Content

## Physical Science

The purpose of science is to better understand the world in which we live. Understanding means that we know more than just memorizing a collection of simple facts. We want to know the underlying cause and effect relationships. Based on observation we see a high degree of regularity in our universe. It is reasonable to assume then that the universe is governed by certain underlying rules which are called laws of nature. Physical science tries to discover these laws and descibe them in a physical theory.

#### Physical theories

Physical theories are ways to represent dynamical systems in the most precise way possible, in order to allow prediction of the future states of a system, given knowledge about the present state. In general a physical theory consists of the following elements:

• Objects, which might stand for entities like electrons, molecules, planets, etc.
• Properties, such as position, mass, charge, velocity, ...
• Propositions, the attribution of a particular predicate to a particular object, e.g. electron e has position P.
• Initial State, a proposition giving all relevant information about the system at a particular instant in time.
• State space, the set of all possible states.
• Operators, a transition rule mapping initial states onto subsequent states.
• Dynamical constraints, a selection criterion which determines which of the possible state transitions will actually take place. Dynamical constraints usually have the form of either
• Differential (or difference) equations, relating the predicted state transition (time derivative of the state) to the present state.
• Conservation principles, stating that a certain global property of the system, e.g. energy, must be conserved during the transition.
• Variation or optimization principles, stating that that transition will occur which minimizes (or maximizes) a certain function of the transition parameters.
• Evolution of the system, a time parameterized trajectory, representing the states of a system at subsequent instants, computable from the initial state, operators and dynamical contraints.

Dynamical systems can be either modelled as deterministic, one outcome possible, or stochastic, multiple outcomes possible each with a certain probability.

#### Determinism and time reversability

In classical physics the laws of nature are time reversable and deterministic, resulting in a state space where each subsequent state has a unique predecessor and a unique successor. The laws of nature together with the initial conditions determine the unique evolution of such a state space.

Classical physics is based on the famous force law of Newton $F=m\ddot{r}$, where $m$ is the mass of an object and $\ddot{r}$ the second derivative of position. A change of motion is the result of a change in the second derivative of position, acceleration caused by a force $F$. It follows that the initial conditions needed to predict the evolution requires two pieces of information: position, $r$ and the first derivative of position, velocity, $\dot{r}$. The initial conditions together with a force law determines the evolution of a classical mechanical system.

Does determinism implies perfect predictability ? The answer is: NO. There is randomness or uncertainty about the future even in a world with deterministic force laws. To illustrate this point, consider P and P' on the arc of a circle associated with the points Q and Q' (where Q is found by drawing a line from the top of the circle through P). Slight errors in the measurement of P produce big errors in the location of Q despite the fact that points on the horizontal line are uniquely determined by those on the circle. The position of P is a real number, but the typical real number can only be expressed as a decimal expansion consisting of an infinite string of digits with no systematic pattern to it, a random sequence. So, by its nature we could never find P exactly and therefore are always uncertain about Q.

Processes in nature that are extremely sensitive to the initial conditions are called non-linear or chaotic systems. Processes that are less sensitive are called linear or deterministic systems. The errors in chaotic systems grow at an exponential rate.

Classical physics is based on the assumption of a deterministic, time-reversible laws of nature. Classical physics does not include any distinction between the past and the future, the laws of Newton are invariant to $-t$. Even the modern theories of relativity and quantum physics are deterministic and time-reversable.

#### Scientific method

Physical theory is developed by a combination of observation, reasoning and experimentation. Reasoning alone is not enough. Each theory must be put to tests in the real world before it has any relevance for the physical sciences. A theory is always an abstraction of a physical situation. Normally this is called a model. Only by abstraction, where we leave out elements of reality, is it possible to deal with the overwhelming complexity of the real word. The validity of a model can never be proved, but one counterexample is enough to falsify a model. There are in principle two ways of arriving at a physical theory. By induction: observations are generalized into theories. The classical physical theories are mainly developed by induction. Another way is by deduction: from a set of axioms the theory is logically derived. Modern physical theories about the far-out regions of the very fast, very large and very small are deductive theories. It is very important to verify these deductive theories with experimental facts. One important principle for these deductive theories is Occam's razor:one should not make more axioms than the minimum needed.

