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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

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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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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

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The quantization of Bohr electron diffraction electron microscopy

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J. Clinton in 1937 Davisson and George P. Thomson received the Nobel for showing the interference patterns electrons, thus proving that the particle behaves like a wave. De Broglie had derived an equation for the electron had a wavelength that depended on his speed.

This wavelength is in range of a few angstroms, the distance separating two atoms in a solid. This wavelength also corresponds to X-ray photons, radiation and was used to study the structure of matter in crystallography.

If the electron had to behave like a wave, was expected to follow the same equations as for x-rays A crystalline solid is arranged in an orderly fashion: the atoms are arranged in equidistant planes. When a wave reaches a solid, is reflected in each of these planes. As each plane is deeper in the sample, it generates each traveling a different space reflection. At the end of the sample if the difference of these paths coincides with a multiple of the wavelength, then the interference produced are constructive, and the signal is stronger. In addition to the distance between planes, the difference between ways depends on the angle of incidence of the wave, so only certain angles where the interference is constructive.

This theory is summarized in an equation, Bragg's Law:

The wavelength depends on the angular momentum of electrons. This angular momentum is communicated by accelerating particles by a voltage V, to acquire a kinetic energy

For the same acceleration voltage value displayed various angles, for all integer multiples n. Davisson conducted their experiments on a nickel foil. Instead of changing the angle of incidence and detection, it was more convenient to vary the accelerating voltage, and maintain a fixed angle of detection at 50 °.

Each time the voltage reaches the relationship deduced from Bragg's Law, there is a peak in the graph

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One

the most direct applications of the wave nature of electrons is the electron microscope. A microscope, in general, is " probe" a sample with test particles, and observe the results after the interactions.

A light microscope uses visible light photons, which after interaction with the sample, are collected by a lens to be finally detected by the eye of a person. There is however a limit of resolution, which is related to the wavelength of light. When objects are the size of this wavelength (400 nm for blue, 700 for red), the light undergoes diffraction phenomena, and it is possible to see sharp objects.

To improve the resolution is therefore necessary to reduce the wavelength of light. However, the eye can not detect light below 400 nm, light sources are required and specific detectors. However

is possible to illuminate the sample with light, but electrons. An electron moving at a constant speed has a length wave is below the angstrom (smaller than the size of an atom). If we analyze the resulting electrons after interaction with the sample, you can generate a sample image, we have an electron microscope.



Since the de Broglie relationship, we know that the wavelength depends on the angular momentum. To communicate this angular momentum, the electron must be accelerated with a voltage V, which reports a kinetic energy

For an accelerating voltage of 1000 V, l = 0.38 å. Thus, an electron microscope allows much higher resolution than that of any optical microscope.

There are two types of electron microscopes:

TEM (Transmission Electron Microscope , transmission electron microscopy): The electrons are transmitted through the sample, which is previously thinned.

SEM (Scanning Electron Microscope , scanning microscope): The electron beam is " sweep" across the sample. At each point, the electrons are absorbed, and the atoms in the sample emit secondary electrons. When to scan and collect these secondary electrons are generated image of the sample.

later delve into this type of microscope. By the time I was interested to note just a straightforward application of the wave nature of particles such as electrons.

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quantum physics: waves are particles

Quantum mechanics provides a theoretical framework for the development of the structure of matter, starting with the same structure of the atom. Much of quantum mechanics developed in parallel with the latter, although however, is much wider, and actually serves to describe any phenomenon in the atomic spatial scale.

As with the atomic structure quantum mechanics is developed from scattered observations, initially related to the light and its interaction with matter.

black body radiation and the ultraviolet catastrophe

late nineteenth century, Maxwell's electromagnetism had shown that visible light was electromagnetic waves. Moreover, there were other waves of the same type, whose only difference was the frequency of oscillation. All these waves meet a relationship between the frequency ν, and wavelength λ, so

λν = C

where c is the speed of wave propagation, in this case, c = 2997 × 10 8 m / s. Speaking of frequencies, or discuss wavelength is equivalent, although the change in one of them involves a change in the opposite direction from the other: an increase in frequency is equivalent to a decrease in wavelength, and vice versa.

