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states of matter

So far we have seen the internal structure isolated atom, and how electrons are arranged in it. The research that led to understand the atom led to the development of quantum mechanics .

However, the atoms are not isolated, but interact with other atoms and molecules, or radiation. The result of these interactions a set of atoms can occur in several different states, depending on the intensity of these interactions:

can form a gas, when interactions are weak, and reduced almost to collide among them, as if they were a bunch of billiard balls completely free to move around the table. The gas is adapted to the volume that is enclosed, occupied entirely.

form a liquid when forming bonds interactions are weak enough to bind molecules or atoms in a short period of time, so that all takes a certain cohesion, but the atoms and molecules retain a high mobility within the set. A liquid is adapted to the volume that contains it, but does not have to occupy it fully, such as when pouring water into a bottle, takes shape, but does not occupy the entire volume of the bottle.

and form a solid when interactions are able to make these lasting bonds between atoms and stable, so completely lost their mobility. A solid is rigid and not adapted to the volume that contains it. A solid has its own shape and volume.

A very important factor that determines whether the interactions are strong or weak, is the kinetic energy: the movement of atoms or molecules, if it is fast or slow. An indicator directly related to this parameter is temperature. In gases, the kinetic energy is high so that the interactions are effective only when the atoms pass very close to each other, resulting in collisions, but no brakes. It makes no sense to talk of a structure.

atoms or molecules in a liquid have a kinetic energy lower than the respective gas, so that the interactions are longer range and duration than in gases. The liquid also has a structure in itself, but may have small clusters of atoms or molecules, called "clusters" with a certain order or microscopic structure.

In a solid, the kinetic energy is so small that the interactions affect both the atoms are stopped, making the interactions between them stable and durable. In this case we can speak of a macroscopic structure of a material.

The crystalline solid

gases and liquids in the interactions between atoms are not stable or lasting. Therefore, the disturbance that would suffer the atoms are not stable, and its internal structure is not affected. In a solid
however, the interaction between atoms is so intense that keeps static, making them durable and stable as well. How do these interactions to the atoms and their internal structure? And how these differences contribute to a single atom properties of the solid?

A solid is a set of static atoms occupy given position. There is a first distinction in the structure of solids, depending on the positions of atoms:

Amorphous materials are materials that have atoms occupying the space of irregular shape: it is not possible to find a repeating pattern. An example of amorphous material is glass.

In crystalline materials, or crystals, the atoms maintain a position following a regular distribution. That is, a crystalline solid is formed by a small group of atoms with a specific structure and this structure is repeated periodically at fixed distances. The vast majority of the materials are crystalline. A well-known example is common salt, which is small cubes of sodium and chlorine which are repeated throughout the material.

An ideal crystal is constructed as an infinite repetition of a structural unit, or unit cell. " This in turn may contain several atoms, arranged in any way. Thus, there are two parts in the unit cell:

Network: The "box", or structure will be repeated throughout the crystal, which is delimited by vectors, which need not be perpendicular or have equal length.

Base: The contents of the structure, which is always the same, and always placed in the same positions and guidance regarding the origin of coordinates of the network.

lattice parameter is called the size of the network, which is what determines the frequency of the crystal. In a crystal can have different periodicities in each axis of space. As an origin of coordinates, it is possible to know the position of all atoms, since all are spaced an integral number of times the lattice parameter.



fundamental Networks A network is parameterized by a vector. They may have different sizes (giving rise to different periodicities in each direction), and need not be 90 º relative to each other. (In the drawings represented a 2-dimensional network defined by two vectors. A three-dimensional network is delimited by 3 vectors).

However, any network is valid. Are valid only to meet certain symmetries. For example, the Pentagon has a symmetry that is valid for a network, since that figure is not able to fill the space without leaving voids.

The symmetry that has a hexagon does allow, however, that a network can do:

Thus, the number of possible networks is limited. In two dimensions, there are only 5 types of networks, depending on the relative length of each vector, and the angle.

Each of these networks are called Network Bravais

To 3 dimensions, there are a few more: 14 Bravais lattices in total, grouped into 7 different systems.

Annex

Art, glass and ducks

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lattice, crystals and ducks

comes from: The lattice

Maurits Escher (1898-1972), MC Escher , was a Dutch artist, whose works more famous buildings include stairs and impossible. Some of his works also deal with how to fill the space, through the repetition of patterns, as occurs in a crystalline solid. It is easy to identify in his works the same kind of structures that occur in a solid, albeit restricted to two dimensions.

