meaning of the science

Science (from the Latin scientia, meaning "knowledge") is, in its broadest sense, any systematic knowledge that is capable of resulting in a correct prediction or reliable outcome. In this sense, science may refer to a highly skilled technique, technology, or practice.

In today's more restricted sense, science refers to a system of acquiring knowledge based on scientific method, and to the organized body of knowledge gained through such research. It is a "systematic enterprise of gathering knowledge about the world and organizing and condensing that knowledge into testable laws and theories". This article focuses upon science in this more restricted sense, sometimes called experimental science, and also gives some broader historical context leading up to the modern understanding of the word "science."

From the Middle Ages to the Enlightenment, "science" had more-or-less the same sort of very broad meaning in English that "philosophy" had at that time. By the early 1800s, "natural philosophy" (which eventually evolved into what is today called "natural science") had begun to separate from "philosophy" in general. In many cases, "science" continued to stand for reliable knowledge about any topic, in the same way it is still used in the broad sense in modern terms such as library science, political science, and computer science. In the more narrow sense of "science" today, as natural philosophy became linked to an expanding set of well-defined laws (beginning with Galileo's laws, Kepler's laws, and Newton's laws for motion), it became more common to refer to natural philosophy as "natural science". Over the course of the 1800s, the word "science" become increasingly associated mainly with the disciplined study of the natural world (that is, the non-human world). This sometimes left the study of human thought and society in a linguistic limbo, which has today been resolved by classifying these areas of study as the social sciences.

Basic classifications

Scientific fields are commonly divided into two major groups: natural sciences, which study natural phenomena (including biological life), and social sciences, which study human behavior and societies. These groupings are empirical sciences, which means the knowledge must be based on observable phenomena and capable of being tested for its validity by other researchers working under the same conditions. There are also related disciplines that are grouped into interdisciplinary and applied sciences, such as engineering and health science. Within these categories are specialized scientific fields that can include elements of other scientific disciplines but often possess their own terminology and body of expertise.

Mathematics, which is classified as a formal science, has both similarities and differences with the natural and social sciences. It is similar to empirical sciences in that it involves an objective, careful and systematic study of an area of knowledge; it is different because of its method of verifying its knowledge, using a priori rather than empirical methods. Formal science, which also includes statistics and logic, is vital to the empirical sciences. Major advances in formal science have often led to major advances in the empirical sciences. The formal sciences are essential in the formation of hypotheses, theories, and laws, both in discovering and describing how things work (natural sciences) and how people think and act (social sciences).

History and etymology

It is widely accepted that 'modern science' arose in the Europe of the 17th century (towards the end of the Renaissance), introducing a new understanding of the natural world. While empirical investigations of the natural world have been described since antiquity (for example, by Aristotle and Pliny the Elder), and scientific methods have been employed since the Middle Ages (for example, by Alhazen and Roger Bacon), the dawn of modern science is generally traced back to the early modern period during what is known as the Scientific Revolution of the 16th and 17th centuries.

The word "science" comes through the Old French, and is derived in turn from the Latin scientia, "knowledge", the nominal form of the verb scire, "to know". The Proto-Indo-European (PIE) root that yields scire is *skei-, meaning to "cut, separate, or discern". Similarly, the Greek word for science is 'επιστήμη', deriving from the verb 'επίσταμαι', 'to know'. From the Middle Ages to the Enlightenment, science or scientia meant any systematic recorded knowledge.Science therefore had the same sort of very broad meaning that philosophy had at that time. In other languages, including French, Spanish, Portuguese, and Italian, the word corresponding to science also carries this meaning.

Prior to the 1700s, the preferred term for the study of nature among English speakers was "natural philosophy", while other philosophical disciplines (e.g., logic, metaphysics, epistemology, ethics and aesthetics) were typically referred to as "moral philosophy". Today, "moral philosophy" is more-or-less synonymous with "ethics". Well into the 1700s, science and natural philosophy were not quite synonymous, but only became so later with the direct use of what would become known formally as the scientific method. By contrast, the word "science" in English was still used in the 17th century (1600s) to refer to the Aristotelian concept of knowledge which was secure enough to be used as a prescription for exactly how to accomplish a specific task. With respect to the transitional usage of the term "natural philosophy" in this period, the philosopher John Locke wrote disparagingly in 1690 that "natural philosophy is not capable of being made a science".

Locke's assertion notwithstanding, by the early 1800s natural philosophy had begun to separate from philosophy, though it often retained a very broad meaning. In many cases, science continued to stand for reliable knowledge about any topic, in the same way it is still used today in the broad sense (see the introduction to this article) in modern terms such as library science, political science, and computer science. In the more narrow sense of science, as natural philosophy became linked to an expanding set of well-defined laws (beginning with Galileo's laws, Kepler's laws, and Newton's laws for motion), it became more popular to refer to natural philosophy as natural science. Over the course of the nineteenth century, moreover, there was an increased tendency to associate science with study of the natural world (that is, the non-human world). This move sometimes left the study of human thought and society (what would come to be called social science) in a linguistic limbo by the end of the century and into the next.

Through the 1800s, many English speakers were increasingly differentiating science (i.e., the natural sciences) from all other forms of knowledge in a variety of ways. The now-familiar expression “scientific method,” which refers to the prescriptive part of how to make discoveries in natural philosophy, was almost unused until then, but became widespread after the 1870s, though there was rarely total agreement about just what it entailed. The word "scientist," meant to refer to a systematically working natural philosopher, (as opposed to an intuitive or empirically minded one) was coined in 1833 by William Whewell.Discussion of scientists as a special group of people who did science, even if their attributes were up for debate, grew in the last half of the 19th century. people actually meant by these terms at first, they ultimately depicted science, in the narrow sense of the habitual use of the scientific method and the knowledge derived from it, as something deeply distinguished from all other realms of human endeavor.

By the twentieth century (1900s), the modern notion of science as a special kind of knowledge about the world, practiced by a distinct group and pursued through a unique method, was essentially in place. It was used to give legitimacy to a variety of fields through such titles as "scientific" medicine, engineering, advertising, or motherhood. Over the 1900s, links between science and technology also grew increasingly strong. As Martin Rees explains, progress in scientific understanding and technology have been synergistic and vital to one another.

