Difference between revisions of "Quantum mechanics"

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(Principles)
(Collapse of The Wave Function: Merge old and new information. I think the examples are helpful.)
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[[File:norm.png|right|150x250px|thumb|An example of a wave that could be a position function.  (Actual position functions are normally much more concentrated.)]]
 
[[File:norm.png|right|150x250px|thumb|An example of a wave that could be a position function.  (Actual position functions are normally much more concentrated.)]]
  
When speaking in the quantum sense, it becomes impossible to make definite statements such as "the particle is here". This is a consequence of the Heisenberg Uncertainty which (simply) states that particles do move. Thus the only useful way one can talk about the position of a particle is the likelihood that it exists at any given spot. The wave function is just this, it tells us the probability of finding the particle in one place or another. This wave function can be "collapsed" if the particle is observed. What this means is that if the particle is measured to be in one spot, then is quickly measured again the probability of finding the particle around that position is drastically increased. Graphically the wave function will peak at the location where the particle was measured. If the particle is not measured again it will take on its original wave function.
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In quantum mechanics, it is impossible to make definite statements such as "the particle is here". This is a consequence of the Heisenberg Uncertainty Principle which (simply put) states that particles move randomly. Thus, the position of a particle is not described as a point but rather as the particle's "position function," which gives the probability that the particle is at any given spot. It peaks at the location where the particle "exists" in the classical sense, and we might [[integral|integrate]] the function over two angstroms around that point and find that there is a 99% chance of finding the particle in that area; however, there is never a 100% chance; the function always has nonzero "tails" everywhere else in the universe. This means that there is always an infinitesimal chance of the particle suddenly "jumping" a foot or even a light-year away from its original location.
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When the particle is actually observed and found to be in a specific location, this wave function has "collapsed" to yield a specific value. However, immediately after that measurement, the particle takes on a new wave function peaking at that location, so scientists cannot keep it collapsed.
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Scientists have tried but failed to explain why a wave function collapses and predict to which place it will collapse.
  
 
===The uncertainty principle===
 
===The uncertainty principle===

Revision as of 17:42, 9 December 2009

Quantum mechanics is the branch of physics that describes the behavior of systems on very small length and energy scales, such as those found in atomic and subatomic interactions. It is essential for understanding certain concepts that classical physics cannot explain, such as the discrete nature of small-scale interactions, wave-particle duality, the uncertainty principle, and quantum entanglement. Quantum mechanics forms the basis for our understanding of many phenomena, including chemical reactions and radioactive decay, as well as all computers and electronic devices today.

Another historical name for "quantum mechanics" was "wave mechanics," which is a more descriptive term because wave equations were developed to describe the position of particles.

History

Until the early 1900's, scientists believed that electrons and protons were small discrete lumps. Thus, electrons would orbit the nucleus of an atom just as planets orbit the sun. The problem with this idea was that, according to classical electromagnetism, the orbiting electron would emit energy as it orbited. This would cause it to lose rotational kinetic energy and orbit closer and closer to the proton, until it collapses into the proton! Since atoms are stable, this model could not be correct.

The idea of "quanta", or discrete units, of energy was proposed by Max Planck in 1900, to explain the energy spectrum of black body radiation. He proposed that the energy of what we now call a photon is proportional to its frequency. In 1905, Albert Einstein also suggested that light is composed of discrete packets (quanta) in order to explain the photoelectric effect.

In 1915, Niels Bohr applied this to the electron problem by proposing that angular momentum is also quantized - electrons can only orbit at certain locations, so they cannot spiral into the nucleus. While this model explained how atoms do not collapse, not even Bohr himself had any idea why. As Sir James Jeans remarked, the only justification for Bohr's theory was "the very weighty one of success".[1]

It was Prince Louis de Broglie who explained Bohr's theory in 1924 by describing the electron as a wave with wavelength λ=h/p. Therefore, it would be logical that it could only orbit in orbits whose circumference is equal to an integer number of wavelengths. Thus, angular momentum is quantized as Bohr predicted, and atoms do not self-destruct. [1]

Eventually, the mathematical formalism that became known as quantum mechanics was developed in the 1920s and 1930s by John von Neumann, Hermann Weyl, and others, after Erwin Schrodinger's discovery of wave mechanics and Werner Heisenberg's discovery of matrix mechanics.