From a base of two ducks, including a parallelogram structure, Escher can fill an entire plane to repeat the parallelogram. Ducks are coupled with each other to keep free holes, and fill the plane to infinity.

Related links Official website MC Escher

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The development of quantum mechanics makes possible the discovery of new effects, impossible from a classical viewpoint. Perhaps the most popular is the tunnel, where making a quick comparison and bad, it's like throwing a ball at a wall to cross it without touching it. This is how it appears from an analysis of a similar situation, through the Schrödinger equation .

quantum Wall

Consider first what is meant by " wall" and what happens to an electron when it arrives. The energy of a particle is always the sum of its kinetic energy and potential energy. Thus, energy will always be equal to or greater than the potential. The cases in which the energy is lower than potential from classical physics, states represent unattainable by a particle. Thus, the point at which the total energy equals the potential represents a "turning point , the particle can not move forward, but must go back. Is the equivalent of a "wall ."

In the figure, the electron energy above is greater than the step, and both are superior, losing a bit of kinetic energy. Below the electron instead he should go back after reaching the point where its energy is equal to the potential energy, and therefore its kinetic energy is zero at that point.

Consider the situation of the second electron from the quantum point of view, with the Schrödinger equation.

The equation to solve is:

whose solution we saw in the previous entry , is a combination of sine and cosine. The combination of sine and cosine, by Euler's formula is equivalent to an exponential function imaginary and more useful for the analysis that follows. Thus, the wave function can be expressed generally as:

As we saw earlier, k (the wave vector ) is related to the angular momentum and therefore the sense that the particle moves. A + means that moves in the direction of x increasing (going from left to right). The sign - describes a movement in the opposite direction. As the picture has been raised, we are in the first case, ie, the electron moves from left to right, whereas in the wave function describes the first term ( k positive). Therefore, in our case, you should choose B = 0 to cancel the second term.

Let's see the two regions of space that defines the step. On the left, (EV) is a positive quantity (total energy greater than the potential), and the wave equation represents a free electron (an imaginary exponential, or a combination of sine and cosine). In contrast, in the right area, (EV) is negative (energy less than the potential energy), the wave vector is imaginary , and the wave function represents a real exponentially decreasing. That is, although it is a forbidden zone as classical physics, in quantum mechanics a particle can exist in that area, but with a diminishing likelihood as deeply into the wall.

The distance that an electron can penetrate into the forbidden zone before your chance is almost nil, depends on the difference between energy and the value of the potential. The greater the difference EV, the probability falls more rapidly. In the extreme case in which V tends to infinity, the penetration depth tends to zero, ie, the electron does not enter into the wall (as we assumed when we spoke of quantum well).

The tunnel effect

Therefore, a particle can penetrate a wall of potential, something impossible according to classical physics. A particle can penetrate a distance (small), although the probability of finding the particle at that location decreases as depth. What if the potential wall finishes before this probability is terminated, or reduced too?. In this case, the "other side " from the wall of the wave function again described by a free electron. It is therefore possible that an electron reaches a barrier, the cross, and appears on the other side of it, with a certain probability, although less than it did before crossing the barrier.

The probability of crossing the barrier depends on the mass of the particle, the barrier height, but above all, of its width. The typical distances that probability is sufficient to tunnel is in the order of angstroms and nanometers. Alpha decay



The emission of alpha particles is related to the tunnel effect. An alpha particle is an atom composed of 2 protons and 2 neutrons, not electrons. Corresponds to a nucleus of helium 4 (4 ^I 2 +). An atom with a large number of protons and neutrons keeps these particles stuck together by the strong interaction. An outline of the potential energy inside an atom is as follows:

There is an area of \u200b\u200bpotential barrier at the transition between the strong interaction domain and the electrostatics. Overcoming this barrier would require making a lot of energy. However, an alpha particle is able to cross the barrier by tunnel effect, breaking up the nucleus to which it belonged.

The scanning tunneling microscope

The tunnel is now the basis of some devices such as diodes, lasers and detectors. But one of the most important is related to the surface microscopy.