Richard Feynman described science in the following way for his students: "The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific 'truth'. But what is the source of knowledge? Where do the laws that are to be tested come from? Experiment, itself, helps to produce these laws, in the sense that it gives us hints. But also needed is imagination to create from these hints the great generalizations — to guess at the wonderful, simple, but very strange patterns beneath them all, and then to experiment to check again whether we have made the right guess." Feynman also observed, "...there is an expanding frontier of ignorance...things must be learned only to be unlearned again or, more likely, to be corrected."

Relationship to continuum mechanics

Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table.

Continuum mechanics the study of the physics of continuous materials	Solid mechanics: the study of the physics of continuous materials with a defined rest shape.	Elasticity: which describes materials that return to their rest shape after an applied stress.
		Plasticity: which describes materials that permanently deform after a large enough applied stress.	Rheology: the study of materials with both solid and fluid characteristics
	Fluid mechanics: the study of the physics of continuous materials which take the shape of their container.	Non-Newtonian fluids
		Newtonian fluids

In a mechanical view, a fluid is a substance that does not support shear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.

Assumptions

Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface.

Fluid mechanics assumes that every fluid obeys the following:

• Conservation of mass
• Conservation of momentum
• The continuum hypothesis, detailed below.

Further, it is often useful (and realistic) to assume a fluid is incompressible – that is, the density of the fluid does not change. Liquids can often be modelled as incompressible fluids, whereas gases cannot.

Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between the fluid and the boundary results also in a zero velocity for the fluid at the boundary. This is called the no-slip condition. For a porous media otherwise, in the frontier of the containing vessel, the slip condition is not zero velocity, and the fluid has a discontinuous velocity field between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition).

The continuum hypothesis

Main article: Continuum mechanics

Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.

The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.

Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above unity are best evaluated using statistical mechanics for reliable solutions.

Navier–Stokes equations

Main article: Navier–Stokes equations

The Navier–Stokes equations (named after Claude-Louis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier–Stokes equations describe the balance of forces acting at any given region of the fluid.

The Navier–Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change the variables of interest. For example, the Navier–Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

General form of the equation

The general form of the Navier–Stokes equations for the conservation of momentum is:

where

• is the fluid density,
• is the substantive derivative (also called the material derivative),
• is the velocity vector,
• is the body force vector, and
• is a tensor that represents the surface forces applied on a fluid particle (the comoving stress tensor).

Unless the fluid is made up of spinning degrees of freedom like vortices, is a symmetric tensor. In general, (in three dimensions) has the form:

where

• are normal stresses,
• are tangential stresses (shear stresses).

The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.

Newtonian versus non-Newtonian fluids

A Newtonian fluid (named after Isaac Newton) is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction). Important fluids, like water as well as most gases, behave — to good approximation — as a Newtonian fluid under normal conditions on Earth.^[2]

By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time – this behaviour is seen in materials such as pudding, oobleck, or sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property — for example, most fluids with long molecular chains can react in a non-Newtonian manner.^[2] Equations for a Newtonian fluid

Main article: Newtonian fluid

The constant of proportionality between the shear stress and the velocity gradient is known as the viscosity. A simple equation to describe Newtonian fluid behaviour is

where

τ is the shear stress exerted by the fluid ("drag")

μ is the fluid viscosity – a constant of proportionality

is the velocity gradient perpendicular to the direction of shear

For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure, not on the forces acting upon it. If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is

where

τ_ij is the shear stress on the i^th face of a fluid element in the j^th direction

v_i is the velocity in the i^th direction

x_j is the j^th direction coordinate

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.

Among fluids, two rough broad divisions can be made: ideal and non-ideal fluids. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. An Ideal fluid is non viscous- offers no resistance whatsoever to a shearing force.

One can group real fluids into Newtonian and non-Newtonian. Newtonian fluids agree with Newton's law of viscosity. Non-Newtonian fluids can be either plastic, bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelatic.

Motion (physics)

In physics, motion is change of location or position of an object with respect to time. Change in motion is the result of an applied force. Motion is typically described in terms of velocity also seen as speed, acceleration, displacement, and time.^[1] An object's velocity cannot change unless it is acted upon by a force, as described by Newton's first law also known as Inertia. An object's momentum is directly related to the object's mass and velocity, and the total momentum of all objects in a closed system (one not affected by external forces) does not change with time, as described by the law of conservation of momentum.

A body which does not move is said to be at rest, motionless, immobile, stationary, or to have

Motion involves change in position, such as in this perspective of rapidly leaving Yongsan Station

constant (time-invariant) position.

Motion is always observed and measured relative to a frame of reference. As there is no absolute reference frame, absolute motion cannot be determined; this is emphasised by the term relative motion.^[2] A body which is motionless relative to a given reference frame, moves relative to infinitely many other frames. Thus, everything in the universe is moving.^[3]

More generally, the term motion signifies any spatial and/or temporal change in a physical system. For example, one can talk about motion of a wave or a quantum particle (or any other field) where the concept location does not apply.

Laws of Motion

Main article: Mechanics

In physics, motion in the universe is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe (such as projectiles, planets, cells, and humans) are described by classical mechanics. Whereas the motion of very small atomic and sub-atomic sized objects is described by quantum mechanics.

Classical mechanics

Classical mechanics

Newton's Second Law

History of ... Branches Statics · Dynamics / Kinetics · Kinematics · Applied mechanics · Celestial mechanics · Continuum mechanics · Statistical mechanics Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics: Lagrangian mechanics Hamiltonian mechanics Fundamental concepts Space · Time · Velocity · Speed · Mass · Acceleration · Gravity · Force · Torque / Moment / Couple · Momentum · Angular momentum · Inertia · Moment of inertia · Reference frame · Energy · Kinetic energy · Potential energy · Mechanical work · Virtual work · D'Alembert's principle [show]Core topics Rigid body · Rigid body dynamics · Motion · Newton's laws of motion · Newton's law of universal gravitation · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement [show]Scientists Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson

v • d • e

Main article: Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science, engineering and technology.

Classical mechanics is fundamentally based on Newton's Laws of Motion. These laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 5, 1687. His three laws are:

1. In the absence of a net external force, a body either is at rest or moves with constant velocity.
2. The net external force on a body is equal to the mass of that body times its acceleration; F = ma. Alternatively, force is proportional to the time derivative of momentum.
3. Whenever a first body exerts a force F on a second body, the second body exerts a force −F on the first body. F and −F are equal in magnitude and opposite in direction.^[4]

Newton's three laws of motion, along with his law of universal gravitation, explain Kepler's laws of planetary motion, which were the first to accurately provide a mathematical model or understanding orbiting bodies in outer space. This explanation unified the motion of celestial bodies and motion of objects on earth.

