# Light and matter

## Photoelectric effect

### Planck's equation

$$E=hf,\quad f={c \over \lambda}$$
$$\therefore E={hc \over \lambda}$$

where  
$E$ is energy of a quantum of light (J)  
$f$ is frequency of EM radiation  
$h$ is Planck's constant ($6.63 \times 10^{-34}\operatorname{J s}$)

### Electron-volts

$$ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$$

*Amount of energy an electron gains when it moves through a potential difference of 1V*

- equivalent unit is Joule seconds (e.g. $h$)

### Photoelectric effect

- some metals becomes positively charged when hit with EM radiation
- this is due to e- being ejected from surface of metal
- *photocurrent* - flow of e- due to photoelectric effect
- causes increase in current in a circuit
- $V_{\operatorname{supply}}$ does not affect photocurrent
- if $V_{\operatorname{supply}} \gt 0$, e- are attracted to collector anode.
- if $V_{\operatorname{supply}} \lt 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$
- not all electrons have the same velocity - depends on ionisation energy (shell)

#### Wave / particle (quantum) models
wave model:  

- cannot explain photoelectric effect
- $f$ is irrelevant to photocurrent
- predicts that there should be a delay between incidence of radiation and ejection of e-

particle model:  

- explains photoelectric effect
- rate of photoelectron release is proportional to intensity of incident light
- shining light on a metal "bombards" it with photons
- no time delay

#### Work function and threshold frequency

- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected
- if $f \lt f_0$, no photoelectrons are detected

- Einstein: energy required to eject photoelectron is constant for each metal
- *work function* $\phi$ - minimum energy required to release photoelectrons
- $\phi$ is determined by strength of bonding

$$\phi=hf_0$$

#### $E_K$ of photoelectrons (stopping energy)

$$E_{\operatorname{k-max}}=hf - \phi$$

where  
$E_k$ is max energy of an emmitted photoelectron  
$f$ is frequency of incident photon (**not** emitted electron)  
$\phi$ is work function ("latent" energy)

Gradient of a frequency-energy graph is equal to $h$  
y-intercept is equal to $\phi$

## Wave-particle duality

### Double slit experiment
Particle 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.

## De Broglie's theory
- theorised that matter may display both wave- and particle-like properties like light
- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$
- 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 ($< r_{\operatorname{proton}}$)
- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit)
- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern
- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
- rearranging this, $2\pi r = n{h \over mv}$ (circumference)
- therefore, stable orbits are those where circumference = whole number of e- wavelengths
- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established

### Photon momentum
- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum
- therefore photons have (theoretical) momentum
- to solve photon momentum, rearrange $\lambda = {h \over mv}$

## Spectral analysis


### Absorption
- Black lines in spectrum show light not reflected  

### Emission
- Coloured lines show light being ejected from e- shells  
- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$  
- Bohr's model describes discrete energy levels
- Energy is conserved (out = in)
- Ionisation energy - minimum energy required to remove an electron
- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)

## Light sources
- **incandescent:** <10% efficient, broad spectrum
- **LED:** semiconducting doped-Si diodes
- - most electrons in *valence band* (energy level)
- - provided energy, electrons can jump to *conduction band* and move through Si as current
- - colour determined by $\Delta E$ between bands (shells), and type of doping
- **laser:** gas atoms are excited
- - *popular inversion* - most gas atoms are excited
- - photons are released if stimulated by another photon of the right wavelength
- **synchrotron:** - magnetically accelerates electrons
- - extremely bright
- - highly polarised
- - emitted in short pulses
- - broad spectrum

## Quantum mechanics

- uncertainty occurs in any measurement
- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality)
- interaction between observer and object
- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it

### Indeterminancy principle

$$\sigma E \sigma t \ge {h \over 4 \pi}$$

where $\sigma n$ is the uncertainty of $n$

**$\sigma E$ and $\sigma t$ are inversely proportional$**

Therefore, position and velocity cannot simultaneously be known with 100% certainty.

### Single-slit diffraction

- one photon passes through slit at any time (controlled by intensity)
- diffraction pattern can be explained by wave front split into wavelets
- diffraction can be represented as uncertainty of photonic momentum


### Comparison with Bohr's model

**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated

**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



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