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.