When a body is heated, it emits radiation. The spectrum of this radiation, ie the amount of radiation emitted by each particular frequency may depend on the temperature and type of the issuing body: when heated an iron bar, it changes color from red to red, yellow and white. This light is emitted by the hot iron itself.

However, there is a body type no matter which size, shape or material but depends only on the specific temperature at which it is. Such a body absorbs all radiation that hits it, hence it is called black body. An example is a black body cavity, with a small hole where the radiation enters. Inside the body, the radiation bounces off the walls without ever leaving the hole through which he entered. However, the fact of being at a temperature, it emits its own radiation hole. This is the black body radiation, which sought to characterize the physical end nineteenth century.

Although the black body more important that we know is the sun: a ball of gas at high temperature, which generates a spectrum of light equivalent to a black body at 5500 K.

underlying assumptions scientists known as thermodynamics and statistical mechanics: a set of molecules is impossible to describe through the path and interactions between each of them (in a container is about 10 ²³ molecules!), so they resorted to calculate the average value and distribution of their speeds in order to calculate measurable quantities such as temperature and pressure. Within

as a black body cavity, there are many waves, so Rayleigh (1842-1919) and Jeans (1877-1946) followed the same pattern: the total energy is distributed equally in all possible wave . The result explained reasonably well the low frequency region (infrared), but nevertheless predicted an increase in the amount of higher frequency radiation, which would involve the issuance of an infinite amount of energy of any object. This result is called the "ultraviolet catastrophe"

Max Planck (1858-1947) in 1900 revised the way you do the calculation. Energy equipartition assumed by Rayleigh and Jeans assumed that each frequency corresponds to the same amount of energy. Planck reviewed this concept to infer that each wave corresponds to an amount of energy proportional to its frequency (E = hν)

According to the development of Rayleigh and Jeans, it could drive an infinite number of waves in a cavity, as will always be some whose wavelength is short enough to get into it, and the equipartition of energy, ensuring that there is always enough energy to excite, however minimal, no limit on the number of waves possible within the activity ca. In addition, to enter more easily into the cavity wavelengths shorter than longer, accumulate more of them, that is, the more energy accumulates in the ultraviolet, or high frequency: the ultraviolet catastrophe

In contrast, Planck's hypothesis , higher frequency waves (= shorter wavelength), we need a greater amount of energy, so there comes a time when there is enough energy to excite waves at high frequencies. Ie a limited number of waves can be excited in the cavity . The constant of proportionality between energy and frequency, proved to be very small: h = 6.62 × 10 -34 [J • s], and now called Planck's constant .

The implication of the result is deeper than just the explanation of black body spectrum. The energy of each wave is quantized. Each frequency requires a minimum energy to excite the vibration. Thus, the total energy of a frequency (ν E) be a whole number of times (n) the minimum energy hν. That is, E = nhν ν.

The photoelectric effect

Another problem of light-matter interaction would further evidence of behavior corpuscular radiation. Philipp von Lennard (1862-1947) studied thin layers of metals when illuminated by radiation.

In a classical setting, the light should excite the electrons, and come to pluck with increasing radiation intensity. With increasing intensity, was also pulled forward to the greater electron kinetic energy. But found instead that a minimum frequency needed to boot the metal electrons. Below this minimum frequency, it was impossible to start electrons, regardless of the intensity of light. Moreover, the kinetic energy of the electrons plucked not dependent intensity, but frequency as well.