Classical mechanics was later further enhanced by Albert Einstein's special relativity and general relativity. Special relativity explains the motion of objects with a high velocity, approaching the speed of light; general relativity is employed to handle gravitation motion at a deeper level.

Quantum mechanics

Main article: Quantum mechanics

Quantum mechanics is a set of principles describing physical reality at the atomic level of matter (molecules and atoms) and the subatomic (electrons, protons, and even smaller particles). These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation energy, this described in the wave–particle duality.

In contrast to classical mechanics, where accurate measurements and predictions can be calculated about location and velocity, in the quantum mechanics of a subatomic particle, one can never specify its state, such as its simultaneous location and velocity, with complete certainty (this is called the Heisenberg uncertainty principle).

In addition to describing the motion of atomic level phenomenon, quantum mechanics is useful in understanding some large scale phenomenon such as superfluidity, superconductivity, and biological systems, including the function of smell receptors and the structures of proteins.

List of "imperceptible" human motions

Humans, like all things in the universe are in constant motion,^[5] however, aside from obvious movements of the various external body parts and locomotion, humans are in motion in a variety of ways which are more difficult to perceive. Many of these "imperceptible motions" are only perceivable with the help of special tools and careful observation. The larger scales of "imperceptible motions" are difficult for humans to perceive for two reasons: 1) Newton's laws of motion (particularly Inertia) which prevent humans from feeling motions of a mass to which they are connected, and 2) the lack of an obvious frame of reference which would allow individuals to easily see that they are moving.^[6] The smaller scales of these motions are too small for humans to sense.

Universe

• Spacetime (the fabric of the universe) is actually expanding. Essentially, everything in the universe is stretching like a rubber band. This motion is the most obscure as it is not physical motion as such, but rather a change in the very nature of the universe. The primary source of verification of this expansion was provided by Edwin Hubble who demonstrated that all galaxies and distant astronomical objects were moving away from us ("Hubble's law") as predicted by a universal expansion.^[7]

Galaxy

• The Milky Way Galaxy, is hurtling through space at an incredible speed. It is powered by the force left over from the Big Bang. Many astronomers believe the Milky Way is moving at approximately 600 km/s relative to the observed locations of other nearby galaxies. Another reference frame is provided by the Cosmic microwave background. This frame of reference indicates that The Milky Way is moving at around 552 km/s.^[8]

Solar System

• The Milky Way is rotating around its dense galactic center, thus the solar system is moving in a circle within the galaxy's gravity. Away from the central bulge or outer rim, the typical stellar velocity is between 210 and 240 km/s (or about a half-million mi/h).^[9]

Earth

• The Earth is rotating or spinning around its axis, this is evidenced by day and night, at the equator the earth has an eastward velocity of 0.4651 km/s (or 1040 mi/h).^[10]
• The Earth is orbiting around the Sun in an orbital revolution. A complete orbit around the sun takes one year or about 365 days; it averages a speed of about 30 km/s (or 67,000 mi/h).^[11]

Continents

• The Theory of Plate tectonics tells us that the continents are drifting on convection currents within the mantle causing them to move across the surface of the planet at the slow speed of approximately 1 inch (2.54 cm) per year.^[12][13] However, the velocities of plates range widely. The fastest-moving plates are the oceanic plates, with the Cocos Plate advancing at a rate of 75 mm/yr^[14] (3.0 in/yr) and the Pacific Plate moving 52–69 mm/yr (2.1–2.7 in/yr). At the other extreme, the slowest-moving plate is the Eurasian Plate, progressing at a typical rate of about 21 mm/yr (0.8 in/yr).

Internal body

• The human heart is constantly contracting to move blood throughout the body. Through larger veins and arteries in the body blood has been found to travel at approximately 0.33 m/s.^[15] Though considerable variation exists, and peak flows in the venae cavae have been found to range between 0.1 m/s and 0.45 m/s.^[16]
• The smooth muscles of hollow internal organs are moving. The most familiar would be peristalsis which is where digested food is forced throughout the digestive tract. Though different foods travel through the body at rates, an average speed through the human small intestine is 2.16 m/h or 0.036 m/s.^[17]
• Typically some sound is audible at any given moment, when the vibration of these sound waves reaches the ear drum it moves in response and allows the sense of hearing.
• The human lymphatic system is constantly moving excess fluids, lipids, and immune system related products around the body. The lymph fluid has been found to move through a lymph capillary of the skin at approximately 0.0000097 m/s.^[18]

Cells

The cells of the human body have many structures which move throughout them.

• Cytoplasmic streaming is a way which cells move molecular substances throughout the cytoplasm.^[19]
• Various motor proteins work as molecular motors within a cell and move along the surface of various cellular substrates such as microtubules. Motor proteins are typically powered by the hydrolysis of adenosine triphosphate, (ATP), and convert chemical energy into mechanical work.^[20] Vesicles propelled by motor proteins have been found to have a velocity of approximately 0.00000152 m/s.^[21]

Particles

• According to the laws of thermodynamics all particles of matter are in constant random motion as long as the temperature is above absolute zero. Thus the molecules and atoms which make up the human body are vibrating, colliding, and moving. This motion can be detected as temperature; high temperatures (which represent greater kinetic energy in the particles) feel warmer to humans, whereas lower temperatures feel colder.^[22]

Subatomic particles

• Within each atom the electrons are speeding around the nucleus so fast that they are not actually in one location, but rather smeared across a region of the electron cloud. Electrons have a high velocity, and the larger the nucleus they are orbiting the faster they move. In a hydrogen atom, electrons have been calculated to be orbiting at a speed of approximately 2,420,000 m/s^[23]
• Inside the atomic nucleus the protons and neutrons are also probably moving around due the electrical repulsion of the protons and the presence of angular momentum of both particles

Physical Measurements

• Reproducibility
  All measurements that are made to the limits of the measuring device necessarily involve some estimation of the final digit in the value for the measured quantity. If this doesn't appear to be the case as in some instruments with digital outputs, it is because the meter is rounding for you and the true limit of "precision" for the instrument is being obscured by this rounding. In any case, reproducibility in measurements is a critical aspect of the measurements and often is reported. In engineering or production this is commonly referred to as "the tolerance." For example, furniture construction might be done to a tolerance of 1/64^th of an inch while machining of an engine component might be done to a tolerance of 1/10,000^th of an inch. In science, reproducibility is referred to as such or as "the uncertainty." The word "tolerance" is not used in the context of scientific measurement but means the same thing.

