From 06084e083f8c5111f8fc0179023797eb3315c0b3 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 22 Aug 2018 09:27:23 +1000
Subject: [PATCH] update methods / physics / spec notes

---
 methods/calculus.md     |  5 +++++
 physics/light-matter.md | 30 ++++++++++++++++++++++++++++++
 spec/calculus.md        | 21 ++++++++++++++-------
 3 files changed, 49 insertions(+), 7 deletions(-)

diff --git a/methods/calculus.md b/methods/calculus.md
index 8cb3521..fb3761f 100644
--- a/methods/calculus.md
+++ b/methods/calculus.md
@@ -72,3 +72,8 @@ Normal line for point $P(q,r)$ on function $f$ is $y=mx+c$ where $m={-1 \over m_
 
 **In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)  
 **In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
+
+## Stationary points
+
+Stationary where $m=0$.  
+Find derivative, solve for ${dy \over dx} = 0$
diff --git a/physics/light-matter.md b/physics/light-matter.md
index fd6828c..a44f583 100644
--- a/physics/light-matter.md
+++ b/physics/light-matter.md
@@ -32,11 +32,13 @@ $$ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$$
 
 #### 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
@@ -117,6 +119,34 @@ Particle model allows potential for photons to interact as they pass through sli
 
 ## 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
\ No newline at end of file
diff --git a/spec/calculus.md b/spec/calculus.md
index 3823f37..a5f3346 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -95,11 +95,8 @@ $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
 
 ## 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:
 
@@ -143,17 +140,22 @@ $\log_b (xy)=\log_b x + \log_b y$
 $\log_b x^n = n \log_b x$  
 $\log_b y^{x^n} = x^n \log_b y$
 
+### 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
 | ------ | ------------- |
 | $\sin x$ | $\cos x$ |
 | $\sin ax$ | $a\cos ax$ |
@@ -161,6 +163,11 @@ $${d(\log_e x)\over dx} = x^-1 = {1 \over x}$$
 | $\cos ax$ | $-a \sin ax$ |
 | $e^x$ | $e^x$ |
 | $e^{ax}$ | $ae^{ax}$ |
+| $ax^{nx}$ | $an \cdot e^{nx}$ |
 | $\log_e x$ | $1 \over x$ |
 | $\log_e {ax}$ | $1 \over x$ |
+| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
+| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
+
+<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->
 
-- 
2.49.0