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Springer eBooks

Schrödinger Equation in One-Electron System: Hydrogenic Atom

Taku Onishi

Published Jan 1, 2022• 0 citations
Publisher
DOI
10.1007/978-981-19-2669-3_7

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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.

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Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.

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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.
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 Schrödinger wave theory to one-electron atom. The simplest one-electron system is the Hydrogen atom, whose Schrödinger 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.
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.