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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)