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Journal of Modern Physics

On the Heisenberg and Schrödinger Pictures

Shigeji Fujita, James R. McNabb III, Akira Suzuki

Published Jan 1, 2014• 1 citation
Publisher

Abstract

A quantum theory for a one-electron system can be developed in either Heisenberg picture or Schrodinger picture. For a many-electron system, a theory must be developed in the Heisenberg picture, and the indistinguishability and Pauli’s exclusion principle must be incorporated. The hydrogen atom energy levels are obtained by solving the Schrodinger energy eigenvalue equation, which is the most significant result obtained in the Schrodinger picture. Both boson and fermion field equations are nonlinear in the presence of a pair interaction.

DOI
10.4236/jmp.2014.55027

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A quantum theory for a one-electron system can be developed in either Heisenberg picture or Schrodinger picture. For a many-electron system, a theory must be developed in the Heisenberg picture, and the indistinguishability and Pauli’s exclusion principle must be incorporated. The hydrogen atom energy levels are obtained by solving the Schrodinger energy eigenvalue equation, which is the most significant result obtained in the Schrodinger picture. Both boson and fermion field equations are nonlinear in the presence of a pair interaction.
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Bohr theory was not satisfactory as it failed to provide an interpretation of relative line intensities and splitting of spectral lines in the presence of electric field and magnetic field in the hydrogen spectrum. This theory also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this chapter, we consider the application of Schrodinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrodinger equation can be exactly solved. Other one-electron system like He + , Li 2+ etc. can also be solved exactly. The chemical system with more than one-electron cannot be solved exactly due to electron-electron repulsion, which is the most problematic term.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
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.
‘Quantum mechanics’ shows how Bohr's old quantum theory was superseded by a newer theory of quantum mechanics. Bohr's quantum theory worked well in single electron sytems, but not in multi-electron systems. Quantum mechanics allowed the development of Schrödinger's equation, which could theoretically predict the determination of electron energy levels in any system. The advantage of quantum mechanics over quantum theory lies in its treatment of electrons as waves. This allowed Schrödinger to apply mathematical boundary conditions to his equation and quantise the energy levels of electrons. Further work by Heisenberg showed that electrons can be thought of as being spread around a shell rather than being in one place.
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.