Analytical solution of the Schrodinger equation for the neutral helium atom in the ground state considering the uncertainty principle, vibrational modes and quantum-electrodynamical effects
We present a direct ab initio solution of the Schrodinger equation for neutral helium and helium-like atoms that reproduces the energy of the singlet S state 1S0. By redefining the two-electron wavefunction with tools from complex analysis and solving the S=L=0 problem via Laplace transforms, we derive an analytic description of the entangled electrons. To respect Heisenbergs uncertainty principle while retaining pointlike electrons, we introduce an adapted, generic electron potential and incorporate it into the equations. Perturbative dark electronic vibrational modes, Breit-Hamiltonian corrections, and QED effects (including the Lamb shift) are included. The ground-state energy agrees with literature within 1.86 meV. The framework also accounts for heliums chemical inertness (closed shell) and yields spatial structure parameters in reasonable accord with reported values, providing an analytic route to describing helium.
We present a direct ab initio solution of the Schrodinger equation for neutral helium and helium-like atoms that reproduces the energy of the singlet S state 1S0. By redefining the two-electron wavefunction with tools from complex analysis and solving the S=L=0 problem via Laplace transforms, we derive an analytic description of the entangled electrons. To respect Heisenbergs uncertainty principle while retaining pointlike electrons, we introduce an adapted, generic electron potential and incorporate it into the equations. Perturbative dark electronic vibrational modes, Breit-Hamiltonian corrections, and QED effects (including the Lamb shift) are included. The ground-state energy agrees with literature within 1.86 meV. The framework also accounts for heliums chemical inertness (closed shell) and yields spatial structure parameters in reasonable accord with reported values, providing an analytic route to describing helium.
For the S states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances r1 and r2, and a conditional amplitude, which depends functionally on the interelectronic distance r12 and parametrically on r1 and r2. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point (r1,r2), such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.
Quantum mechanics (QM) and quantum electrodynamics (QED) are often touted as the most successful theories ever. In this paper, this claim is critically evaluated by a test of internal consistency for the ability to calculate the conjugate observables of the nature of the free electron, ionization energy, elastic electron scattering, and the excited states of the helium atom using the same solution for each of the separate experimental measurements. It is found that in some cases quantum gives good numbers, but the solutions are meaningless numbers since each has no relationship to providing an accurate physical model. Rather, the goal is to mathematically reproduce an experimental or prior theoretical number using adjustable parameters including arbitrary wave functions in computer algorithms with precision that is often much greater (e.g. 8 significant figures greater) than possible based on the propagation of errors in the measured fundamental constants implicit in the physical problem. Given the constraints of adherence to physical laws and internal consistency, an extensive literature search indicates that quantum mechanics has never solved a single physical problem correctly including the hydrogen atom and the next member of the periodic chart, the helium atom. Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to the same problem. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of − e moving in the Coulombic field of the proton and the wave equation as modified after Schrodinger, a classical approach is explored which yields a model which is remarkably accurate and provides insight into physics on the atomic level. The proverbial view deeply seated in the wave-particle duality notion that there is no large-scale physical counterpart to the nature of the electron is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems. Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [1-7] that successfully applied physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrodinger boundary condition: “ 0 Ψ→ as →∞ r ,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound 1 = n -state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution, modeled as a superposition of accelerating charges, does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math as in the case of the most famous equations of Newton, Maxwell, Einstein, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used. The accurate solution of the helium atom is confirmed by the agreement of predicted and observed conjugate parameters using the same unique physical model in all cases.