View STM: Scanning tunneling microscope

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The STM tunneling: The scanning tunneling microscope

The scanning tunneling microscope (Scanning Tunneling Microscope) was invented in 1981 by Gerg Binnig and Heinrich Rohrer. Received the Nobel Prize for it in 1986. The STM uses the ability of electrons to traverse a potential barrier to record the electrical current that occurs between a tip (or probe), and the sample. An electron in a metal has a particular energy. The solid surface represents a potential barrier to be " jump" to get out of it. By bringing the metal (the probe), and applying an electric field to direct the electron to the metal, it creates a barrier with a distance small enough for the electron tunneling can be no need to jump the barrier.

The current can be recorded with an ammeter is proportional to the probability that the tunnel occurs. Which in turn depends on the distance between the tip and sample. In this way, recording the electrical current, information is obtained the distance that the tip is.

A simple schematic of an assembly of a STM tip is near a sample records the intensity tunnel occurs. The intensity controls a piezoelectric (material that varies in length by applying an electric field) acting on the tip to zoom in or removed from the sample, which is mounted on a table x, and the moves. Thus, to take a trip on y, the tip scans the sample, recording the intensity that occurs at each point.

There are several ways of operating in a VTS, but the most common is to maintain a constant tunneling current. This is achieved by maintaining distance between tip and sample constant. As the tip scans the sample in the x-axis and and , will control the tunneling current. When the tip reaches a point where the sample has valleys and outgoing, this current will vary. This variation indicates that you zoom the tip of the sample in the z axis to return to get the same stream, which is done through the piezoelectric. One computer had to be recorded as pan and zoom the point on the axis z , at that point ( x, y ) of the sample, so that after completing the sweep has a map that shows the variations in z axis. Ie a graph is related to the topography, the shape of the sample.

Another way to act is to maintain constant tip position, and record the power at each sample point, which varies as it passes through valleys and outgoing. However, there is a risk of crashing the tip into a projection of the sample, spoiling both the one and other. The end result also gives information on the topography of the sample.

Usually, the results are presented in map form x, y colors where the color code represents the z-axis values \u200b\u200b .

microscope Act, allowing " see "the surface of the sample. The accuracy of this instrument is that it can see the atoms of a solid. It is useful to study the surfaces of solids and their electronic properties. However, samples must be conductive. Another requirement for the STM is to be run dry, leading to integrate the system into vacuum chambers, with the hassle to change samples involved.

To see some images obtained by STM , you can do in the website nanotechnology.

View tunneling The

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Schrödinger's equation in action

The Schrödinger equation determines the wave function of a system. In particular, applied to an electron, described how he behaves when a particular environment described by the potential energy V (x) and boundary conditions. In the words of this potential and conditions, the equation is more or less easy to resolve. Let's see a couple of simple cases, where important consequences are deduced.

The free electron

The simplest situation is when an electron is not subjected to any kind of interaction. In this case, the potential energy is zero, and there is talk of a " free electron. " The Schrödinger equation is time independent then as

The solution to this equation is:

ψ (x) = A • sin (kx) + B · cos (kx)

For each value of k, there is an eigenfunction, or system state. The meaning of the variable k is related to the angular momentum. Substituting the wave function in the Schrödinger equation, one can deduce the value of energy as a function of the variable k:

from which it follows k is proportional to the angular momentum. The variable k is called the wave vector , since if we recall the expression of time for a photon,

k is related to the wavelength.

The relationship between E and k is called dispersion relation , and describes the possible states of the electron. In the case of free electron, all its kinetic energy is only due to their movement. The electron is far from any interaction and there is no special condition on E ko, and hence, the electron can have any value of energy kinetics. Ie a free electron is not quantized your energy, your energy is a continuous , and there are infinite possible states for him.

The infinite quantum well

The second example is called the quantum well. The total energy of the electron is always the sum of its kinetic energy and potential. If we limit a region of space where the potential is zero, but outside it is infinite, then the electron is confined, stuck in a well, limited between the positions x = 0 and x = L .

makes no sense to raise the Schrödinger equation in a zone of infinite potential, since the electron will never have a total energy greater than this, but you can put into the well. However, it is now necessary to include boundary conditions: the probability of finding the electron in an area where there is no point to consider is that no, that is:

ψ (0) = 0
ψ (L) = 0

The solution to the Schrödinger equation in the well region remains the same as for the free electron. By applying the conditions boundary, however:

In the first condition, it follows that all the eigenfunctions are described only by the sine function, not the sum of sine and cosine, as in the free electron. But the most important condition is the second: it can only be zero when A • sin (kL) = 0 . One solution is A = 0 . But when within a periodic function, it also occurs when kL is an integer multiple of π. Thus, it is revealed that the wave vector k is quantized: there can be any state, but only those whose wave vector is an integer multiple of π / L. As a result, energy is also quantized:

words, even when there are infinite possible states for an electron in an infinite well, the energy of these can not be anything, but is quantized.

