---
title: "Lecture Note 01"
author: "sin-iu ho"
date: "3 March 2020"
header-includes:
output:
pdf_document
---
# Lecture Note 01
###### tags: `Quantum Physics II`
###### date: `3 March 2020`
---
## Review of last semester
- Wave Function
- princ. o' superposition
- related to prob. (statisical interpr.) (diff.from.CM)
- uncertainty princ. (diff.from.CM)
* *spin correlation* upper limit?
- time-ind. sol. of Schröd.eqn.
- exact sol. がある
- inf.square well
- fin.square well (wave can penetrate)
- delta func. (wave can penetrate)
- harmonic
- Formalisation
- **Hilbert Space** (a **[square-int.](https://en.wikipedia.org/wiki/Square-integrable_function) vector space**)
- stric.spk, plain wave is not square-int.
- Let $\psi = \dfrac{1}{N}e^{-i(Et-kx)}$, which is a plain wave (V.), $\psi$ is not square-int because $\int_{-L}^Ldx|\psi|^2 = 1$ with $N = 1/\sqrt{2L}$.
- Dirac
- Heisenberg's Matrix
- state as vector
- operator as matrix
If you were to contruct a theory which differs from the one taught in textbokks and but doesn't contain a matrix, then there ... might be flaw
- predict the H-atom Spectrum as Bohr did-- significant milestone of QM
- angular momentum operator $J$
- spin $S$ and orb.$L$ act on diff. space (how diff?)
- imagine time-space as lattice, (non-relativisticly)
$L$ need more "latice", rel. to a "point" as origin,
$S$ need only one, need no origin.
rotational invariance/
Ask: from Schro.eqn. we cannot see origin chosen?
Ans: It needn't, but rotaional symmetry is not guaranteed, sol. would be more obscure.
- Lowrence force (acted on charges moving in E-M field) correspd. to (which part of) Schro.eqn.
Use gauge field instead of E field and B field
A-B effect!!!
- Identical Particles
- (superficially discussed)
- **symmetry & conservation laws** important both in CM and QM

This semester: Application (Part II)
---
### Ptb. theory
Hamiltonian in 2 parts
$H = H^0 + \lambda H^1$
$H^0$ large, treated normally like before
we can assume $\lambda\to 0$ if we like
then the convergence is very good
a bit like Taylor expan.
$\lambda H^1$ very small, relatively

Many kinds of interactions (grav., weak, strong & sometimes EM), if small enough, can use Ptb.
If too strong, or intermediate states too many, then Ptb. doesn't work
"intermediate states too many", such as the case where the pre-factor is larger then 2;
$E_0 = \sum^\infty_{i=0}c_i\lambda^2_i, \lambda_i\ll1$
? electron moves in fixed $\mathbf{E}$ field > spin orb. coupling > spin up/down > spectrum split (fine-structure?)
**Lamb shift** 21 cm
**Landé g-factor**
dirac eqn. systematically contain those

### Variational Princ.
find in all posible wave func and choose the one with the least energy, which is ground state.
sounds idiot, but it can help one to pick out quasi-ground state
prepare a ground state that can be distinguished from the first excited state -- cool down the system by the temp. corresp. to the small energy gap btwn. the two.

### WKB approx.
idea: reduce to a SHO, solve the ground state
useful in nuclear physics, scattering (incl.bio phys.)

### Others
entangle -- Bell's inequal?
...

### Why learn this
though num. sol. is getting , you still need some knolwged and intuition

### What's beyond
1. num. appl. (based on approx.)
2. fund. interac. & fund. par.
3. scattering (exp.&th.)
4. quantum info. (comb. w/ 1) genreal qunantum cumputor, crptography
* large scale: universe itslf.
* small scale: COSMOLOGY early universe (inflation)(LHC)((cosmic neutrinos)(ICE cube) TeV
* larger: nuclear MeV
* atomic eV
* solid state, condensed matter
* --> new materials (solar cells batteries)
* topological materials
*
)
.... later, diverged into various, trivial subjects
all in all, lots of subjects are related to QM

triangle riraoij

$\hat{H}_0\left|\phi_q^i\right>=E^0_p\left|\phi_q^i\right>$