• Stating Results
  The following are some terms that are frequently used in reporting and comparing results: In some experiments, a physical measurement will be made and this "experimental" value will be compared to a value that is widely accepted by other scientists. This accepted value is usually called the "accepted" or "true" value. These values will usually have uncertainties associated with them, but the uncertainty will not always be stated. If it is not stated, it is implied that the last digit is the uncertain digit. For example, if the density of aluminum is given to be 2.698 g/cm³, it is implied that the uncertainty is .001 g/cm³. The experimental value may also be compared with a theoretical or predicted value that is based on hypothesis or theory.
  • Accuracy
    The accuracy of an experiment is the measurement of how close the result is to the accepted or predicted value. This can be stated in several ways.

    

    

  • Precision
    The Precision is a measurement of how reproducible or how well the result of an experiment is known. The precision of the measurement is referred to as the "uncertainty" and has the same units as the measured value. The result of an experiment would be stated as:

    

    The "uncertainty" can be expressed as an "absolute" uncertainty or a "relative" uncertainty and frequently is seen both ways. For example, suppose the result of a length measurement using a meter stick is 14.7 cm. And, further, suppose that as a result of the way the meterstick was marked, it was possible to estimate that the uncertainty in this value was 0.1 cm, then the result of the measurement would be reported as follows:

    

    The "± 0.1 cm" is the uncertainty in this measurement and it is an "absolute" uncertainty. Often, it is more useful to have the "relative" uncertainty expressed because this states how big the uncertainty is compared to the quantity being measured.

    

    So, in this example, the relative uncertainty is:

    

    Please notice that "precision" and "accuracy" are not the same thing! It is possible to have a very precise measurement that is has a very large "error" because something was wrong with the measuring device or the person using it was not using it properly.

    Random uncertainties: An example of something that naturally varies is the number of apples on a tree. Suppose that there is an orchard consisting of dwarf apple trees that are as uniform as possible. Even though every effort is made to have these trees be uniform there will be natural, random variation in the number of apples that will mature on each that can have a basis in anything from pollination to insect infestation. The total number of apples on any tree can be counted exactly, but the number varies from tree to tree. It might be very helpful to have a representative number for the number of apples on a tree in this orchard. What should that number be? In fact, a number is really not the answer; we probably ought to have a range so that we not only know about how many apples to expect per tree but we also have a good measure of just how variable this can be. In-other-words, the answer to the question will be the result as expressed above and again here:

    

    In this example the "measured value" would be the AVERAGE of a number of sample counts. The "uncertainty" is usually given by the "sample standard deviation". This quantity is often represented by a lower case Greek letter sigma, s_n-1, with the n-1 indicating that this is the "sample" rather than "population" value. When you have a choice with your calculator, use the s_n-1 function. On TI-8X series calculators this function is represented by "Sx", and "s" is reserved for the population standard deviation. No attempt will be made here to describe how to compute the sample standard deviation if your calculator will not do it for you. The equation that follows is presented so that there is no confusion about what value is expected in our laboratory work.

    

    (Where "N" is the number of samples, X_i is a particular sample, "i" represents the position in the list of the results for that sample and X is the average of the individual samples.)

    The significance of the standard deviation as a measure of uncertainty is that the range it describes around the average includes a predictable number of the samples. So, for example, if our sample of trees in the orchard gives as a result of our counting and calculation:

    

    The "147" is the average of the number of apples counted on the trees in our sample and the "9" is the sample standard deviation. So the RANGE, 138 ® 156, contains 68% of the values used to calculate the average. Another interpretation of this range is that there are 68 chances out of 100 of obtaining a value in this range if one were to count another tree in the orchard. For our purposes in this course, we are going to be a little casual about this 68% and simply refer to this as approximately equivalent to 2/3 or two chances out of three.

    If, instead of counting, one is using a tool to make some measurement, then the separate results of repeated measurements are averaged and the sample standard deviation calculated in the same way.

    Systematic error: This is the result of the measuring device having a built in error - or - the person using it, not being aware of how to use the device properly. This can be something as simple as forgetting to "tare" (set to zero, usually) the electronic scales or it might be the result of using a cheap ruler on which the inscribed distances are, say, 2.3% too short. These kinds of errors are very hard to detect sometimes. They don't always have an effect on what one is trying to discover, but we often "calibrate" equipment to test for the presence of systematic error, because they can lead to serious problems if undiscovered.

Physical optics

In physics, physical optics, or wave optics, is the branch of optics which studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effects such as quantum noise in optical communication, which is studied in the sub-branch of coherence theory.

The physical optics approximation

Physical optics is also the name of an approximation commonly used in optics, electrical engineering and applied physics. In this context, it is an intermediate method between geometric optics, which ignores wave effects, and full wave electromagnetism, which is a precise theory. The word "physical" means that it is more physical than geometric or ray optics and not that it is an exact physical theory.

This approximation consists of using ray optics to estimate the field on a surface and then integrating that field over the surface to calculate the transmitted or scattered field. This resembles the Born approximation, in that the details of the problem are treated as a perturbation.

In optics, it is a standard way of estimating diffraction effects. In radio, this approximation is used to estimate some effects that resemble optical effects. It models several interference, diffraction and polarization effects but not the dependence of diffraction on polarization. Since it is a high frequency approximation, it is often more accurate in optics than for radio.

In optics, it typically consists of integrating ray estimated field over a lens, mirror or aperture to calculate the transmitted or scattered field.

In radar scattering it usually means taking the current that would be found on a tangent plane of similar material as the current at each point on the front, i. e. the geometrically illuminated part, of a scatterer. Current on the shadowed parts is taken as zero. The approximate scattered field is then obtained by an integral over these approximate currents. This is useful for bodies with large smooth convex shapes and for lossy (low reflection) surfaces.

The ray optics field or current is generally not accurate near edges or shadow boundaries, unless supplemented by diffraction and creeping wave calculations.

