Analyzing the Probability Density Distribution of an Electron in Free Space

W Gough

V. F. Krotov

The conditions for observation of the particle coordinates, required by logic of the Special Relativity and filtering the quantum field effects, are described. A general relation between the corresponding density of probability and the wave function is found. It is a relativistic invariant describing probability of the particle emergences in space -time. This density is concretized for bosons, both scalar and vector (including photon), charged and neutral, also electron. The Heisenberg's uncertainty relations have been approved in regards to relativistic particle. As applied to the quantum field, this new construction is transformed to new characteristic of the particles distribution in space-time, which complete distribution throughout impulses. The operators of these distributions and the invariant relativistic description for free quantum fields have been obtained. These new properties of the particles and fields are proposed for experimental investigations.

D A Simmons, Donald J Hinson, A. Ioannidou-Phili

Donald J Hinson

Gong Jian-pin

Using numerical method the picture of probability distribution of the electron in Stark effect is analyzed,the relationship between the direction of the electric dipole moment and the probability density distribution is also discussed.

Roei Remez, Aviv Karnieli, Sivan Trajtenberg‐Mills +4 more

We investigate, both experimentally and theoretically, the interpretation of the free-electron wave function using spontaneous emission. We use a transversely wide single-electron wave function to describe the spatial extent of transverse coherence of an electron beam in a standard transmission electron microscope. When the electron beam passes next to a metallic grating, spontaneous Smith-Purcell radiation is emitted. We then examine the effect of the electron wave function transversal size on the emitted radiation. Two interpretations widely used in the literature are considered: (1) radiation by a continuous current density attributed to the quantum probability current, equivalent to the spreading of the electron charge continuously over space; and (2) interpreting the square modulus of the wave function as a probability distribution of finding a point particle at a certain location, wherein the electron charge is always localized in space. We discuss how these two interpretations give contradictory predictions for the radiation pattern in our experiment, comparing the emission from narrow and wide wave functions with respect to the emitted radiation's wavelength. Matching our experiment with a new quantum-electrodynamics derivation, we conclude that the measurements can be explained by the probability distribution approach wherein the electron interacts with the grating as a classical point charge. Our findings clarify the transition between the classical and quantum regimes and shed light on the mechanisms that take part in general light-matter interactions.