This graph represents the values \u200b\u200bof energy that can have a particle in an infinite well represented by the two blue vertical lines. Superimposed on each energy, is making the wave function for that state.

The most important of these two examples is to understand that a particle is confined between two potential barriers can not have any energy, but the will quantized. By contrast, when a particle is free any interaction, there are no conditions for its energy, but is continuing .

The Schrödinger atom

What is the potential energy of each electron in an atom? This potential is none other than the electrostatic, which is inversely proportional to the distance:

where r is the distance in the radial axis, and Z the number of protons in the atom. In this potential can clearly see two separate areas, depending on the electron energy:

An area where an electron would be considered "free " and therefore outside the atom and continuous power ( Zone A), and an area where the electron is confined the potential (Zone B), and therefore its quantized energy. The solution of the Schrödinger equation for this potential is complicated, but it finally pops quantization Bohr and Sommerfeld had to apply to describe the electrons of the atom.

The boundary between areas A and B is called "vacuum level" is the reference energy for which is covered by the other., And is the difference between an electron bound or confined within the atom, and an electron that has escaped. According to this reference is taken, the energies of bound electrons are negative, while the free electrons are positive.

To remove an electron from an atom, energy must give at least equal to that of his tie. With that, his power is left in the vacuum level. If you are given more extra energy is used in setting in motion the electron kinetic energy that the longer a free electron has no special status.

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Copenhagen Interpretation

After the development of quantum mechanical wave Schrödinger and the matrix Heisenber , Paul Dirac made a step further by explaining how the two theories were the same, with a different description. Moreover, Dirac deepened development including electromagnetic radiation and relativistic effects in quantum electrodynamics where it came from, and the prediction of the existence of particles exactly equal to those already known, except that its charge was reversed: Dirac discovered and antiparticles.

Parallel to the development, work is also in the interpretation of quantum theory. If classical mechanics is everyday situations and systems, which are translated into equations, quantum physics, although the first discoveries (black body radiation, photoelectric effect, etc ...) respond to this scheme, to further develop the mechanical presents the opposite case, in which mathematics to develop solutions that must find a physical interpretation. (The antiparticles are a case, for example)

was not until the Solvay conference 1927 in Brussels that the interpretation was finally established, due mainly to Niels Bohr. An interpretation is not free from debate, and scientists of the stature of Einstein shared it, and tried again and again to test it with experiments, trying to prove some kind of contradiction.

Probability and wave function

The interpretation of the wave function is not trivial to do. Depending on the problem, the system can be in different states represented by their position, angular momentum angular momentum, or any other observable quantity; states described by its energy.

But the Heisenberg uncertainty principle shows that it is impossible to determine with precision the same values. Max Born (1882-1970) concluded that the wave function is then the probability of finding a system in a given state. For an atom, representing the probability of finding an electron at a given position, but will not describe how it orbits around the nucleus, just as classical physics describes the trajectory of a planet around the sun and loses sense

the concept of orbit which had been handled, and there is the concept of orbital, which is the region of space which is likely to find the electron. Instead of a circular orbit where a planet occupies a certain position, there are regions of space, more likely than others to find the electron there.

Complementarity: particle duality - Wave

Almost from the beginning of the development of quantum mechanics, it became clear that waves and particles appear to be two properties that can have a single system. For a classical physicist, these two properties are mutually exclusive, and therefore one of the two must be wrong.

However, a quantum physicist wave and particle properties are not exclusive but complementary. Which of the two properties are revealed in an experiment depends on the experiment in question. The same experiment will never show both properties, so that you can not discriminate whether one is more correct than the other.

The wave function can represent this complementarity. A wave function that covers a vast region of space is incompatible with the idea of \u200b\u200ba particle occupying a position. One speaks of a delocalized wave function, and the system will behave more like a wave and a particle. And it can happen otherwise, that the wave function is found highly concentrated in a small region of space, then the system will behave like a particle, and there is talk of a localized wave function.

