Quantum mechanics

Interpretation of wave function
Probablity densitity(in a sense) of finding a particle, so it should be normalized.

Schrodinger equation:
Non-relativistic formalism, coordinate representation, Time evlution won't change the normalize condition.

left side: first-order time derietive

right side:

kinetic part: second-order space derivative

potential part

Expectation
Ensemble average but time average.

Obsevarble is the average of corresponding
Mechanical Quantity

Uncertainty Principle
Product of the standard derviation of position and momentum is gerater than half of hbar

Seperation of varibles
When potential is only the function of position, the Schrodinger equation can be separated into time part and position part.

Time-independent Schordinger equation
Hamitonlian operator's enginvalue equation, coordinate repersentation

Time part
Solution is exponent of t

Time-independent Schrodinger equation
Solution depends on the relative magnitude between the energy and the potential at infinity

Bound state
Energy is discrete, and bounded below

Harmonic oscillator

Infinity deep square potential well
Since wave funciton is continous, so it must be stational wave inside the well.

Scattering state
Energy is continous, but the transimition(reflection) rate is determinable. The state cannot be normalized. The basis(usually wave solution) are Dirac normalized.

Free particle

Finity deep square potential well

Dirac-delta potential well
Wave function is continous, derivative of the wave function is uncontinous at the infinity

Mixed

What are we concerned?
Energy level, other mechanical quantity's average in certain state, like the <x> of a excited state

Hillbert space, state vector, inner product
<a|b>=conjugate a times b
Commute
Two operators share same eigenstate(degnerate ? )

Hermitian Operator
1. enginvalues are real
2. enginstates are orthanoal
3. enginstates are completeness
(cannot prove in infintite dimension space)

General statistical intepretation:
if you measure Q(x, p) for a particle in the state Psi(x,t), The result must be a eigenvalue q_n of the operator Q(x, p), corresponding probablity is |Cn|^2 or |Cz|^2dz

Hisenberg's equation
In Hisenberg picture, the state vector are time-independent
The observable satisifies
dA/dt = i/hbar [H,A(t)]+\partal A(t)/\partial t

Uncertainty principle
sigma_a*sigma_b >= (<[A,B]>/2i)^2

Quantum number
In general, if we want to find a particle in 3-dimension,
there are three coordinates and three momentums. Due to the uncertainty principle, we have only three freedom of degrees left.
We use quantum numbers n, l, m to denotes them.

Angular momentum
Deteremin a particle's angular momentum:
1. Magnitude of angular momentum
2. z-component of angular momentum

Trick
[Lx, Ly]=i\hbar Lz,
L+ = Lx + iLy
L- = Lx - iLy
L^2 = Lx^2+Ly^2+Lz^2

<L^2>=l(l+1)\hbar^2

<Lz> = m\hbar

Spin
Besides the quantum numbers mentioned above, There exist two more internal freedom of degree which may caused physical effect.

Spin 1/2
Two enginstates up and down

Addition of angular momentum

Identical particle:
The kinds of particles share same internal property, like mass, charge, spin.

Identical particles are undistingable, the wave function is either antisymmetry or antisymmerty under exchange