The theory of physical optics has some defects in the evaluation of the scattered fields.^[1] For example the diffracted fields, which are created by the edge discontinuities, are obtained by the edge point contributions of the physical optics integrals. Y. Z. Umul has proposed a modified theory that leads to exact solutions to wave diffraction problems

Wave–particle duality

In physics and chemistry, wave–particle duality is the concept that all matter exhibits both wave-like and particle-like properties. Being a central concept of quantum mechanics, this duality addresses the inadequacy of classical concepts like "particle" and "wave" in fully describing the behavior of quantum-scale objects. Orthodox interpretations of quantum mechanics explain this ostensible paradox as a fundamental property of the Universe, while alternative interpretations explain the duality as an emergent, second-order consequence of various limitations of the observer. This treatment focuses on explaining the behavior from the perspective of the widely used Copenhagen interpretation, in which wave–particle duality is one aspect of the concept of complementarity, that a phenomenon can be viewed in one way or in another, but not both simultaneously.

The idea of duality originated in a debate over the nature of light and matter dating back to the 1600s, when competing theories of light were proposed by Christiaan Huygens and Isaac Newton: light was thought either to consist of waves (Huygens) or of corpuscles [particles] (Newton). Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles also have a wave nature (and vice versa).^[1] This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. In fact, according to traditional formulations of non-relativistic quantum mechanics, wave–particle duality applies to all objects, even macroscopic ones; but because of their small wavelengths, the wave properties of macroscopic objects cannot be detected

Brief history of wave and particle viewpoints

Aristotle was one of the first to publicly hypothesize as to the nature of light, proposing that it was a disturbance in the element air (hence it was a wave-like phenomenon). On the other hand, Democritus – the original atomist – argued that all things in the universe, including light, were composed of indivisible sub-components (light being some form of solar atom).^[3] At the beginning of the 11th century, the Arabic scientist Alhazen wrote the first comprehensive treatise on optics; describing refraction, reflection, and the operation of a pinhole lens via rays of light traveling from the point of emission to the eye. He asserted that these rays were composed of particles of light. In 1630, René Descartes popularized and accredited in the West the opposing wave description in his treatise on light, showing that the behavior of light could be re-created by modeling wave-like disturbances in his universal medium (plenum). Beginning in 1670 and progressing over three decades, Isaac Newton developed and championed his corpuscular hypothesis, arguing that the perfectly straight lines of reflection demonstrated light's particle nature; only particles could travel in such straight lines. He explained refraction by positing that particles of light accelerated laterally upon entering a denser medium. Around the same time, Newton's contemporaries – Robert Hooke, Christian Huygens, and Augustin-Jean Fresnel – mathematically refined the wave viewpoint, showing that if light traveled at different speeds in different media (such as water and air), refraction could be easily explained as the medium-dependent propagation of light waves. The resulting Huygens–Fresnel principle was extremely successful at reproducing light's behavior and, subsequently supported by Thomas Young's discovery of double-slit interference, effectively disbanded the particle light camp.^[4]

Thomas Young's sketch of two-slit diffraction of waves, 1803.

The final blow against corpuscular theory came when James Clerk Maxwell discovered that he could combine four simple equations, which had been previously discovered, along with a slight modification to describe self propogating waves of oscillating electric and magnetic fields. When the propagation speed of these electromagnetic waves was calculated, the speed of light fell out. It quickly became apparent that visible light, ultraviolet light, and infrared light (phenomenon thought previously to be unrelated) were all electromagnetic waves of differing frequency. The wave theory had prevailed – or at least it seemed.

While the 19th century had seen the success of the wave theory at describing light, it had also witnessed the rise of the atomic theory at describing matter. In 1789, Antoine Lavoisier securely differentiated chemistry from alchemy by introducing rigor and precision into his laboratory techniques; allowing him to deduce the conservation of mass and categorize many new chemical elements and compounds. However, the nature of these essential chemical elements remained unknown. In 1799, Joseph Louis Proust advanced chemistry towards the atom by showing that elements combined in definite proportions. This led John Dalton to resurrect Democritus' atom in 1803, when he proposed that elements were invisible sub components; which explained why the varying oxides of metals (e.g. stannous oxide and cassiterite, SnO and SnO₂ respectively) possess a 1:2 ratio of oxygen to one another. But Dalton and other chemists of the time had not considered that some elements occur in monatomic form (like Helium) and others in diatomic form (like Hydrogen), or that water was H₂O, not the simpler and more intuitive HO – thus the atomic weights presented at the time were varied and often incorrect. Additionally, the formation of HO by two parts of hydrogen gas and one part of oxygen gas would require an atom of oxygen to split in half (or two half-atoms of hydrogen to come together). This problem was solved by Amedeo Avogadro, who studied the reacting volumes of gases as they formed liquids and solids. By postulating that equal volumes of elemental gas contain an equal number of atoms, he was able to show that H₂O was formed from two parts H₂ and one part O₂. By discovering diatomic gases, Avogadro completed the basic atomic theory, allowing the correct molecular formulas of most known compounds – as well as the correct weights of atoms – to be deduced and categorized in a consistent manner. The final stroke in classical atomic theory came when Dimitri Mendeleev saw an order in recurring chemical properties, and created a table presenting the elements in unprecedented order and symmetry. But there were holes in Mendeleev's table, with no element to fill them in. His critics initially cited this as a fatal flaw, but were silenced when new elements were discovered that perfectly fit into these holes. The success of the periodic table effectively converted any remaining opposition to atomic theory; even though no single atom had ever been observed in the laboratory, chemistry was now an atomic science.

The turn of the century and the paradigm shift

Particles of electricity?

At the close of the 19th century, the reductionism of atomic theory began to advance into the atom itself; determining, through physics, the nature of the atom and the operation of chemical reactions. Electricity, first thought to be a fluid, was now understood to consist of particles called electrons. This was first demonstrated by J. J. Thomson in 1897 when, using a cathode ray tube, he found that an electrical charge would travel across a vacuum (which would possess infinite resistance in classical theory). Since the vacuum offered no medium for an electric fluid to travel, this discovery could only be explained via a particle carrying a negative charge and moving through the vacuum. This electron flew in the face of classical electrodynamics, which had successfully treated electricity as a fluid for many years (leading to the invention of batteries, electric motors, dynamos, and arc lamps). More importantly, the intimate relation between electric charge and electromagnetism had been well documented following the discoveries of Michael Faraday and Clerk Maxwell. Since electromagnetism was known to be a wave generated by a changing electric or magnetic field (a continuous, wave-like entity itself) an atomic/particle description of electricity and charge was a non sequitur. And classical electrodynamics was not the only classical theory rendered incomplete.

