Zusammenfassung der Ressource
PHYS2041 Quantum Mechanics
- Wave-particle duality
Anmerkungen:
- every object has wave-like and particle-like properties (microscopic objects 'are’ particles and waves at the same time)
- De Broglie wavelength
Anmerkungen:
- De Broglie wavelength
\[ \lambda = \frac{h}{p} \]
h = 6.24 x10-34 Js
- non-relativistic particles
Anmerkungen:
- Momentum \[ p = mv \] \(m \) -mass (kg)
\(v = |v| \) -speed
\(h\) - plank's constant \(6.62607004\times10-34 Js \)
wavelength \[ \lambda = \frac{h}{mv} \]
- particles of light
Anmerkungen:
- photons = quanta of E.M radiation
\[ p = hk = h \omega/c \rightarrow \lambda = \frac{h}{p} = \frac{2 \pi h}{p} = \frac{2 \pi h}{\omega} =Tc \]
\(\lambda \) - wavelength\(T\) -oscillation period \(\omega \) - frequency\(k = 2 \pi / \lambda \) - wave-number
- Energy of photon
Anmerkungen:
- \(E = h \mu \)
\( \lambda = \frac{h}{p} \)
\[ E = \frac{hc}{\lambda} = pc \]
\( \mu \) - period
- kinetic Energy
Anmerkungen:
- \[\frac{1}{2} mv^2 = \frac{1}{2} pv = \frac{p^2}{2m} \]
- momentum >= 0
Anmerkungen:
- Energy is never zero
Always ground amount of energy
p =mv = kg m/s
- quantised
Anmerkungen:
- comes in discrete portions
-Enger in light particles
- Black body radiation
Anmerkungen:
- how heated bodies radiate
- Rayleigh-Jeans intensty spectrum
result
Anmerkungen:
- \[ I(\lambda ) = \frac{8 \pi}{ \lambda^4} k_{B} T \]
- E.M. radiation
Anmerkungen:
- -Field that permeates all space
Max Planck (1900): Energy of E.M. radiation isquantised (comes in discrete portions): \[ E = nh \omega \]\(n = 0,1,2,3,... \) - number of excitation quantah - planks constant\( \omega \) - frequency
- classically
Anmerkungen:
- Each standing wave or oscillator mode
has two degrees of freedom classically,
and should have an average thermal
energy .
\[ k_{B} T \] (classically)
ultraviolet catastrophe
- Planck’s (quantum) radiation law
Anmerkungen:
- \[ I(\lambda ) = \frac{8 \pi hc}{ \lambda^{5} \left(e^{\frac{hc}{ \lambda k_{B} T}} -1\right)} \]
- Photo-electric effect
- Atomic spectra
Anmerkungen:
- emission spectrum of atoms consists of just
few (discrete) narrow spectral lines at certain wavelengths
- Hydrogen atom spectrum
- Bohr's Rule
Anmerkungen:
- 2π x (electron mass) x (electron orbital speed) x (orbit radius) = (any integer) x h
- The energy lost
by the electron is carried away by a photon:
photon energy = (e’s energy in larger orbit) - (e’s energy in smaller orbit)
- The wave function
Anmerkungen:
- Can only describe quantum systems when closed system (pure states).
Open systems are described by density matrix.
- The Schrodinger
Equation
Anmerkungen:
- \[ ih \frac{ \Psi}{dt} = -\frac{h^2}{2m} \frac{d^2 \Psi}{dx^2} + V(x,t) \Psi \]
- The particle must be
somewhere
Anmerkungen:
- \[ \int_{- \infty}^{\infty} |\Psi( x,t)|^2 dx = 1 \]
- Normalisation
- probabilty density
Anmerkungen:
- \[ <x> = \int_{-\infty}^{+\infty} x |\Psi (x, t)|^2 dx \]
expectation value of x^2
\[ <x^2> = \int_{-\infty}^{+\infty} x^2 |\Psi (x, t)|^2 dx \]
- mean variance of particle position, standard deviation.
\[ \alpha_{x} = \sqrt{<(\Delta x)^2>} = \sqrt{ <x^2> - <x>^2} \]
- Expectation or mean values
Anmerkungen:
- \[ \langle O \rangle = \int dx \psi*O(x,p) \psi \]
- coordinate representation
- momentum operator
Anmerkungen:
- \[ \hat{p} = -ih \frac{d}{dx} \]
- infinite well
- Energy
Anmerkungen:
- \[E_n = \frac{h^2}{2m}(\frac{\pi}{a})^2n^2\]
- wave function
- harmonic oscillator
- length scale
Anmerkungen:
- \[l_{ho} = \sqrt{\ hbar /m \omega} \]
- Properties of raising and lowering operators
Anmerkungen:
- \[ \hat{a}_+ \psi_n = \sqrt{n+1}\psi_{n+1} \]
\[ \hat{a}_- \psi_n = \sqrt{n}\psi_{n-1} \]