Collapse of the wave function and Schrödinger's cat

We saw the wave function, mathematically, is the sum (or superposition) of a series of eigenfunctions for which could solve the Schrödinger equation. These eigenfunctions also represent each of the possible states of a system, but when he makes an observation (Or experimental measurement), one and only one of these states is revealed in the experiment, with a certain probability.

This means that before making a measurement, the system is undefined: the state is a mixture of all possible states. But nevertheless, to observe and interact with the system as the system of study, the interaction makes the system opts for a particular state, a collapse of the wave function of a particular state, which will maintain to which is referred to another type of interaction differently.

The best known example is employed to illustrate this collapse is the thought experiment known as Schrödinger's cat. However, Schrödinger did not support precisely this interpretation, and was trying to illustrate how absurd it was:

In one case, a cat locked together with a radioactive source, a Geiger counter, a hammer and a gas container poisonous. The disintegration of the source is a quantum process, which has a 50% probability of occurring. If it happens, the Geiger counter triggers a device by which the hammer broke the bottle of gas, and the cat dies. If not, the cat remains alive. Everything is put in a box, and the only way to know if the cat is alive or dead is opening it. Therefore, the cat has a 50% chance of being alive or dead. According to the interpretation quantum, while not open the box, the cat is both alive and in a dead state, which seems ridiculous.

is however more useful to understand the collapse, talk of Stern and Gerlach experiment . Remember: an electron has a spin that can have two states: up (s = 1 / 2) and down (s =- 1 / 2). Is made through an area with a nonuniform magnetic field. Before crossing, it is unclear whether the state of the electron spin, has a 50% chance of being in one or another. But through it, the spin is oriented (in fact, the superposition of states collapses up and down one of them) so that the electron is deflected of his career in one way or another.

If now the electron, with a particular spin, we will go through another device Stern - Gerlach, with the magnetic field at the same address as above, the collapse will not happen, because the state had already been determined, and therefore, has a 100% chance of passing through the magnetic field, the final state is the same as the original. Under no circumstances may change to the opposite state.

Suppose instead that an electron moving in axis with a spin up after passing through a magnetic field uniform in the z axis , passes through another device Stern - Gerlach, but with the magnetic field in the x-axis . In this case, the spin was determined with respect to the z axis , but it was not on the x axis , and therefore returns to the initial situation: have a 50% probability that the spin is oriented towards way or another in the x-axis .

"God does not play dice"

(Albert Einstein to Niels Bohr on quantum mechanics)

This interpretation of quantum mechanics was primarily exposed by the Danish Niels Bohr, thus bearing the name of the Copenhagen interpretation. Its main feature is that it is based on probabilities, but are inherent in nature.

gas description, or systems with many particles, is based on statistics, probability and chance. But they are likely based on ignorance, and practically impossible to handle a large number of equations, which are used to find average values \u200b\u200bthat determine the properties of the system.

instead Quantum mechanics is probabilistic in nature. There are no unknowns that prevent us from determining their properties, but those odds are the only properties we can know. This view of the nature of the Copenhagen interpretation did not like, among others, Einstein, who over time had tried to prove that hidden variables exist that prevented hear and determine properties beyond their probabilities. Which today has not been possible to demonstrate.

"Stop telling God what to do"

(Niels Bohr responding to Albert Einstein)

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Wave mechanics

The Heisenberg matrix mechanics was successful as it could derive the results known as quantum physics, but from general principles apply to any system. However, the development is cumbersome, and quite abstract, making it the Heisenberg theory unattractive.

A Erwin Schrödinger (1887-1961) disliked much abstraction, preferring to develop mechanics through concepts more real. left the Louis de Broglie theory , which could be considered the particles as waves. If this was his behavior (at least one of them), then that system must be mathematically described by equations for waves. By developing

mathematician Joseph Fourier (1768-1830) , it is known that any function can be described as an infinite combination of sine and cosine are precisely describing simple waves. Thus, a system and its evolution is described by a sum of waves. The mechanics of Schrödinger wave mechanics is named, and summarized in a single partial differential equation, the equation of time-dependent Schrödinger:

(simplified version, where And is the wave function describing the system, and V (x) is the potential energy of the system. The equation can be generalized to the 3-dimensional space, using Cartesian geometry, cylindrical or spherical)

But what does this equation?

mechanics is to find the evolution of a system, based on the factors that affect it. Newtonian mechanics is to find the path in the space of a mobile, knowing how some forces act through the three laws of Newton.