#### Space-Time

From everyday life we know that we have the ability to distinguish objects from an environment. We observe an object at a distinct position in that environment. An instant later we are able to recognize that the position of the object has changed. We call this change a phenomenon or an event. We formulate this basic experience into an axiom for physics.

#### Postulate 1

Each object occupies a unique set of positions in space at a certain instant of time. Each instant of time is followed by a next instant of time at which the object can occupy a different set of positions in space. This change in position with time is a physical phenomena and called the motion of the object.

This axiom introduces the fundamental concepts of physics: object, position, space, time and motion. We should define these concepts precisely, but formal description of these concepts is not that easy. In physics these concepts are given a more precise content by applying a certain geometry for space and time. This geometry depends on the specific physical theory.

#### Central Theories

There are several important theories that are central to physics.

### Central theories

#### Concepts

Classical mechanics Newtonian mechanics, Lagrangian mechanics, Hamilton mechanics, Chaos theory, Fluid dynamics, Continuum mechanics Dimension, Space, Time, Motion, Length, Velocity, Mass, Momentum, Force, Energy, Angular momentum, Torque, Conservation law, Harmonic oscillator, Wave, Work, Power
Electromagnetism Electrostatics, Electricity, Magnetism, Maxwell's equations,Optics Electric charge, Current, Electric field, Magnetic field, Electromagnetic field, Electromagnetic radiation, Magnetic monopole
Thermodynamics and Statistical mechanics Heat engine, Kinetic theory Boltzmann's constant, Entropy, Free energy, Heat, Partition function, Temperature
Quantum mechanics Path integral formulation, Schrodinger equation, Quantum field theory Hamiltonian, Identical particles, Planck's constant, Quantum entanglement, Quantum harmonic oscillator, Wave function, Zero-point ener
Theory of relativity Special relativity, General relativity Equivalence principle, Four-momentum, Reference frame, Space-time, Speed of light

## Measurement

To study physical objects and phenomena observation and measurement are crucial. Certain properties are identified. These properties are measured quantitatively and therefore also called physical quantities. A quantity is expressed by a single number and called a scalaror by a mathematical object consisting of a set of numbers in the case of a vectors or a tensor.

There are different set of numbers, such as: natural, integer, rational, real and complex number sets. All these number sets have imposed an ordering of its elements. Two other important properties of a number set is its density and scarity. We define a number set as dense if between any two numbers $A$ and $B$ there exists a number $C$ such that $A < B < C$. The rational numbers are dense. A number set is called complete or continuous if it contains no "holes". The real numbers are complete, the rational numbers not. We define a variable as: discrete (isomorphic with the natural numbers), dense (isomorphic with rational numbers) or continuous (isomorphic with real numbers). Physical variables are most often assumed to be continuous.

#### Reference frames

All measurement in physics are done by an (fictitious) observer assuming a certain geometry of space and time. This observer always measures by assuming a certain reference frame. A reference frame is a set of axes relative to practically rigid body in which an observer measures the position and distances of all points, as well as the orientation.

Different types of reference frames are defined

• A fixed frame is a reference frame that is fixed relative to the entity under study.
• A moving frame is a reference frame that moves (translate / rotate) with the entity that is observed.
• An inertial frame either fixed or moving with a constant velocity.
• A non-inertial frame is a reference frame that is accelerating.

#### Fundamental quantities

For studying the properties of physical objects and phenomena several fundamental quantities are defined. These quantities are measured (i.e. compared with a standard) in units which have a precise operational definition. For these units the standard metric system SI, System International, is used. In physics the following base quantities are defined to study the properties of physical phenomena:

### Fundamental quantities

#### Symbol

Lenght (L) Meter m
Mass (M) Kilogram kg
Time (T) Second s
Electric current (Q) Ampere A
Thermodynamic temperature ($\Theta$) Kelvin K
Amount of substance Mole mol
Luminous intensity Candela cd

For each fundamental quantity a precise measurement procedure has been defined, Bureau International des Poids et Mesures.

