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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