Albert Einstein (1879-1950), one of his famous articles in the wonderful year 1905, deduced the solution to this problem. Planck independently, reached the same conclusion about the quantum of radiation, including the same value for the constant h. When one of these few came to the surface of the metal, an electron absorbs energy, and if this is enough to leave the metal, it does. Otherwise, it is issued. Thus, only radiation with minimal energy (which depends on the material) is capable of starting electrons. Moreover, when energy fixed, the electrons lose some to leave the material, and the remaining (also a fixed amount) is used to acquire a high speed, or kinetic energy.

with radiation below the minimum frequency, increased intensity of the radiation, ie, a greater number of light quanta, not cause it to boot as electrons because no energy is minimal. On the other hand, radiation with enough energy, produces electrons with a kinetic energy set, and increased intensity, will cause it to pull up more electrons, but all with the same kinetic energy. Is to increase the frequency of light with increasing energy electron kinetic uprooted.
This approach is very easily understood if one considers the radiation and fixed-energy particles collide with electrons to snatch the material.

Compton effect

The last of the experiments that highlight the particle properties of light is Compton scattering, discovered by Arthur Compton (1892-1962), when working with X-ray scattering In the scattering, detected as the wavelength of the scattered radiation increased, ie the radiation energy lost. Once again, the phenomenon is understandable if one takes the radiation and particles, and discusses the game against another particle, taking into account the conservation of energy and momentum.


The quantum of radiation has an energy given by its frequency. Also has an angular momentum, as any particle, which depends on its wavelength. In the shock of the energy and angular momentum are transferred to the particle. The result is a loss of radiation energy, which is revealed in its incidence, or increase its wavelength.

The explanation for these phenomena are based on the quantization of light energy, which makes it behave like a particles, called photons. In the next post we will see how matter can behave in turn as a wave

Annexes

-
The photoelectric effect - Compton effect

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The photoelectric effect Compton Effect

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In 1905 Albert Einstein explained the photoelectric effect, independently deducting Planck radiation consisted of particles with an energy proportional to its frequency, and contributing to the development of quantum physics. For this development, Einstein received Nobel Prize in 1928.

In the experiment, a beam of light strikes a metal. The action of radiation is electrons start of the plate, that the application of an electric field, leading to another plate "collector", to record an electric current in an ammeter, a way of quantifying the electrons that are emitted

What was found experimentally was:

- The electrons are emitted when the frequency of the light reaches a minimum value. Below this threshold, not output, regardless of the intensity of light.
- by applying a voltage of "braking" V between the collector and emitter, electrons can be slowed by this potential energy. Increasing this voltage, the intensity decreases, reaching a maximum value above which no electrons reach the collector, and the intensity is zero. This maximum value depends on the frequency, and is greater, the higher the frequency of radiation.

The explanation is simple if you consider light as a particle, a photon. Has a particular energy proportional to its frequency. This is absorbed by the electron, which must:
a) First out of the material. This costs an energy, called the work function.
b) Second, to move. With energy after spending the surplus needed to leave the material, the electron is set in motion with a kinetic energy.

hν = Φ + E _{c
c E = hν-Φ}

Given a material with a work function Φ a photon should at least provide the energy to start the electron. If the photon energy is less than the work function, the electron can not be torn from the material. If the energy is greater than this value, then the excess energy is transformed into energy the electron kinetic moves. In the limiting case where all the energy is used to overcome the work function, but no kinetic energy to move:

The minimum frequency to produce the photoelectric effect is given by Φ / h

applying the brake electric potential V, the electron loses energy on its way, so that only those who will traverse the shortest path between the emitter and collector plate. By increasing the retarding potential, then you can get to the point that even these latter are able to reach. Then eliminating the potential energy of the electron in an amount equal to the kinetic energy that left the electron.


In the limiting case where the voltage stops all electrons, the kinetic energy equals the potential energy created by braking voltage, eV, where e is the electron charge.

_{c eV = E = hν-Φ}

In this way the observed phenomena are explained the photoelectric effect.

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particles is perhaps most evident in the behavior as a particle in an electromagnetic wave, and that mathematical analysis of the problem is exactly like that if two particles collide, except that the expressions of energy and angular momentum of the photon not include the mass, but the wavelength.

A photon with wavelength λ, collides with an electron at rest. The photon energy and angular momentum exchanges with him, and is deflected at an angle θ, while the electron is equally diverted, in addition to acquiring a certain speed.

