Quantum mechanics

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Quantum mechanics

Indentical particle

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

Quantum mechanics in 3D

Addition of angular momentum

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

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

= m\hbar

=l(l+1)\hbar^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.

Formalism

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

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

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

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

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

Stational Schrodinger equation

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

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

Mixed

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.

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

Finity deep square potential well

Free particle

Bound state Energy is discrete, and bounded below

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

Harmonic oscillator

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

Time part Solution is exponent of t

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

Basic assumption

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

Expectation Ensemble average but time average.

Obsevarble is the average of corresponding Mechanical Quantity

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

right side:

potential part

kinetic part: second-order space derivative

left side: first-order time derietive

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