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Classical Probability Density
The classical probability density is the probability density function that represents the likelihood of finding a particle in the vicinity of a certain location subject to a potential energy in a classical mechanical system. These probability densities are helpful in gaining insight into the correspondence principle and making connections between the quantum system under study and the classical limit. Mathematical background Consider the example of a simple harmonic oscillator initially at rest with amplitude . Suppose that this system was placed inside a light-tight container such that one could only view it using a camera which can only take a snapshot of what's happening inside. Each snapshot has some probability of seeing the oscillator at any possible position along its trajectory. The classical probability density encapsulates which positions are more likely, which are less likely, the average position of the system, and so on. To derive this function, consider the fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all parts of the system move sinusoidally with that same frequency). If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. Overview Free vibrations of an elastic body are called ''natural vibrations'' and occur at a frequency called the natural frequency. Natural vibrations are different from forced vibrations which happen at the frequency of an applied force (forced frequency). If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. This phenomenon is known as resonance. In a mass-spring system, with mass ''m'' and spring stiffness ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior. Quantum theory A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the Planck constant normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf. WKB approximation). More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constant , so the "deformation parameter" / can be effectively taken to be zero (cf. Weyl quantization.) Thus ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correspondence Principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations. The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom. The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter. Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position And Momentum Space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all '' position vectors'' r in space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all ''momentum vectors'' p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of asslength] imesup>−1. Mathematically, the duality between position and momentum is an example of ''Pontryagin duality''. In particular, if a function is given in position space, ''f''(r), then its Fourier transform obtains the function in momentum space, ''φ''(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function. These quantities and ideas tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bouncing Ball
The physics of a bouncing ball concerns the physical behaviour of bouncing balls, particularly its motion before, during, and after impact against the surface of another body. Several aspects of a bouncing ball's behaviour serve as an introduction to mechanics in high school or undergraduate level physics courses. However, the exact modelling of the behaviour is complex and of interest in sports engineering. The motion of a ball is generally described by projectile motion (which can be affected by gravity, drag, the Magnus effect, and buoyancy), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure). To ensure fair play, many sports governing bodies set limits on the bounciness of their ball and forbid tampering with the ball's aerodynamic properties. The bounciness of balls has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (for example, in rotation) or the rate of change of the phase of a sinusoidal waveform (for example, in oscillations and waves), or as the rate of change of the argument of the sine function. Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity.(UP1) One turn is equal to 2''π'' radians, hence \omega = \frac = , where: *''ω'' is the angular frequency (unit: radians per second), *''T'' is the period (unit: seconds), *''f'' is the ordinary frequency (unit: hertz) (sometimes ''ν''). Units In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. The unit hertz (Hz) is dimensionally equivalent, but by convention it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spring Constant
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. Hooke's equation holds (to some extent) in many other situations where an elasticity (physics), elastic body is Deformation (physics), deformed, such as wind blowing on a tall building, and a musician plucking a string (music), string of a guitar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic scale, microscopic particles like atoms and molecules, to macroscopic scale, macroscopic particles like powder (substance), powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion (physics), motion. The term ''particle'' is rather general in meaning, and is refined as needed by various scientific fields. Anything that is composed of particles may be referred to as being particulate. However, the noun ''particulates, particulate'' is most frequently ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