Radiation quantization

Black-body radiation, the emission of electromagnetic energy due to an object's heat, could not be explained from classical arguments alone. The equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned equally among the object's vibrational modes. This worked well when describing thermal objects, whose vibrational modes were defined as the speeds of their constituents atoms, and the speed distribution derived from egalitarian partitioning of these vibrational modes closely matched experimental results. Speeds much higher than the average speed were suppressed by the fact that kinetic energy is quadratic – doubling the speed requires four times the energy – thus the number of atoms occupying high energy modes (high speeds) quickly drops off because the constant, equal partition can excite successively fewer atoms. Low speed modes would ostensibly dominate the distribution, since low speed modes would require ever less energy, and prima facie a zero-speed mode would require zero energy and its energy partition would contain an infinite number of atoms. But this would only occur in the absence of atomic interaction; when collisions are allowed, the low speed modes are immediately suppressed by jostling from the higher energy atoms, exciting them to higher energy modes. An equilibrium is swiftly reached where most atoms occupy a speed proportional to the temperature of the object (thus defining temperature as the average kinetic energy of the object).

But applying the same reasoning to the electromagnetic emission of such a thermal object was not so successful. It had been long known that thermal objects emit light. Hot metal glows red, and upon further heating white (this is the underlying principle of the incandescent bulb). Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws. This became known as the black body problem. Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was trivial to assume that it would perform equally well in describing the radiative emission of such objects. But a problem quickly arose when determining the vibrational modes of light. To simplify the problem (by limiting the vibrational modes) a lowest allowable wavelength was defined by placing the thermal object in a cavity. Any electromagnetic mode at equilibrium (i.e. any standing wave) could only exist if it used the walls of the cavities as nodes. Thus there were no waves/modes with a wavelength larger than twice the length (L) of the cavity.

Standing waves in a cavity

The first few allowable modes would therefore have wavelengths of : 2L, L, 2L/3, L/2, etc. (each successive wavelength adding one node to the wave). However, while the wavelength could never exceed 2l, there was no such limit on decreasing the wavelength, and adding nodes to reduce the wavelength could proceed ad infinitum. Suddenly it became apparent that the short wavelength modes completely dominated the distribution, since ever shorter wavelength modes could be crammed into the cavity. If each mode received an equal partition of energy, the short wavelength modes would consume all the energy. This became clear when plotting the Rayleigh–Jeans law which, while correctly predicting the intensity of long wavelength emissions, predicted infinite total energy as the intensity diverges to infinity for short wavelengths. This became known as the ultraviolet catastrophe.

The solution arrived in 1900 when Max Planck hypothesized that the frequency of light emitted by the black body depended on the frequency of the oscillator that emitted it, and the energy of these oscillators increased linearly with frequency (according to his constant h, where E = hν). This was not an unsound proposal considering that macroscopic oscillators operate similarly: when studying five simple harmonic oscillators of equal amplitude but different frequency, the oscillator with the highest frequency possesses the highest energy (though this relationship is not linear like Planck's). By demanding that high-frequency light must be emitted by an oscillator of equal frequency, and further requiring that this oscillator occupy higher energy than one of a lesser frequency, Planck avoided any catastrophe; giving an equal partition to high-frequency oscillators produced successively fewer oscillators and less emitted light. And as in the Maxwell–Boltzmann distribution, the low-frequency, low-energy oscillators were suppressed by the onslaught of thermal jiggling from higher energy oscillators, which necessarily increased their energy and frequency.

The most revolutionary aspect of Planck's treatment of the black body is that it inherently relies on an integer number of oscillators in thermal equilibrium with the electromagnetic field. These oscillators give their entire energy to the electromagnetic field, creating a quantum of light, as often as they are excited by the electromagnetic field, absorbing a quantum of light and beginning to oscillate at the corresponding frequency. Planck had intentionally created an atomic theory of the black body, but had unintentionally generated an atomic theory of light, where the black body never generates quanta of light at a given frequency with an energy less than hν. However, once realizing that he had quantized the electromagnetic field, he denounced particles of light as a limitation of his approximation, not a property of reality. Never to accept quantum theory, even a genius like Planck could not live with wave–particle duality.

The photoelectric effect illuminated

Yet while Planck had solved the ultraviolet catastrophe by using atoms and a quantized electromagnetic field, most physicists immediately agreed that Planck's "light quanta" were unavoidable flaws in his model. A more complete derivation of black body radiation would produce a fully continuous, fully wave-like electromagnetic field with no quantization. However, in 1905 Albert Einstein took Planck's black body model in itself and saw a wonderful solution to another outstanding problem of the day: the photoelectric effect. Ever since the discovery of electrons eight years previously, electrons had been the thing to study in physics laboratories worldwide. Nikola Tesla discovered in 1901 that when a metal was illuminated by high-frequency light (e.g. ultraviolet light), electrons were ejected from the metal at high energy. This work was based on the previous knowledge that light incident upon metals produces a current, but Tesla was the first to describe it as a particle phenomenon.

The following year, Philipp Lenard discovered that (within the range of the experimental parameters he was using) the energy of these ejected electrons did not depend on the intensity of the incoming light, but on its frequency. So if one shines a little low-frequency light upon a metal, a few low energy electrons are ejected. If one now shines a very intense beam of low-frequency light upon the same metal, a whole slew of electrons are ejected; however they possess the same low energy, there are merely more of them. In order to get high energy electrons, one must illuminate the metal with high-frequency light. The more light there is, the more electrons are ejected. Like blackbody radiation, this was at odds with a theory invoking continuous transfer of energy between radiation and matter. However, it can still be explained using a fully classical description of light, as long as matter is quantum mechanical in nature^[5]

