The incompatibility between two observables in quantum theory is described by Heisenberg's uncertainty principle. In this work, we study spatial confinement effects on Heisenberg's uncertainty principle for a hydrogen atom located at the center of an impenetrable spherical cavity with radius
ro. Both the radial and vector representation of the uncertainty principle are considered. For this, we solve the Schr\"odinger equation numerically within a finite-differences approach. We find that for small cavity sizes the values of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\mathrm{\ensuremath{\Delta}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{r}$ (radial) bunch according to the number of nodes and that for Rydberg states, i.e., large excitation, they become more coherent, satisfying exactly Heisenberg's uncertainty principle, in contrast to the vector description. However, for the vector case, we find that $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{r}}$ degenerates for small cavity sizes and bunches according to the principal quantum number
n for large cavities. We find that the behavior of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ is responsible for the breaking of the energy degeneracy for confined quantum systems. This occurs when the confinement radius is of the order of the orbital size, as determined by the electron average distance $\ensuremath{\langle}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\ensuremath{\rangle}$. In addition, we estimate the critical cavity size for which relativistic effects become relevant and verify that the relativistic corrections to the energy, obtained from first-order perturbation theory, become important when the total energy of the atom surpasses 93.845 Hartree, corresponding to 10% of the speed of light, which is fulfilled for cavity sizes
ro<1 a.u.
The purpose of this study is to analyze the probability and uncertainty of electron linear momentum in with the Heisenberg uncertainty approach. Measurement of the position and momentum of atomic electrons is probabilistic. The probability and uncertainty of electron linear momentum are analysed analytically, and simulations of hydrogenic atoms' normalized radial wave function are performed. they can be viewed as hydrogenic atoms with only one electron orbital. The probability and uncertainty of electron linear momentum in decrease with increasing values of the principal quantum number n ≤ 3. While the uncertainty of the electron position is increasing. The results of this study are in accordance with the characteristics of position and linear momentum that are not commutable. The increase in the value of the main quantum number means that the electron's position against this is getting farther, and the speed in the orbital is getting smaller.
While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.