William Hamilton (1805-1865) developed a mechanical equivalent from different concepts. Instead of dealing with the intuitive concept of force (a force a change in the movement of a particle, an acceleration ), used the more abstract and general energy. A system has a kinetic energy due to their movement. When nothing interacts with the system, that is their only power. However, when yes there is an interaction, there is an exchange of energy through a potential energy . This exchange is what produces the forces in Newton's description.

Hamilton mechanics first determines which is the total energy of system: the sum of kinetic energy and potential. This amount is given a name: the Hamiltonian . To find the path makes use of a general principle of physics: the principle of least action , by which the evolution of a system will be such that its total energy is the minimum possible . Of all the possible paths that may have a phone, make one that minimize their energy.

The Schrödinger wave equation following the same philosophy. The first term on the left side of the equation represents the kinetic energy, while the second the potential energy is the Hamiltonian, but in quantum version.



When a system does not depend on time, is a stationary system, and the equation to solve is it called Schrödinger equation independent of time.

While Hamilton's classical mechanics try to find the function And as to the value of E minimal, quantum mechanics is to calculate all the functions And _n with corresponding energy E _n , since according to the approach taken by de Broglie and Schrödinger, and thanks to Fourier's mathematical development, the overall description of the system is a combination of all these wave functions, each with its own energy.

operators

If we see the Schrödinger equation independent of time, mathematically it is a problem known as eigenvalues \u200b\u200b , and had already developed Fourier: the solution to the equation are all of these functions (called eigenfunctions ) And ₁ , And ₂ , And ₃ ... such that when subjected to a series of operations , is a number of times 1 E, E ₂ , E ₃ ... ( eigenvalue or eigenvalue ) the same function And 1 , And ₂ , And ₃ ... Only those functions are valid, and comprehensive solution is a sum of all the eigenfunctions.

Thus, we must talk about operators. Schrödinger equation, we have said that represents the quantum version of Hamiltonian H , an amount that contains the sum of kinetic and potential energy:


If we assimilate the classical and quantum expressions of the Hamiltonian, then we must identify the linear momentum p with operator acting on the wave function by calculating its derivative. The potential energy, a function of position x would be a operator that multiplies the expression V (x) by the wave function.

The physical meaning of the operators is to calculate a magnitude observable in a measurement process. More specifically, given a wave function And , the sum of several eigenfunctions ( And = A And 1 + B 2 And + C And ₃ ... ) , the result is actually the probability that the measure corresponds to a system in the state And ₁ , And 2, or 3 And ... For example, the Hamiltonian operator H , observable results in total system energy E n , each of the possible states and the probability measure such value.

The wave function thus represents a likely respect to the state it is the system and a measure of the system reveals one and only one of these states, with a certain probability.

This is one of the most important pillars for the interpretation of quantum mechanics, before the measurement process, system status is not defined, but there is a chance that after making a measurement, the measurement result is one in particular. However, after being measured, the system remains in that particular state. For a system since it is not possible to determine which state will be revealed in a measurement process. However, it can determine the probability that the result appears.

vs matrix mechanics wave mechanics

An important outcome of the matrix mechanics Heisenbserg was the uncertainty principle that appeared during the measurement process. The order in which measurements are made varies the result, so that if you measure position and momentum, the amount [x • p - p • x] is nonzero (and in particular, a complex number .) This result suggested the need to use matrices representing x and p , since matrix multiplication is not commutative.

now developed with the operators x and the difference p xp - px applied to a wave function Y .


These operators therefore have no commutative property, but there is a distinct difference from zero, and equal to that obtained by Heisenberg. This coincidence therefore suggested that treatment operators is equivalent to treatment with matrices. In fact, the elements that form matrices can be calculated from the operators and wave functions.

