What shape the probability distribution for finding a particular particle in 3D space usually takes at any given time, for free particles not subject to any external influence? and does this shape changes with the type of particle?
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2 Answers
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The wave function can be essentially any square-integrable function (depending on the problem). For instance, consider the harmonic oscillator. The energy eigenstates consist of Hermite polynomials multiplied by Gaussians, and so the probability distributions associated with the energy eigenstates are products of polynomials and Gaussians.
However, any linear combination of these functions, provided that the resulting function is square-integrable, is a valid quantum harmonic oscillator state. So while there are special states (the energy eigenstates) that have particular functional forms (and, for the ground state of the harmonic oscillator, we do get a Gaussian as the probability distribution function), really any square-integrable function on the real line is a valid quantum state.
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It is not random, but determined by the specific problem. E.g., the ground state of a harmonic oscillator is described by a Gaussian distribution. However, for other types of potential or even other eigenstates of the oscillator, the distribution is different.
In other words, the number of different distributions is not limited. The ones cited in the OP are some model distributions that are frequently used in physics and statistics, mainly because they are easy to manipulate and/or have appear as limiting cases in many practical problems (e.g., Gaussian/normal distribution is appears in the Central limit theorem, as a distribution of a sum of many small random values).
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$\begingroup$ But don’t particles in free space, as they are not subject to any particular external influence, have a usaual type of distributions? $\endgroup$– NellCommented Mar 17, 2022 at 16:30