• second:The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
• meter: The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
• kilogram:The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.
• ampere:The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 x 10–7 newton per metre of length.
• temperature:The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
• mole:The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
• candela:The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

In nature there are three important physical constants which provide natural dimensions and units (see Physical Measurement Laboratory):

• velocity $LT^{-1}$ with unit speed of light $c=3\times10^8 m/s$
• action $ML^2T^{-1}$ with unit contant of Planck $hslash=1.05\times10^{-34}kgm^2/s$
• attraction $L^3M^{-1}T^{-2}$ with unit gravitational constant $G=6.67\times10^{-11}m^3kg/s^2$

Besides the base quantities there are derived quantities from the base quantities.

#### Dimension analysis

Physicists hold very strongly to the assumption that the laws of physics possess objective reality: the laws of physics are the same for all observers. One immediate consequence of this assumption is that a law of physics must take the same form in all possible systems of units that a prospective observer might choose to employ. The only way in which this can be the case is if all laws of physics are dimensionally consistent: i.e., the quantities on the left- and right-hand sides of the equality sign in any given law of physics are the same.

#### Conversion

To convert numbers from one unit to another unit a conversion fraction is used as in the following example. $1 \text{year}.\frac{{365 \text{days}}} {{1\text{year}}}.\frac{{24 \text{hours}}} {{1\text{day}}}.\frac{{60 \text{min} }} {{1\text{hour}}}.\frac{{60 s }} {{1\text{min} }} = 3.15 \times 10^{ - 7} s$

#### SI prefix

A unit can be prefixed with one of the following scalers of powers of ten:

### SI prefixes

#### Scale

$10^{24}$ yotta Y quadrillion
$10^{21}$ zetta Z trilliard
$10^{18}$ exa E trillion
$10^{15}$ peta P billiard
$10^{12}$ tera T billion
$10^9$ giga G milliard
$10^6$ mega M million
$10^3$ kilo k thousand
$10^2$ hecto h hundred
$10^1$ deca da ten
$10^{-1}$ deci d tenth
$10^{-2}$ centi c hundredth
$10^{-3}$ milli m thousandth
$10^{-6}$ micro $\mu$ millionth
$10^{-9}$ nano n milliardth
$10^{-12}$ pico p billionth
$10^{-15}$ femto f billiardth
$10^{-18}$ atto a trillionth
$10^{-21}$ zepto z trilliardth
$10^{-24}$ yocto y quadrillionth

#### Significance

Scientific notation is the most reliable way of expressing a number to a given number of significant figures. In scientific notation, the power of ten is insignificant. So writing data 5.0 x $10^{1}$ means data in 2 significant figures (i.e. a number between 4.9 x $10^{1}$ and 5.1 x $10^{1}$) ,while 5 x $10^{1}$ means one significant figure (i.e. a number between 4 x $10^{1}$ and 6 x $10^{1}$).

When handling significant figures in calculations, two rules are applied:

Multiplication and division: round the final result to the least number of significant figures of any one term $\frac{{(15.03)(4.87)}} {{1.987}} = 36.8$ The number is rounded to 3 significant figures, because least number of significant figures was found in the term, 4.87. The other terms, 15.03 and 1.987, each had 4 significant figures.

Addition and subtraction: round the final result to the least number of decimal places, regardless of the significant figures of any one term \frac{\begin{align} 1&.003 \\ 1&.2 \\ 1&.484 \\ \end{align} } {{3.687}} rounds off to 3.7 since the least number of decimal places found in the given terms was 1

Although final answers should be expressed with an appropriate number of digits of accuracy, you should still keep all the digits for intermediate computations.