The equivalence between two descriptions of quantum mechanics showed the British Paul Dirac (1902-1984) in 1925

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Schrödinger quantum mechanics

The discovery of de Broglie about the wave nature of particles has an associated new phenomenon in the classical world of common sense, it is surprising and counterintuitive principle.

was Werner Heisenberg (1901-1976) who came to him, but in a completely different way. Until then, the development of mixed quantum classical physics, with the addition of quantum principles, the result of experimental results. However, a full understanding of nature requires a broader theory, which is deducted from these postulates, we must develop a quantum mechanics.

classical mechanics of Newton's three laws. From these, it is possible to describe any situation, and come to an equation of motion, ie to solve an equation that gives the time evolution of the system studied. This system applies both to describe the oscillation of a spring, as the orbit of a planet around the Sun

Quantum mechanics tries to do exactly the same thing from a common point that is able to describe the evolution of any system at the quantum level. Instead of treating each system owners (photoelectric effect, Compton effect, electron diffraction, the Bohr model for the orbits), it is to get the general behavior of any system based on a certain basis.

The development of these mechanics, Heisenberg came to a rather surprising mathematical result. The development included some mathematical operations that represent the experimental observation of the system. The result was that if you made two comments, for example, the position and angular momentum, the order in which it influences the final result. Mathematically, if an observation A, and one B, this meant that A • B is different from B • A . In fact, its ( A • B - B • A ) is a complex number. This result occurs only for certain amounts related, such as position and momentum, or energy and time.

From the school we are taught that multiplication is commutative. However, this is true when we talk about numbers . When it comes for example, matrices, then this property is not insured. This is how the mechanics of Heisenberg is based on matrix algebra, and hence the development of Heisenberg matrix mechanics call him .

Although mathematically it is possible to understand that A • B is not equal to B • A , what is the point that in the real world? implies that the order in which they performed a measure influences the final result . One can measure the position of a particle with a given accuracy. However, when measuring the angular momentum accuracy is limited, and is impossible to determine as good as they want. Although the phenomenon is related to the measurement process and experimentation, it actually has nothing to do. The experiment not limit the possibility of determining the desired precision values, but rather, is nature itself which limits the accuracy of the experiment . Even with measurement systems ideally perfect.

The Heisenberg uncertainty principle can be understood taking into account what is actually a process of experimental observation. A system is observed through a measurement system. This system, in order to take any steps needed interact with the system of observation, ie, it requires an energy exchange , and that change of power, or status of the meter, is related to some property of the studio system.

However, as the meter has a change of state, the system also observed status change, so to make another type of measurement on the system, it is no longer in the same state, and this second measure is not independent , but depends on the first.

determinism that until that moment was unquestioned, said known initial conditions, it was possible determine with accuracy the evolution of a system. However, Heisenberg demonstrated that it is impossible to know with any precision you want the initial conditions of a system, and therefore can not determine its evolution.

Early in the development of quantum physics to calculate the trajectory of an electron around its nucleus was meaningless because there was no means to observe experimentally. Louis de Broglie casts doubt on the possibility of doing so by stating that the electron is a wave. But Heisenberg completely eliminate this possibility, since it is impossible even imagine an experiment to measure the position and speed of the electron with the necessary precision.
The uncertainty principle is part of nature. But as there is no wave motion of particles in the classical world, the principle is not reflected in these scales. Taking for example an electron and measure its position with an uncertainty of 1 angstrom (the size of an atom), then the uncertainty in the speed will be:

is, if you measured the speed of the electron and was around 5% of the light (so you can ignore relativistic effects), then the uncertainty represents a 46% of measurement. If the speed is much smaller extent, this percentage was much higher. And if given a value far below the uncertainty itself ... is the same as not knowing anything about his speed.

The same indeterminacy of an angstrom in the position of the Earth (mass ~ 10 24 Kg) in its orbit around the Sun, gives an uncertainty in the rate of 6 • 10 -48 m / s. The travel speed of the Earth is 30,000 m / s, so an uncertainty of about 10 -48 m / s is negligible, it represents the order of 10 % -51 of action: it can say that the position and speed of the Earth are perfectly determined.

Werner Heisenberg received the Nobel prize in 1932

Annex

Heisenberg and Gamma rays

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Heisenberg uncertainty principle and quantum physics

comes from: quantum mechanics uncertainty principle Heisenberg

devised a thought experiment to explain the uncertainty principle. We want to see through an electron microscope Gamma rays. This radiation has a wavelength of about 10 -12 meters.

The gamma ray detector of a particular size, covering a viewing angle 2α . The resolution of such a system determines the minimum size or that can be distinguished, and is related to the wavelength and the angle of detection

This resolution we determine the uncertainty in the position. If we place an electron in a particular position, radiation does not distinguish it from another that is at a distance less than D x .