Before the collision, both the energy and the time correspond entirely to the photon and the electron is at rest. After the collision, the electron acquires a momentum p = m _and and v, which is related to the kinetic energy E = p _{c and} 2 / 2m e . The total energy and momentum must be equal to the initial, so the match:

The change in wavelength is not very large, ie, λ is similar to λ '. In these circumstances the first term on law is negligible, and can be ignored, so in the end, the equation for Compton scattering of photons by collisions with electrons is:

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quantum numbers

The description of the hydrogen atom by Bohr was a great success, despite being burdened with a series of ad hoc assumptions, the result of empirical observations. Also marks the beginning of a new way of describing nature.

physics (and science in general) is to describe the nature based on observable and measurable phenomena. Planetary orbits, for example, are described in terms of distance and guidance on a fixed point (the Sun) for each time step: describe what your path. Bohr's description, however, despite considering the atom as a small solar system (with the peculiarity of the quantification of the orbits), do not get these paths. It focuses instead on obtaining the energies that have electrons in their orbits, and how it changes it, because ultimately, is this phenomenon that can be observed and measured experimentally (the later development of quantum mechanics showed that even makes sense to calculate the trajectory of the electron). Thus, the description of an electron in an atom, is not to know its history, but what is your state , the orbit that takes information that is contained in the quantum number n .

Bohr atomic model explains perfectly the absorption and emission lines of a hydrogen atom. However, he had problems for atoms with more electrons. And even new lines were discovered in the hydrogen atom for which there was no solution.

azimuthal number

Arnold Sommerfeld (1868-1951) proposed the inclusion of new quantum numbers. Of planetary motion, which is known as wider orbit is elliptical. Sommerfeld proposed azimuthal quantum number, l , as a measure of how it was elliptical orbit, its eccentricity. Sommerfeld found that the value varying from l l = 0 to the value l = n-1 being l = 0 a perfectly circular orbit. Thus, the number n no longer represents an orbit, without determining the distance from the core, means a layer which can contain from l = 0 to l = n-1 orbits, all the same distance from the nucleus

For example, for first layer ( n = 1), the only possible value of l is 0. That is, the first layer contains a single orbit is circular. layer n = 2 contains the 2 orbits l = 0 (circular), and l = 1 (elliptical) to n = 3, l = 0, l = 1 and l = 2, (3 orbits with varying degrees of eccentricity), and so on.

magnetic Number

Pieter Zeeman (1865-1943) discovered in 1890 the effect that bears his name. Found that in a gas inside a magnetic field, some lines are unfolded, and appeared triplets around a common line. After inclusion of the azimuthal number, Bohr made new calculations by introducing the magnetic quantum number, m .

An electron orbiting a nucleus is an electric current, and as such, produces a magnetic field perpendicular to the plane in which the electron moves. It is a small magnet . By applying an external magnetic field, this magnet oriented, but this orientation is also quantized, so that it can take values \u200b\u200branging from m m =- l, to m = l .

The emergence of more and more lines, and more and more quantum numbers, is but proof of fine structure in the management of the electrons in the atom. Electrons are placed in layers, which differ in a quantity of energy. Within each layer, there is a difference in energy between each orbit, but is much smaller than that between layers, hence were discovered only when it increased the accuracy of the experiments. Moreover, the Zeeman effect reveals the existence of orbits that can be separated in energy by applying a magnetic field, revealing an internal structure of the orbits.

Spin Number

After inclusion of the numbers and azimuthal magnetic , the triplets were explained the Zeeman effect. However, the Zeeman effect also had other collections of lines (doublets), which were not explained by these numbers, called anomalous Zeeman effect .

Wolfgang Pauli (1900-1958) sensed the existence of a fourth quantum number, but was unable to realize the idea. Were instead George Uhlenbeck and Goudsmit Sam who proposed the quantum number spin, s, whose characteristic is not due to the orbit occupied in the atom, but the own rotation of the electron on itself.

