# The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron.

Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet.

In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

Classical physics draws a distinction between particles and energy, holding that only the latter exhibit waveform characteristics, whereas quantum mechanics is based on the observation that matter has both wave and particle aspects and postulates that the state of every subatomic particle can be described by a wavefunction — a mathematical expression used to calculate the probability that the particle, if measured, will be in a given location or state of motion.

# The description of nature is essentially probabilistic, with the probability of an event related to the square of the amplitude of the wave function.

One of the mathematical constructs that enables experimenters to predict very accurately certain experimental results is sometimes called a probability wave.

In its mathematical form it is analogous to the description of a physical wave, but its " crests " and " troughs " indicate levels of probability for the occurrence of certain phenomena ( e. g., a spark of light at a certain point on a detector screen ) that can be observed in the macro world of ordinary human experience.

It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object.

The wave function is complex and since its square modulus is associated with the probability of observing the object, the complex character of the wave function is associated to the phase.

According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate.

Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.

If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates.

Some wave functions produce probability distributions that are constant, or independent of time-such as when in a stationary state of constant energy, time vanishes in the absolute square of the wave function.

Unlike the earlier Bohr model of the atom, however, the wave model describes electrons as " clouds " moving in orbitals, and their positions are represented by probability distributions rather than discrete points.

The emitted wave frequency of the thermal radiation is a probability distribution depending only on temperature, and for a black body is given by Planck's law of radiation.

Knowing the wave function allows one to calculate the probability density for that electron to be found at some location.

In the case of tunneling, the tip and sample wave functions overlap such that when under a bias, there is some finite probability to find the electron in the barrier region and even on the other side of the barrier.

Let us assume the bias is V and the barrier width is W. This probability, P, that an electron at z = 0 ( left edge of barrier ) can be found at z = W ( right edge of barrier ) is proportional to the wave function squared,

In one dimension for one particle, the probability of finding a particle in space at some time t ( not shown ) is distributed as a ( complex-valued ) waveform through space, mathematically described by the particle's wave function ψ.

For the plane wave ( left ), the wave-like probability distribution ψ the particle is likely to be found anywhere, due to the periodicity of the wave.

Therefore, if the target can significantly change its location in something less than 30 minutes, the probability of having destroyed it is drastically lowered.

If the target can change its position significantly during the 30 minutes the missile is in the air on its way, the probability of the missile destroying the target is drastically reduced.

Strictly speaking, this means that the probability for each possible outcome of the experiment can be computed by multiplying together the probabilities of the possible outcomes of the single binomial trials.

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:

If a defendant has been convicted and can prove that his lawyer did not adequately handle his case and that there is a reasonable probability that the result of the trial would have been different had the lawyer given competent representation, he is entitled to a new trial.

* Pattern: A character who has walked the pattern can walk in shadow to any possible universe, and while there can manipulate probability.

This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus.

Solomonoff used this algorithm, and the code lengths it allows, to define a " universal probability " of a string on which inductive inference of the subsequent digits of the string can be based.

We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound.

It can be said that the quantum state is measured to be in the correct state with high probability.

The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain.

In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

Pollution can occur from a number of sources: garbage and industrial debris ; the transportation of petroleum in tankers, increasing the probability of large oil spills ; small oil spills created by large and small vessels, which flush bilge water into the ocean.

A halting probability can be interpreted as the measure of a certain subset of Cantor space under the usual probability measure on Cantor space.

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6 / π < sup > 2 </ sup > ( see pi ), which is about 61 %.

Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law.

For example, the probability that an alpha particle striking a beryllium target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

There is a well-known relationship between probability and entropy which states that Af, where **zq is the probability that state ( i.e., volume for an ideal gas ) could be reached by chance alone.

Sample areas in the new investigations were selected strictly by application of the principles of probability theory, so as to be representative of the total population of defined areas within calculable limits.

Rather than viewing the abortive recovery in 1959-60 as a reason for believing we have lost prospects for growth '', he said `` it should be viewed as a lesson well learned which will increase the probability of substantial improvement in this recovery ''.

There must not only be greater good than evil objectively in view, but also greater probability of actually doing more good than harm.

If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify ( in the protocol for the experiment or observational study ) that the responses be transformed to stabilize the variance.

In the case of new evidence, there must be a high probability that its presence or absence would have made a material difference in the trial.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ( sometimes termed the atom ’ s “ electron cloud ” ) tends toward a generally spherical zone of probability describing where the atom ’ s electrons will be found.

The idea is that the dealer's second card has a fairly high probability ( nearly one-third ) to be ten-valued, giving the dealer blackjack and disappointment for the player.

* For the frequentist a hypothesis is a proposition ( which must be either true or false ), so that the frequentist probability of a hypothesis is either one or zero.

After the 1920s, " inverse probability " was largely supplanted by a collection of methods that came to be called frequentist statistics.

Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments.