Information about 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 the wave function, expressed as a function of position. The wave function is a complex-valued function of a continuous variable.
For a state ψ, the associated probability density function is ψ*ψ, which is equal to |ψ|2. This is sometimes called just probability density[1], and may be found and used without normalization.
Probability amplitude:
Probability density:
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
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.
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 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
and measured in units of (probability)/(area × time) = r−2t−1.
The probability flux satisfies a quantum continuity equation, i.e.:
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.
It is easy to show that for a plane wave function,
the probability flux is given by
The bi-linear form of the axiom has interesting consequences as well.
For a state ψ, the associated probability density function is ψ*ψ, which is equal to |ψ|2. This is sometimes called just probability density[1], and may be found and used without normalization.
Probability amplitude:
Probability density:
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
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.
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 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
and measured in units of (probability)/(area × time) = r−2t−1.
The probability flux satisfies a quantum continuity equation, i.e.:
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.
It is easy to show that for a plane wave function,
the probability flux is given by
The bi-linear form of the axiom has interesting consequences as well.
Notes
quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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Heisenberg uncertainty principle, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously.
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A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e.
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.
Formally, a probability distribution has density f, if f
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Formally, a probability distribution has density f, if f
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normalisable1: the probability of the particle to occupy any place must equal 1. Mathematically, in one dimension this is expressed as
in which the integration parameters A and
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in which the integration parameters A and
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wave function, proposed by Max Born.
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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In mathematics, an integrable function is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e.
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In mathematics, an integrable function is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e.
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In physics, a wave packet is an envelope or packet containing an arbitrary number of wave forms. In quantum mechanics the wave packet is ascribed a special significance: it is interpreted to be a "probability wave" describing the probability that a particle or particles in a
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resonance is the tendency of a system to oscillate at maximum amplitude at a certain frequency. This frequency is known as the system's resonance frequency. When damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which
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Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to
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In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate
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In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate
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A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.
Formally, a probability distribution has density f, if f
..... Click the link for more information.
Formally, a probability distribution has density f, if f
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Probability is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of
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conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. Any particular conservation law is a mathematical identity to certain symmetry of a physical system.
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A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e.
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Max Born
Max Born
Born November 11 1882
Breslau, Germany
Died January 5 1970 (aged 89)
Göttingen, Germany
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Max Born
Born November 11 1882
Breslau, Germany
Died January 5 1970 (aged 89)
Göttingen, Germany
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