Wave Functions and Probability

Carl Friedrich von Weizsäcker

JAYDIP DATTA

In this note the statistical thermodynamics is correlated to wave function through mathematical probability. The note can be subdivided into two following portions.STATISTICS -THE BASIC OF STATISTICAL THERMODYNAMICS: A CORRELATION and PROBABILITY -A THE BASICS OF WAVE FUNCTION .Randomisation , Wave Function , Probability Density Function , Multiplicative Probability , Partition function , Correlation ( K.P ) coefficient , Probability , Weights , Stirling’s Approximation. Updated to Boltzmann -sigmoid function .

Diederik Aerts

Leon Cohen

Bandlimited functions are functions whose Fourier transform is confined to a finite band of frequencies. We generalize this concept to representations other than the Fourier transform and show that this leads to a variety of inequalities in arbitrary representations. Several special cases are considered, including frequency, dilation, and the chirplet transform, among others. Examples are given to illustrate each result. We apply the results to quantum mechanical wave functions and probability distributions. For bounded momentum wave functions, we obtain explicit bounds on the position wave function and its derivatives, as well as bounds on the position probability distribution. We also consider the dual problem in which the position wave function is bounded, as in the case of a particle in a box with an arbitrary wave function, and obtain bounds on the corresponding momentum wave function and momentum probability distribution. The case of wave functions that are sums of a finite number of energy eigenfunctions is also developed, and bounds on the associated probability distributions are obtained. A number of specific examples are considered, including a truncated Gaussian wave function and a quantum bump wave function.

Chol Jong, Jin-Sim Kim, Shin-Hyok Jon +1 more

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

Silviu Guiaşu