If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum hν, then the photoelectric effect could be explained very simply. Low-frequency light only ejects low-energy electrons because each electron is excited by the absorption of a single photon. Increasing the intensity of the low-frequency light (increasing the number of photons) only increases the number of excited electrons, not their energy, because the energy of each photon remains low. Only by increasing the frequency of the light, and thus increasing the energy of the photons, can one eject electrons with higher energy. Thus, using Planck's constant h to determine the energy of the photons based upon their frequency, the energy of ejected electrons should also increase linearly with frequency; the slope of the line being Planck's constant. These results were not confirmed until 1915, when Robert Andrews Millikan, who had previously determined the charge of the electron, produced experimental results in perfect accord with Einstein's predictions. While the energy of ejected electrons reflected Planck's constant, the existence of photons was not explicitly proven until the discovery of the anti-bunching effect, of which a modern experiment can be performed in undergraduate-level labs^[6]. This phenomenon could only be explained via photons, and not through any semi-classical theory (which could alternatively explain the photoelectric effect). When Einstein received his Nobel Prize in 1921, it was not for his more difficult and mathematically laborious special and general relativity, but for the simple, yet totally revolutionary, suggestion of quantized light. While Einstein's "light quanta" would not be called photons until 1925, but even in 1905 they represented the quintessential example of wave–particle duality. Waves of electromagnetic radiation can only exist as discrete elements, thus acting as a wave and a particle simultaneously.

Developmental milestones

Huygens and Newton

The earliest comprehensive theory of light was advanced by Christiaan Huygens, who proposed a wave theory of light, and in particular demonstrated how waves might interfere to form a wavefront, propagating in a straight line. However, the theory had difficulties in other matters, and was soon overshadowed by Isaac Newton's corpuscular theory of light. That is, Newton proposed that light consisted of small particles, with which he could easily explain the phenomenon of reflection. With considerably more difficulty, he could also explain refraction through a lens, and the splitting of sunlight into a rainbow by a prism. Newton's particle viewpoint went essentially unchallenged for over a century.^[7]

Young, Fresnel, and Maxwell

In the early 1800s, the double-slit experiments by Young and Fresnel provided evidence for Huygens' wave theories. The double-slit experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. The wave view did not immediately displace the ray and particle view, but began to dominate scientific thinking about light in the mid 1800s, since it could explain polarization phenomena that the alternatives could not.^[8]

In the late 1800s, James Clerk Maxwell explained light as the propagation of electromagnetic waves according to the Maxwell equations. These equations were verified by experiment by Heinrich Hertz in 1887, and the wave theory became widely accepted.

Planck's formula for black-body radiation

Main article: Planck's law

In 1901, Max Planck published an analysis that succeeded in reproducing the observed spectrum of light emitted by a glowing object. To accomplish this, Planck had to make an ad hoc mathematical assumption of quantized energy of the oscillators (atoms of the black body) that emit radiation. It was Einstein who later proposed that it is the electromagnetic radiation itself that is quantized, and not the energy of radiating atoms.

Einstein's explanation of the photoelectric effect

Main article: Photoelectric effect

The photoelectric effect. Incoming photons on the left strike a metal plate (bottom), and eject electrons, depicted as flying off to the right.

In 1905, Albert Einstein provided an explanation of the photoelectric effect, a hitherto troubling experiment that the wave theory of light seemed incapable of explaining. He did so by postulating the existence of photons, quanta of light energy with particulate qualities.

In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit. Presumably, the light was knocking electrons out of the metal, causing current to flow. However, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light available with the technology of the time caused no current at all. According to the classical theory of light and matter, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. Yet, oddly, this was not so.

Einstein explained this conundrum by postulating that the electrons can receive energy from electromagnetic field only in discrete portions (quanta that were called photons): an amount of energy E that was related to the frequency f of the light by

where h is Planck's constant (6.626 × 10⁻³⁴ J seconds). Only photons of a high enough frequency (above a certain threshold value) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. More intense light above the threshold frequency could release more electrons, but no amount of light (using technology available at the time) below the threshold frequency could release an electron. To "violate" this law would require extremely high intensity lasers which had not yet been invented. Intensity-dependent phenomena have now been studied in detail with such lasers ^[9]

Einstein was awarded the Nobel Prize in Physics in 1921 for his discovery of the law of the photoelectric effect.

De Broglie's wavelength

Main article: Matter wave

In 1924, Louis-Victor de Broglie formulated the de Broglie hypothesis, claiming that all matter,^[10][11] not just light, has a wave-like nature; he related wavelength (denoted as λ), and momentum (denoted as p):

This is a generalization of Einstein's equation above, since the momentum of a photon is given by p = and the wavelength by λ = , where c is the speed of light in vacuum.

De Broglie's formula was confirmed three years later for electrons (which differ from photons in having a rest mass) with the observation of electron diffraction in two independent experiments. At the University of Aberdeen, George Paget Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns. At Bell Labs Clinton Joseph Davisson and Lester Halbert Germer guided their beam through a crystalline grid.

De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis. Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.

Heisenberg's uncertainty principle

Main article: Heisenberg uncertainty principle

In his work on formulating quantum mechanics, Werner Heisenberg postulated his uncertainty principle, which states:

where

Δ here indicates standard deviation, a measure of spread or uncertainty;

x and p are a particle's position and linear momentum respectively.

is the reduced Planck's constant (Planck's constant divided by 2π).

Heisenberg originally explained this as a consequence of the process of measuring: Measuring position accurately would disturb momentum and vice-versa, offering an example (the "gamma-ray microscope") that depended crucially on the de Broglie hypothesis. It is now thought, however, that this only partly explains the phenomenon, but that the uncertainty also exists in the particle itself, even before the measurement is made.

In fact, the modern explanation of the uncertainty principle, extending the Copenhagen interpretation first put forward by Bohr and Heisenberg, depends even more centrally on the wave nature of a particle: Just as it is nonsensical to discuss the precise location of a wave on a string, particles do not have perfectly precise positions; likewise, just as it is nonsensical to discuss the wavelength of a "pulse" wave traveling down a string, particles do not have perfectly precise momenta (which corresponds to the inverse of wavelength). Moreover, when position is relatively well defined, the wave is pulse-like and has a very ill-defined wavelength (and thus momentum). And conversely, when momentum (and thus wavelength) is relatively well defined, the wave looks long and sinusoidal, and therefore it has a very ill-defined position.