When the gamma ray photon strikes the electron angular momentum tells _and P, as in the Compton effect . The Gamma ray meanwhile bounced out to the detector, may fall between two limiting cases, marked in the drawing as ₁ p and p 2

Both give the same signal photons in the detector, even under an exchange angular momentum than the electron. In both cases, the total angular momentum in the x-axis (electron + photon) is equal to the photon momentum p before the crash:

may seem that this result is due to limitations of the experiment . But not, it is the experiment that is limited by nature. Let's try the case wholly ideal in which the spatial resolution of the microscope is perfect , ie Dx = 0, because we use a Gamma-ray radiation does not, but cosmic rays, or others a wavelength as close to zero as we like .

The result is that the indeterminacy of the moment is growing up to be infinite: it is impossible to determine the angular momentum. The shorter the wavelength of the photon, the greater its angular momentum, and the greater the amount that can pass the electron after the collision.

Now suppose we decrease the size of the detector to make it as small as we want. This means that the detection angle is zero, and consequently, the uncertainty of the angular momentum is zero, we have determined with perfect accuracy the angular momentum, Dp = 0 .

However, spatial resolution, the uncertainty in the position Dx is the shooting at infinity: it is impossible to determine the position of the electron.

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Gamma rays: particles are waves

speculated for centuries, and was argued pro and con, on whether the light was a particle or a wave. The emergence of electromagnetic theory Maxwell seemed to close out the debate in favor of wave theory, but nevertheless re-opened with the discovery of energy quanta.

could say that reopening the debate was not new. What if it was a big novelty was the assumption made by Louis Victor de Broglie (1892-1987), according to which a particle like an electron can also behave like a wave, and suffer the same phenomena, such as interference and diffraction. For this contribution he received the Nobel Prize in 1929.

Based on the most famous equation of Einstein derived the expression for the moment kinetics of a photon, and its relationship with its wavelength. From there, he assumed that this formula was general, so that for a particle with mass m, moving at a speed v, you could associate a wavelength.

According to this result, a particle has a shorter wavelength than the greater its speed or mass. The wave diffraction occurs when waves are in their change objects of comparable size to its wavelength. In the case of an electron, whose mass is around 9.10 -31 kg, at speeds as high as 1% of the speed of light (3.10 6 m / s), the electron has a wavelength of l = 2.45 å. This is the typical size between atoms in a solid, and it was expected that the electrons suffer this phenomenon. Thompson

Davisson and independently confirmed the interference and diffraction of electrons, causing an odd situation, because if JJ Thomson in 1906 received the Nobel to show that the electron was a particle, in 1937 his son GP pson Thom (1892-1975) was received (shared with Davisson) to demonstrate experimentally that the electron is a wave.

If all particles have an associated wavelength, if can behave like a wave. So why not see this behavior in everyday life?. The reason is that the wavelength depends on the mass and velocity of the particle. Either a ball, a few grams (say 200 gr) at a speed of 1 m / s, has a wavelength of l = 3.10 -33 meters (size of an atom: 10 ^{- 10 m.} atomic nucleus size 10 -15 m). Wavelength is too small to find an obstacle that the diffract. Thus, in the classical world in which we live, it is impossible to detect wave behavior. Only when we got off at the quantum level of electrons is possible to see these phenomena. Annexes

The Bohr quantization of electron diffraction

electron microscopy

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comes from: quantum physics: particles are waves

When we talked Bohr model of for the hydrogen atom, we said he needed impose a postulate, ie, a condition which did not show, but that allowed him to explain the observed phenomena. This condition was that the electrons are orbiting around the nucleus in orbits whose angular momentum was an integral number of times the reduced Planck constant.

De Broglie's hypothesis, and evidence that an electron is a wave can solve this pitfall. When a wave is confined in an enclosed space, can only wavelengths space size is. A guitar string, for example, can vibrate with multiple wavelengths along its length. Similarly, if we apply now to an electron, its wavelength must be such that n is a number of times the orbit crosses:

orbit l = 2 r = p n L

The wavelength can be expressed as the angular momentum:



The first term corresponds to the angular momentum of the electron orbit, which is the Bohr quantization condition needed .

See also: quantization of the orbits of the Bohr atom