This paper applies the Scretching–Schrödinger Equation (SSE) to hydrogen, helium, and lithium as foundational atomic test systems for connecting wave mechanics, matrix mechanics, and spectroscopic closure. The study begins with the standard time-independent Schrödinger equation as the eigenvalue foundation for atomic structure, energy levels, quantum numbers, and electron configurations. Hydrogen is treated as the exact one-electron benchmark, while helium and lithium are treated as many-electron systems requiring effective Hamiltonians, electron–electron interaction terms, screening, configuration structure, and approximate spectroscopic interpretation. The paper compares the Schrödinger and Heisenberg formulations of quantum mechanics. The Schrödinger representation supplies wavefunctions, eigenstates, orbital quantum numbers, and coordinate-space probability amplitudes. The Heisenberg representation supplies the equivalent operator and matrix formulation, where observables evolve through commutators and transition amplitudes are represented through matrix elements. This comparison is extended into a new SSE matrix formulation in which the ordinary Hamiltonian is modified by a spectroscopic closure term: H^SSE=H^0+V^SQC\hat{H}_{SSE}=\hat{H}_{0}+\hat{V}_{SQC}H^SSE=H^0+V^SQC Here, H^0\hat{H}_{0}H^0 is the standard atomic Hamiltonian, and V^SQC\hat{V}_{SQC}V^SQC is the Scretching Quantum Chain closure operator. The corrected SSE derivation is preserved by treating hydrogen as the calibration benchmark, where V^SQC(H)≈0\hat{V}_{SQC}(H)\approx 0V^SQC(H)≈0 for the exact hydrogenic limit. The paper also carefully distinguishes the single Lyman-α\alphaα oscillator strength, f(1s→2p)=0.4162f(1s\rightarrow2p)=0.4162f(1s→2p)=0.4162, from the full Thomas–Reiche–Kuhn sum rule, ∑ifi=1\sum_i f_i=1∑ifi=1, so that individual transitions are not confused with the total oscillator-strength conservation law. The SQC layer links atomic wavefunction matrix elements to measurable spectroscopic quantities, including oscillator strength fff, transition dipole moment ∣μ∣2|\mu|^2∣μ∣2, Einstein spontaneous-emission coefficient A21A_{21}A21, transition frequency ν\nuν, wavelength λ\lambdaλ, and spectral-line structure. In this framework, hydrogen provides the exact reference model, helium tests the effect of electron correlation and two-electron screening, and lithium extends the method to a three-electron atom with inner-shell shielding and valence-electron spectral transitions. The OpenStax University Physics III: Optics & Modern Physics source provides the broader optical and modern-physics bridge by connecting Maxwell’s electromagnetic-wave theory with photon behavior, matter waves, wave–particle duality, atomic spectra, lasers, molecular spectra, superconductivity, nuclear physics, and particle physics. The LibreTexts quantum-mechanics source adds the formal operator and Hamiltonian framework, including time-dependent and time-independent wavefunctions, the three-dimensional Schrödinger equation, hydrogenic structure, spin–orbit coupling, kinetic-energy corrections, Lamb shift, Zeeman effect, and the Heisenberg uncertainty principle. Greiner’s textbook treatment is incorporated as a formal comparison source for quantization, radiation laws, matter waves, the Schrödinger equation, harmonic oscillators, charged particles in magnetic fields, mathematical foundations, matrix mechanics, Schrödinger, Heisenberg, and interaction representations, perturbation theory, spin, identical particles, and conceptual foundations. The resulting paper presents hydrogen, helium, and lithium as a three-stage atomic ladder for developing the SSE: hydrogen verifies the zero-correction benchmark, helium introduces many-electron correction structure, and lithium extends the method to screened valence-electron spectroscopy. The final contribution is a unified Schrödinger–Heisenberg–Scretching framework in which wavefunctions, operators, matrices, oscillator strengths, transition dipoles, Einstein coefficients, and spectral frequencies are placed into one mathematically consistent atomic-spectroscopic chain.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.
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.
In this paper, we accurately estimate the ground-state energy and the atomic radius of the helium atom and a helium-like Hookean atom by employing the uncertainty principle in conjunction with the variational approach. We show that with the use of the uncertainty principle, electrons are found to be spread over a radial region, giving an electron cloud. Our calculation also shows how the Coulomb interaction between electrons affects their distribution. This leads to a physical picture of how electrons are located with respect to each other in these atoms. Finally, we also obtain through our calculations a general formula for the estimate of ground-state energy and radius of two electron atoms and ions with atomic number Z.