Principles

  • Every system can be described by a wave function, which is generally a function of the position coordinates and time. All possible predictions of the physical properties of the system can be obtained from the wave function. The wave function can be obtained by solving the Schrodinger equation for the system.
  • An observable is a property of the system which can be measured. In some systems, many observables can take only certain specific values.
  • If we measure such an observable, generally the wave function does not predict exactly which value we will obtain. Instead, the wave function gives us the probability that a certain value will be obtained. After a measurement is made, the wave function is permanently changed in such a way that any successive measurement will certainly return the same value. This is called the collapse of the wave function.

Collapse of The Wave Function

An example of a wave that could be a position function. (Actual position functions are normally much more concentrated.)

In quantum mechanics, it is impossible to make definite statements such as "the particle is here". This is a consequence of the Heisenberg Uncertainty Principle which (simply put) states that particles move randomly. Thus, the position of a particle is not described as a point but rather as the particle's "position function," which gives the probability that the particle is at any given spot. It peaks at the location where the particle "exists" in the classical sense, and we might integrate the function over two angstroms around that point and find that there is a 99% chance of finding the particle in that area; however, there is never a 100% chance; the function always has nonzero "tails" everywhere else in the universe. This means that there is always an infinitesimal chance of the particle suddenly "jumping" a foot or even a light-year away from its original location.

When the particle is actually observed and found to be in a specific location, this wave function has "collapsed" to yield a specific value. However, immediately after that measurement, the particle takes on a new wave function peaking at that location, so scientists cannot keep it collapsed.

Scientists have tried but failed to explain why a wave function collapses and predict to which place it will collapse.

The uncertainty principle

As a result of the wave nature of a particle, neither position nor momentum of a particle can never be precisely known. Whenever its position is measured more accurately (beyond a certain limit), its momentum becomes less certain, and visa versa. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the Heisenberg Uncertainty Principle[2]:

where:

This is what causes Brownean motion of dust particles in the air. Due to the uncertainty principle, each individual air molecule moves randomly. At any given time, more molecules are randomly hitting each dust particle on one side than the other; therefore, it randomly floats in one direction.

Interpretations

Several interpretations have been advanced to explain how wavefunctions "collapse" to yield the observable world we see.

  • The "hidden variable" interpretation[3] says that there is actually a deterministic way to predict where the wavefunction will collapse; we simply have not discovered it. Von Neumann attempted to prove that there is no such way; however, John Stuart Bell pointed out an error in his proof.
  • The many-worlds interpretation says that each particle does show up at every possible location on its wavefunction; it simply does so in alternate universes. Thus, myriads of alternate universes are invisibly branching off of our universe every moment.
  • The currently prevailing interpretation, the Copenhagen interpretation, states that the wavefunctions do not collapse until someone observes the particle at a certain location; until someone observes it, it exists in a quantum indeterminate state of simultaneously being everywhere in the universe. However, Schrodinger, with his famous cat experiment, raised the obvious question: who, or what, constitutes an observer? What distinguishes an observer from the system being observed? In essence, the Copenhagen Interpretation requires a soul or something else to distinguish observers from inanimate matter.

Applications

An important aspect of Quantum Mechanics is the predictions it makes about the radioactive decay of isotopes. Radioactive decay processes, controlled by the wave equations, are random events. A radioactive atom has a certain probability of decaying per unit time. As a result, the decay results in an exponential decrease in the amount of isotope remaining in a given sample as a function of time. The characteristic time required for 1/2 of the original amount of isotope to decay is known as the "half-life" and can vary from quadrillionths of a second to quintillions of years.

See also

Concepts in quantum mechanics

Important contributors to quantum mechanics

External Links

For an excellent discussion of quantum mechanics, see: http://www.chemistry.ohio-state.edu/betha/qm/