The existence of an angular momentum (rotation) intrinsic electron was evidenced by the famous experiment of Stern and Gerlach : an electron beam through a nonuniform magnetic field. The interaction of the magnetic field with angular momentum causes the electrons to deviate from its path. According to classical physics, each electron would have an orientation of angular momentum with respect to the random magnetic field, so that everyone would suffer a distinct deviation, and the beam is open continuously over an area. However, it was observed that the beam is split into two beams clearly defined.

This result suggests that the intrinsic angular momentum of the electron is quantized, and can only have two possible values \u200b\u200bof s: +1 / 2 and -1 / 2 , most often referred to colloquially spin up or spin down . This makes the value of spin rotation in a very strange phenomenon. The spin is to describe the symmetry of the rotation. Take

ace in the deck. If broken 360 degrees back to its position initial. This is equivalent to a spin s = 1 . If you take a French king in a deck, when rotated 180 degrees (half turn), will be like in its original position. This is an example of spin s = 2.

The spin s = 1 / 2 means that it is necessary to rotate 720 ° (two laps) to regain the starting position. Rotation is very difficult to imagine. The closest thing is a movement like the following:

1 - Take an electron in the palm
2 - Rotate the hand inward, passing the electron under the arm to complete the turn. Now the electron has a spin, but the arm is in a forced position. The entire assembly electron-arm is not in the same position as at the beginning.
3 - Raise the arm (without pulling the electron), at the height of the head while touring the hand.
4 - The electron and arm are now in the same position as at the beginning, for which the electron has two turns. This would be something like the spin s = 1 / 2

With the inclusion of the quantum number completes the description of the possible states of the electrons in atoms. The combination of these four numbers identify the status and power they have, and may explain any emission or absorption line as the transition of an electron from one state to another NLMs n'l'm state's' .

The Bohr-Sommerfeld model can explain the experimental observations. However, based on some assumptions that are not shown, the quantization of varying amounts.

Until now, we can speak of a quantum physics, which simply applies quantization rules of physics and classical mechanics. The emergence of quantum mechanics can deepen the concepts, and give rise to themselves, these rules of quantization, and a series of new effects and implications, without equivalent in the classical world.

Annex

Electronic configuration of atoms

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Electronic configuration of atoms

comes from: quantum numbers

Once you know how to describe the electrons need to know how to sort the atom. To do so, exposed the Pauli exclusion principle , for which two electrons can have the same quantum numbers. Thus, there can be no more than two electrons in one orbital. Of the 4 quantum numbers, 3 of them describe the specific orbital, while the fourth is referred only to the electron (spin up / spin down).

and electrons are occupying each orbital, from the lowest energy at most.

traditionally to refer to a particular orbital, you specify the main number n, followed by a letter that represents the azimuthal number l:

To l = 0 , point s
To l = 1, point p
To l = 2, letter d
To l = 3, point f

Does not specify the number and m s as they are orbit equivalent to each other, which have equal energy and are only observed when a magnetic field, Zeeman effect.

order to know the energy that filled orbitals used a "recipe " builds a table of the orbital, and join with arrows like this:

This prescription, ordering is as follows: 1s 2s

2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p ...

Electrons will fill these orbitals, and until it filled one, do not start the next. For example, an atom of Sodium (Na), with 11 electrons, it begins to fill the 1s orbital with two electrons. The next two are placed in layer 2s. In the layer fit 2p 6 electrons, as well attend 3 orbital energy (for m =- 1, m = 0 and m = 1), and the last electron is placed in the 3s orbital. When writing the electron configuration is indicated with a superscript the number of electrons there. The configuration of sodium is:

[Na] = 1s 2 ^{2s 2 2p} 6 3s 1

The 3s orbital has still room for one more electron. The same happens to the atoms of Hydrogen, Lithium, Potassium, Rubidium, Cesium and Francium: Its external orbital is an orbital s , and has room for one more electron. This gives them some chemical properties similar to them. Other atoms have in common other orbitals, and therefore have other properties.

The periodic table of elements the atoms ordered according to their electronic configuration, which gives them their chemical properties. Currently, the table comes up just past the element 105, whose external orbital (which is still being filled) is 5f.