Abstract The Heisenberg uncertainty relations are cornerstones of quantum theory. Yet care is required in applying them. While they determine the lowest energy level in some systems, such as the isotropic three-dimensional harmonic oscillator, they fail to do so in other systems, notably, the hydrogen atom. This failure has been overlooked in many textbooks and pedagogic articles. The failure arises due to the singularity of the Coulomb potential. To elucidate the point, we present four subfamilies of position probability distributions. By delineating the limitations in applying position-momentum uncertainty to the hydrogen atom, we hope to improve the teaching and learning of modern physics.
Abstract The Heisenberg uncertainty relations are cornerstones of quantum theory. Yet care is required in applying them. While they determine the lowest energy level in some systems, such as the isotropic three-dimensional harmonic oscillator, they fail to do so in other systems, notably, the hydrogen atom. This failure has been overlooked in many textbooks and pedagogic articles. The failure arises due to the singularity of the Coulomb potential. To elucidate the point, we present four subfamilies of position probability distributions. By delineating the limitations in applying position-momentum uncertainty to the hydrogen atom, we hope to improve the teaching and learning of modern physics.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
Background: There has been altercation for and against Heisenberg uncertainty principles, whose earlier mathematical form has assumed other forms. This is despite the fact that the original form ought to be for the hydrogen atom only. Methods: the method is purely theoretical and computational, involving the use of spectroscopic data in the literature. Aim and objectives: The aim of the research was to show that half of reduced Planck’s constant (ℏ/2) cannot be restricted to a lower limit to the product of the uncertainty in the momentum and the uncertainty in the position. The objectives were: 1) to undertake the derivation of an equation unifying the Heisenberg uncertainty principle and atomic principles; and 2) to give evidence that half of the reduced Planck’s constant (ℏ/2) is strictly for hydrogen. Results and discussion: With hydrogen apart from other elements, fractions and multiples of ℏ/2, 0.0053–0.0264 corresponding to the radii ranging between a0/100 and a0/20 (a0 is the Bohr’s radius), and 5-26 corresponding to the radii ranging between 4a0 and 49a0 were demonstrated; the probability of locating the position and momentum increases with decreasing radii. Conclusion: The equations for the determination of the fractions and multiples of ℏ/2 were derived. Computation gave values less than and higher than ℏ/2; the latter was found to be restricted to hydrogen only. The possibility of linking the product of uncertainties in position and momentum to Pa may be reserved for a future study.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.
With hydrogen apart from other elements, fractions and multiples of ℏ/2, 0.0053–0.0264 corresponding to the radii ranging between a0/100 and a0/20 (a0 is the Bohr’s radius), and 5-26 corresponding to the radii ranging between 4a0 and 49a0 were demonstrated; the probability of locating the position (a) and momentum (p) increases with decreasing radii. The equations for the determination of the fractions and multiples of ℏ/2 were derived. Computation gave values less than and higher than ℏ/2; the latter was found to be restricted to hydrogen atom only. A relation between uncertainties and pa is for future study.
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in 1/D in similar systems with a non-standard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-D limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The D-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the D-dimensional hydrogenic atom at the limit of large D. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.