The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this context, the special case ( $N=4$ , $s=4$ ) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$ . In the present article we derive alternatively $p(κ)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s ε_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(κ)$ transforms to the Boltzmann law $P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty$ where $\langle ε\rangle = E/N$ . Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this context, the special case ($N=4$, $s=4$) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$. In the present article we derive alternatively $p(κ)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s ε_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(κ)$ transforms to the Boltzmann law $P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty$ where $\langle ε\rangle = E/N$. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.

arXiv:2111.09246Nov 17, 20210 citations

States, Measurements and Probabilities

Emilio d’Emilio, Luigi E. Picasso

UnitextJan 1, 20110 citations

Coherent states and Bayesian duality

S. Twareque Ali, Jean‐Pierre Gazeau, Bárbara Heller

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.

arXiv:0803.2269Journal of Physics A Mathematical and TheoreticalAug 5, 200823 citations

States, Measurements and Probabilities

Emilio d’Emilio, Luigi E. Picasso

UNITEXT for physicsJan 1, 20170 citations

Quantum State Analysis : Probability theory as logic in Quantum mechanics

Anders Månsson

Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.

Jan 1, 20070 citations

PROBABILITIES, OBSERVATIONS AND PREDICTIONS

Arthur H. Copeland

University of California Press eBooksDec 31, 19565 citations

The statistical problem of the distribution of

s

quanta of equal energy

ε_0

and total energy

E

among

N

distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by

p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s

In this context, the special case (

N=4

s=4

) presented indicates that the alternative concept of most probable state is not valid for finite values of

s

and

N

. In the present article we derive alternatively

p(κ)

by distributing

s

quanta over

N

particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $;

s ε_{o} = E \sim

fixed, where the energy of the particles becomes continuous,

p(κ)

transforms to the Boltzmann law

P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty

where

\langle ε\rangle = E/N

. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.