (a.k.a. anything that can happen will happen). This is an open research question. #openquestion
- Inflation (probably) does not extend infinitely into the past, a.k.a. our universe had a beginning. Technical section I did not follow (#unresolved), but the takeaway is that, “new physics (other than inflation) is needed to describe what happens at this boundary”. In other words, don’t extrapolate current physics to think the universe started as a singularity (See Matt Strassler’s post on this issue).

**Geometric Series**

Consider:

so

**Power Series**

- for
- for (Binomial series)

**Taylor Series**

**Comparison test**

Let be sequences of positive reals and s.t. and converges, then converges

**Ratio test**

If for large , , then converges. Derived from comparison test with geometric series.

**Integral test**

If is *decreasing positive* and converges, converges

Using newton’s method to approximate this to the 2nd order:

(for moderately large *n*)

The last step in the above makes use of the **Gaussian integral**:

using **parametric differentiation**: