a44f583e777e725cad8152aab71389f76dc5413d
   1# Light and matter
   2
   3## Photoelectric effect
   4
   5### Planck's equation
   6
   7$$E=hf,\quad f={c \over \lambda}$$
   8$$\therefore E={hc \over \lambda}$$
   9
  10where  
  11$E$ is energy of a quantum of light (J)  
  12$f$ is frequency of EM radiation  
  13$h$ is Planck's constant ($6.63 \times 10^{-34}\operatorname{J s}$)
  14
  15### Electron-volts
  16
  17$$ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$$
  18
  19*Amount of energy an electron gains when it moves through a potential difference of 1V*
  20
  21- equivalent unit is Joule seconds (e.g. $h$)
  22
  23### Photoelectric effect
  24
  25- some metals becomes positively charged when hit with EM radiation
  26- this is due to e- being ejected from surface of metal
  27- *photocurrent* - flow of e- due to photoelectric effect
  28- causes increase in current in a circuit
  29- $V_{\operatorname{supply}}$ does not affect photocurrent
  30- if $V_{\operatorname{supply}} \gt 0$, e- are attracted to collector anode.
  31- if $V_{\operatorname{supply}} \lt 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$
  32
  33#### Wave / particle (quantum) models
  34wave model:  
  35
  36- cannot explain photoelectric effect
  37- $f$ is irrelevant to photocurrent
  38- predicts that there should be a delay between incidence of radiation and ejection of e-
  39
  40particle model:  
  41
  42- explains photoelectric effect
  43- rate of photoelectron release is proportional to intensity of incident light
  44- shining light on a metal "bombards" it with photons
  45- no time delay
  46
  47#### Work function and threshold frequency
  48
  49- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected
  50- if $f \lt f_0$, no photoelectrons are detected
  51
  52- Einstein: energy required to eject photoelectron is constant for each metal
  53- *work function* $\phi$ - minimum energy required to release photoelectrons
  54- $\phi$ is determined by strength of bonding
  55
  56$$\phi=hf_0$$
  57
  58#### $E_K$ of photoelectrons
  59
  60$$E_{\operatorname{k-max}}=hf - \phi$$
  61
  62where  
  63$E_k$ is max energy of an emmitted photoelectron  
  64$f$ is frequency of incident photon (**not** emitted electron)  
  65$\phi$ is work function ("latent" energy)
  66
  67Gradient of a frequency-energy graph is equal to $h$  
  68y-intercept is equal to $\phi$
  69
  70## Wave-particle duality
  71
  72### Double slit experiment
  73Particle model allows potential for photons to interact as they pass through slits. However, an interference pattern still appears when a dim light source is used so that only one photon can pass at a time.
  74
  75## De Broglie's theory
  76- theorised that matter may display both wave- and particle-like properties like light
  77- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$
  78- impossible to confirm de Broglie's theory of matter with double-slit experiment, since wavelengths are much smaller than for light, requiring an equally small slit ($< r_{\operatorname{proton}}$)
  79- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit)
  80- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern
  81- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
  82- rearranging this, $2\pi r = n{h \over mv}$ (circumference)
  83- therefore, stable orbits are those where circumference = whole number of e- wavelengths
  84- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established
  85
  86### Photon momentum
  87- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum
  88- therefore photons have (theoretical) momentum
  89- to solve photon momentum, rearrange $\lambda = {h \over mv}$
  90
  91## Spectral analysis
  92
  93
  94### Absorption
  95- Black lines in spectrum show light not reflected  
  96
  97### Emission
  98- Coloured lines show light being ejected from e- shells  
  99- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$  
 100- Bohr's model describes discrete energy levels
 101- Energy is conserved (out = in)
 102- Ionisation energy - minimum energy required to remove an electron
 103- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 104
 105## Light sources
 106- **incandescent:** <10% efficient, broad spectrum
 107- **LED:** semiconducting doped-Si diodes
 108- - most electrons in *valence band* (energy level)
 109- - provided energy, electrons can jump to *conduction band* and move through Si as current
 110- - colour determined by $\Delta E$ between bands (shells), and type of doping
 111- **laser:** gas atoms are excited
 112- - *popular inversion* - most gas atoms are excited
 113- - photons are released if stimulated by another photon of the right wavelength
 114- **synchrotron:** - magnetically accelerates electrons
 115- - extremely bright
 116- - highly polarised
 117- - emitted in short pulses
 118- - broad spectrum
 119
 120## Quantum mechanics
 121
 122- uncertainty occurs in any measurement
 123- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality)
 124- interaction between observer and object
 125- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it
 126
 127### Indeterminancy principle
 128
 129$$\sigma E \sigma t \ge {h \over 4 \pi}$$
 130
 131where $\sigma n$ is the uncertainty of $n$
 132
 133**$\sigma E$ and $\sigma t$ are inversely proportional$**
 134
 135Therefore, position and velocity cannot simultaneously be known with 100% certainty.
 136
 137### Single-slit diffraction
 138
 139- one photon passes through slit at any time (controlled by intensity)
 140- diffraction pattern can be explained by wave front split into wavelets
 141- diffraction can be represented as uncertainty of photonic momentum
 142
 143
 144### Comparison with Bohr's model
 145
 146**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated
 147
 148**Quantum mechanical model** - electron clouds rather than discrete shells (electrons are not particlces). We can only calculate probability of an electron being observed at a particular position
 149
 150
 151
 152774 abc melbourne