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author: Andrew Lorimer
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+<!-- \usepackage{multicols} -->
# Light and Matter
## Planck's equation
-$$f={c \over \lambda}$$
-
-$$E=hf={hc \over \lambda}$$
+$$f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c$$
$$h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s}$$
## Force of electrons
-$$F=evB$$
+$$F={2P_{\text{in}}\over c}$$
+
+$$\text{photons per second}={\text{total energy} \over \text{energy per photon}}={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}$$
## 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 $I\rightarrow 0$
+- $V_{\operatorname{sup}} > 0$: e- attracted to collector anode
+- $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
- $v$ 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
+### Threshold frequency $f_0$
+- minimum $f$ 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
+### Work function $\phi$
+- minimum $E$ required to release photoelectrons
- magnitude of $y$-intercept of frequency vs $E_K$ graph
- $\phi$ is determined by strength of bonding
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_
+### Stopping potential ($V$ for minimum $I$)
<!-- $$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ -->
$$V=h_{\text{eV}}(f-f_0)$$
## De Broglie's theory
$$\lambda = {h \over \rho} = {h \over mv}$$
-$$\rho = {hf \over c} = {h \over \lambda}$$
-$$E = \rho c$$
+$$\rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c$$
-- 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
+- cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
+<!-- - confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit) -->
+- confirmed by similar e- and x-ray diff patterns
## 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
+- e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
+- rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
+- if $2\pi r \ne n{h \over mv}$, no standing wave
+- if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
+- calculating $h$: $\lambda = {h \over \rho}$
## 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 $= hc$
+- $E$ and $f$ of photon: $E_2 - E_1 = hf = hc$
- 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. $\therefore, \sigma E \propto {1 \over \sigma t}$
+measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
+ <!-- $\therefore, \sigma E \propto {1 \over \sigma t}$ -->
+
+<!-- $$\sigma E \sigma t \ge {h \over 4 \pi}$$ -->
-$$\sigma E \sigma t \ge {h \over 4 \pi}$$
+$$\sigma \rho \sigma x = {h \over 4\pi}$$
## Wave-particle duality
wave 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
+- double slit: photons interact. 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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