#### 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
## 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
+
774 abc melbourne
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## Chain rule for $(f\circ g)$
-$$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$
-
-Leibniz notation:
-
$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
Function notation:
$\log_b x^n = n \log_b x$
$\log_b y^{x^n} = x^n \log_b y$
+### Index identities
+$b^{m+n}=b^m \cdot b^n$
+$(b^m)^n=b^{m \cdot n}$
+$(b \cdot c)^n = b^n \cdot c^n$
+
### $e$ as a logarithm
$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
$$\ln x = \log_e x$$
### Differentiating logarithms
-$${d(\log_e x)\over dx} = x^-1 = {1 \over x}$$
+$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
-## Solving $e^x$
+## Solving $e^x$ etc
-| $f(x)$ | $f^\prime(x)$ |
+| $f(x)$ | $f^\prime(x)$ |xs
| ------ | ------------- |
| $\sin x$ | $\cos x$ |
| $\sin ax$ | $a\cos ax$ |
| $\cos ax$ | $-a \sin ax$ |
| $e^x$ | $e^x$ |
| $e^{ax}$ | $ae^{ax}$ |
+| $ax^{nx}$ | $an \cdot e^{nx}$ |
| $\log_e x$ | $1 \over x$ |
| $\log_e {ax}$ | $1 \over x$ |
+| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
+| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
+
+<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->