PHYS2041 Quantum Mechanics

Lucy Lowe
Mind Map by Lucy Lowe, updated more than 1 year ago
Lucy Lowe
Created by Lucy Lowe almost 3 years ago
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Physics Mind Map on PHYS2041 Quantum Mechanics, created by Lucy Lowe on 07/24/2017.

Resource summary

PHYS2041 Quantum Mechanics
1 Wave-particle duality

Annotations:

  • every object has wave-like and particle-like properties (microscopic objects 'are’ particles and waves at the same time)
1.1 De Broglie wavelength

Annotations:

  • De Broglie wavelength \[ \lambda = \frac{h}{p} \] h = 6.24  x10-34 Js
1.1.1 non-relativistic particles

Annotations:

  • Momentum \[ p = mv \] \(m \) -mass (kg) \(v = |v| \) -speed \(h\) - plank's constant \(6.62607004\times10-34 Js \) wavelength \[ \lambda = \frac{h}{mv} \]
1.1.1.1 particles of light

Annotations:

  • 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
1.1.1.1.1 Energy of photon

Annotations:

  • \(E = h \mu \) \( \lambda = \frac{h}{p} \) \[ E  = \frac{hc}{\lambda} = pc \] \( \mu \) - period
1.1.1.2 kinetic Energy

Annotations:

  • \[\frac{1}{2} mv^2 = \frac{1}{2} pv =  \frac{p^2}{2m} \]
1.1.2 momentum >= 0

Annotations:

  • Energy is never zero Always ground amount of energy p =mv = kg m/s
2 quantised

Annotations:

  • comes in discrete portions -Enger in light particles
3 Black body radiation

Annotations:

  • how heated bodies radiate 
3.1 Rayleigh-Jeans intensty spectrum result

Annotations:

  • \[ I(\lambda ) = \frac{8 \pi}{ \lambda^4} k_{B} T \]
3.2 E.M. radiation

Annotations:

  • -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
3.2.1 classically

Annotations:

  • 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
3.3 Planck’s (quantum) radiation law

Annotations:

  • \[ I(\lambda ) = \frac{8 \pi hc}{ \lambda^{5} \left(e^{\frac{hc}{ \lambda k_{B} T}} -1\right)} \]
4 Photo-electric effect
5 Atomic spectra

Annotations:

  • emission spectrum of atoms consists of just few (discrete) narrow spectral lines at certain wavelengths
5.1 Hydrogen atom spectrum
5.2 Bohr's Rule

Annotations:

  • 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)
6 The wave function

Annotations:

  • Can only describe quantum systems when closed system (pure states). Open systems are described by density matrix.
6.1 The Schrodinger Equation

Annotations:

  • \[ ih \frac{ \Psi}{dt} = -\frac{h^2}{2m} \frac{d^2 \Psi}{dx^2} + V(x,t) \Psi \]
6.1.1 The particle must be somewhere

Annotations:

  • \[ \int_{- \infty}^{\infty} |\Psi( x,t)|^2 dx = 1 \]
6.2 Normalisation
6.2.1 probabilty density

Annotations:

  • \[ <x> = \int_{-\infty}^{+\infty} x |\Psi (x, t)|^2 dx \] expectation value of x^2 \[ <x^2>  = \int_{-\infty}^{+\infty} x^2 |\Psi (x, t)|^2 dx \]
  • mean variance of particle position, standard deviation. \[ \alpha_{x} = \sqrt{<(\Delta x)^2>} = \sqrt{ <x^2> - <x>^2} \]
6.3 Expectation or mean values

Annotations:

  • \[ \langle O \rangle  = \int dx \psi*O(x,p) \psi \]
6.4 coordinate representation
6.4.1 momentum operator

Annotations:

  • \[ \hat{p} = -ih \frac{d}{dx} \]
7 infinite well
7.1 Energy

Annotations:

  • \[E_n = \frac{h^2}{2m}(\frac{\pi}{a})^2n^2\]
7.2 wave function
8 harmonic oscillator
8.1 length scale

Annotations:

  • \[l_{ho} = \sqrt{\ hbar /m \omega} \]
8.2 Properties of raising and lowering operators

Annotations:

  • \[ \hat{a}_+ \psi_n = \sqrt{n+1}\psi_{n+1} \] \[ \hat{a}_- \psi_n = \sqrt{n}\psi_{n-1} \]

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