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+\usepackage{multicols}
# Light and Matter
+## Planck's equation
+
+$$f={c \over \lambda}$$
+
$$E=hf={hc \over \lambda}$$
+$$h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s}$$
+
$$ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$$
+## Force of electrons
+
+$$F=evB$$
+
+## Photoelectric effect
+
+- $V_{\operatorname{supply}}$ does not affect photocurrent
+- if $V_{\operatorname{supply}} > 0$, e- are attracted to collector anode
+- if $V_{\operatorname{supply}} < 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$
+- $v$ of e- depends on ionisation energy (shell)
+- max current depends on intensity
+
+### Threshold frequency
+- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected
+- $x$-intercept of frequency vs $E_K$ graph
+- if $f < f_0$, no photoelectrons are detected
+
+### Work function
+- *work function* $\phi$ - minimum energy required to release photoelectrons
+- magnitude of $y$-intercept of frequency vs $E_K$ graph
+- $\phi$ is determined by strength of bonding
+
+$$\phi=hf_0$$
+
+### Kinetic energy
+
+$$E_{\operatorname{k-max}}=hf - \phi$$
+
+voltage in circuit or stopping voltage = max $E_K$ in eV
+equal to $x$-intercept of volts vs current graph (in eV)
+
+### Stopping potential
+
+_Smallest voltage to achieve minimum current_
+
+<!-- $$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ -->
+$$V=h_{\text{eV}}(f-f_0)$$
+
+## De Broglie's theory
+
+$$\lambda = {h \over \rho} = {h \over mv}$$
+$$\rho = {hf \over c} = {h \over \lambda}$$
+$$E = \rho c$$
+
+- 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
+
+## X-ray and electron interaction
+
+- 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)
+- if $2\pi r \ne n{h \over mv}$, interference occurs, standing wave cannot be established
+
+## Spectral analysis
+
+<!--  -->
+- $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
+- $f$ of a photon emitted or absorbed can be calculated from energy difference: $E_2 – E_1 = hf$ or $= hc$
+- Ionisation energy - min $E$ required to remove e-
+- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
+
+## Indeterminancy principle
+
+measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it. $\therefore, \sigma E \propto {1 \over \sigma t}$
+
+$$\sigma E \sigma t \ge {h \over 4 \pi}$$
+
+## Wave-particle duality
+wave model:
+
+- cannot explain photoelectric effect
+- $f$ is irrelevant to photocurrent
+- predicts delay between incidence and ejection
+- speed depends on medium
+
+particle model:
+
+- explains photoelectric effect
+- rate of photoelectron release $\propto$ intensity
+- no time delay - one photon releases one electron
+- double slit: photons interact as they pass through slits. interference pattern still appears when a dim light source is used so that only one photon can pass at a time
+- light exerts force
+- light bent by gravity
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