De Broglie himself had proposed a pilot wave construct to explain the observed wave–particle duality. In this view, each particle has a well-defined position and momentum, but is guided by a wave function derived from Schrödinger's equation. The pilot wave theory was initially rejected because it generated non-local effects when applied to systems involving more than one particle. Non-locality, however, soon became established as an integral feature of quantum theory (see EPR paradox), and David Bohm extended de Broglie's model to explicitly include it. In Bohmian mechanics,^[12] the wave–particle duality is not a property of matter itself, but an appearance generated by the particle's motion subject to a guiding equation or quantum potential.

Wave behavior of large objects

Since the demonstrations of wave-like properties in photons and electrons, similar experiments have been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in 1929.^[13] Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.

A dramatic series of experiments emphasizing the action of gravity in relation to wave–particle duality were conducted in the 1970s using the neutron interferometer^[14]. Neutrons, one of the components of the atomic nucleus, provide much of the mass of a nucleus and thus of ordinary matter. In the neutron interferometer, they act as quantum-mechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything, including light (see tests of general relativity and the Pound-Rebka falling photon experiment), the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.

In 1999, the diffraction of C₆₀ fullerenes by researchers from the University of Vienna was reported.^[15] Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength is 2.5 pm, whereas the diameter of the molecule is about 1 nm, about 400 times larger. As of 2005, this is the largest object for which quantum-mechanical wave-like properties have been directly observed in far-field diffraction.

In 2003 the Vienna group also demonstrated the wave nature of tetraphenylporphyrin^[16]—a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a near-field Talbot Lau interferometer.^[17][18] In the same interferometer they also found interference fringes for C₆₀F_48., a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms.^[16] Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e. to certain decoherence mechanisms.^[19][20]

Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle's Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius, a scale at which current theories of physics may break down or need to be replaced by more general ones.^[21]

Treatment in modern quantum mechanics

Wave–particle duality is deeply embedded into the foundations of quantum mechanics, so well that modern practitioners rarely discuss it as such. In the formalism of the theory, all the information about a particle is encoded in its wave function, a complex valued function roughly analogous to the amplitude of a wave at each point in space. This function evolves according to a differential equation (generically called the Schrödinger equation), and this equation gives rise to wave-like phenomena such as interference and diffraction.

The particle-like behavior is most evident due to phenomena associated with measurement in quantum mechanics. Upon measuring the location of the particle, the wave-function will randomly "collapse" to a sharply peaked function at some location, with the likelihood of any particular location equal to the squared amplitude of the wave-function there. The measurement will return a well-defined position, a property traditionally associated with particles.

Although this picture is somewhat simplified (to the non-relativistic case), it is adequate to capture the essence of current thinking on the phenomena historically called "wave–particle duality". (See also: Mathematical formulation of quantum mechanics.)

Alternative views

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Particle-only view

The pilot wave model, originally developed by Louis de Broglie and further developed by David Bohm into the hidden variable theory proposes that there is no duality, but rather particles are guided, in a deterministic fashion, by a pilot wave (or "quantum potential") which will direct them to areas of constructive interference in preference to areas of destructive interference. This idea is held by a significant minority within the physics community.^[22]

At least one physicist considers the “wave-duality” a misnomer, as L. Ballentine, Quantum Mechanics, A Modern Development, p. 4, explains:

When first discovered, particle diffraction was a source of great puzzlement. Are "particles" really "waves?" In the early experiments, the diffraction patterns were detected holistically by means of a photographic plate, which could not detect individual particles. As a result, the notion grew that particle and wave properties were mutually incompatible, or complementary, in the sense that different measurement apparatuses would be required to observe them. That idea, however, was only an unfortunate generalization from a technological limitation. Today it is possible to detect the arrival of individual electrons, and to see the diffraction pattern emerge as a statistical pattern made up of many small spots (Tonomura et al., 1989). Evidently, quantum particles are indeed particles, but whose behaviour is very different from classical physics would have us to expect.

Wave-only view

At least one scientist proposes that the duality can be replaced by a "wave-only" view. Carver Mead's Collective Electrodynamics: Quantum Foundations of Electromagnetism (2000) analyzes the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particle-like behavior to quantization effects and eigenstates. According to reviewer David Haddon:^[23]

Mead has cut the Gordian knot of quantum complementarity. He claims that atoms, with their neutrons, protons, and electrons, are not particles at all but pure waves of matter. Mead cites as the gross evidence of the exclusively wave nature of both light and matter the discovery between 1933 and 1996 of ten examples of pure wave phenomena, including the ubiquitous laser of CD players, the self-propagating electrical currents of superconductors, and the Bose–Einstein condensate of atoms.

Albert Einstein, who, in his search for a Unified Field Theory, did not accept wave-particle duality, wrote:^[24].

This double nature of radiation (and of material corpuscles)...has been interpreted by quantum-mechanics in an ingenious and amazingly successful fashion. This interpretation...appears to me as only a temporary way out...

And theoretical physicist Mendel Sachs, who completed Einstein's unified field theory, writes:^[25]

Instead, one has a single, holistic continuum, wherein what were formerly called discrete, separable particles of matter are instead the infinite number of distinguishable, though correlated manifestations of this continuum, that in principle is the universe. Hence, wave-particle dualism, which is foundational for the quantum theory, is replaced by wave (continuous field) monism.

The many-worlds interpretation (MWI) is sometimes presented as a waves-only theory, including by its originator, Hugh Everett who referred to MWI as "the wave interpretation"^[26].

'Three Wave Hypothesis' of R. Horodecki relates the particle to wave ^[27]. ^[28]. The hypothesis implies that a massive particle is an intrinsically spatially as well as temporary extended wave phenomenon by a nonlinear law. According to M. I. Sanduk this hypothesis is related to a hypothetical bevel gear model ^[29]. Then both concepts of particle and wave may be attributed to an observation problem of the gear ^[30].

Relational approach to wave–particle duality

Relational quantum mechanics is developed which regards the detection event as establishing a relationship between the quantized field and the detector. The inherent ambiguity associated with applying Heisenberg's uncertainty principle and thus wave–particle duality is subsequently avoided [1]. See Zheng et al. (1992, 1996)^[31].

Applications

Although it is difficult to draw a line separating wave–particle duality from the rest of quantum mechanics, it is nevertheless possible to list some applications of this basic idea.

• Wave–particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can be used to view objects much smaller than what is visible using visible light.

• Similarly, neutron diffraction uses neutrons with a wavelength of about one ångström, the typical spacing of atoms in a solid, to determine the structure of solids.