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Encyclopedia > Probability amplitude

In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. For example, each particle has a probability amplitude describing its position. This amplitude is then called wave function. This is a complex-valued function of the position coordinates. A simple introduction to this subject is provided in Basics of quantum mechanics. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... Partial plot of a function f. ... In quantum physics, the Heisenberg uncertainty principle states that one cannot assign values (with full precision) to certain pairs of observable variables of a single elementary particle at the same time even with infinitely precise instruments. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...


For a probability amplitude ψ, the associated probability density function is ψ*ψ, which is equal to |ψ|2. This is sometimes called just probability density1, and may be found used without normalization. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In quantum mechanics, wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must equal 1. ...


If |ψ|2 has a finite integral over the whole of three-dimensional space, then it is possible to choose a normalising constant, c, so that by replacing ψ by cψ the integral becomes 1. Then the probability that a particle is within a particular region V is the integral over V of |ψ|2. Which means, according to the Copenhagen interpretation of quantum mechanics, that, if some observer tries to measure the quantity associated with this probability amplitude, the result of the measurement will lie within ε with a probability P(ε) given by In calculus, the integral of a function is a generalization of area, mass, volume and total. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ...

Probability amplitudes which are not square integrable are usually interpreted as the limit of a series of functions which are square integrable. For instance the probability amplitude corresponding to a plane wave corresponds to the 'non physical' limit of a monochromatic source of particles. Another example: The Siegert wave functions describing a resonance are the limit for of a time-dependent wave packet scattered at an energy close to a resonance. In these cases, the definition of P(ε) given above is still valid. In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... The wave packet is one of the most widely misunderstood and misused concepts in physics. ... The Tacoma Narrows Bridge (shown twisting) in Washington collapsed spectacularly, under moderate wind, in part because of resonance. ...


The change over time of this probability (in our example, this corresponds to a description of how the particle moves) is expressed in terms of ψ itself, not just the probability function |ψ|2. See Schrödinger equation. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...


In order to describe the change over time of the probability density it is acceptable to define the probability flux (also called probability current). The probability flux j is defined as: In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. ... In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. ...

and measured in units of (probability)/(area*time) = r-2t-1.


The probability flux satisfies a quantum continuity equation, i.e.: All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

where P(x,t) is the probability density and measured in units of (probability)/(volume) = r-3. This equation is the mathematical equivalent of probability conservation law. In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... The word probability derives from the Latin probare (to prove, or to test). ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...


It is easy to show that for a plane wave function, In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

the probability flux is given by

The bi-linear form of the axiom has interesting consequences as well.


Notes


  Results from FactBites:
 
Amplitude - Wikipedia, the free encyclopedia (367 words)
Amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle.
The amplitude of sound waves and audio signals conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the diaphragm of a speaker) is described.
The square of the amplitude is proportional to the intensity of the wave.
Probability amplitude - Wikipedia, the free encyclopedia (404 words)
In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity.
Probability amplitudes which are not square integrable are usually interpreted as the limit of a series of functions which are square integrable.
For instance the probability amplitude corresponding to a plane wave corresponds to the 'non physical' limit of a monochromatic source of particles.
  More results at FactBites »

 
 

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