Abstract
The Hydrogen Atom is considered. Its energy spectrum and energy eigenfunctions are derived. The energy splittings due to a homogeneous magnetic field (Zeeman effect) are analyzed. Corrections to the spectrum due to relativistic effects and due to the magnetic field generated by the motion of the electron (Fine Structure) are computed. Corrections due to the effective coupling between the proton and electron spin-magnetic moments (Hyperfine Structure) are also computed. As an example of other atoms the Helium atom is considered and its ground state energy is estimated.
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Notes
- 1.
- 2.
Radiation absorption and emmission phenomena will be discussed in a future chapter.
- 3.
Note that an alternative normalization is sometimes used in the literature in which the 1Â /Â s! factor in the generating formula for Laguerre polynomials is missing. Nevertheless, apart from a modification of the wave function normalization factor, we obtain ultimately the same Hydrogen energy eigenfunctions.
- 4.
You may use the property of associated Laguerre polynomials \(\int _0^{\infty }dx x^{\alpha }\,e^{-x}\,\left( L_n^{\alpha }(x)\right) ^2\,=\,{\Gamma (n+\alpha +1)}/{n!}\,.\)
- 5.
In a subsequent chapter we are going to consider the general problem of a charged particle in a magnetic field. There, we will see that in addition to the interaction term linear in the magnetic field there is also an interaction term quadratic in the magnetic potential. Such a term turns out to be negligible in atoms subject to magnetic fields of moderate size. Nevertheless, it can become relevant in the presence of very strong magnetic fields, as the ones encountered in astrophysical environments.
- 6.
In the discussion of corrections to the Hydrogen atom Hamiltonian we may also include the fact that the proton is not immovable by simply replacing the electron mass with the reduced mass \(m=m_em_p/(m_e+m_p)\).
- 7.
These are eigenstates of \(L^2,\,L_z,\,S^2,\,S_z\) and, of course, \(\hat{H}_0\).
- 8.
A constant in the energy can always be subtracted away and has no physical effect. Such a constant becomes relevant only in the framework of the theory of gravity (General Relativity).
- 9.
\(2|E_1|=m_ee^4\hbar ^2=(m_ec^2)\frac{e^4}{\hbar ^2c^2}\,=\,\alpha ^2(m_ec^2)\).
- 10.
- 11.
Helium exists in the form of two isotopes, namely as \({}^3\mathrm{He}\), with a nucleus of two protons and a neutron, and as \({}^4\mathrm{He}\), with a nucleus of two protons and two neutrons. The two types of nuclei have drastically different properties due to the fact that the former is a fermion, while the latter a boson.
References
D. Griffiths, Introduction to Quantum Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2017)
E. Merzbacher, Quantum Mechanics, 3rd edn. (Wiley, Hoboken, 1998)
G. Baym, Lectures in Quantum Mechanics, Lecture Notes and Supplements in Physics (ABP, 1969)
F. Levin, An Introduction to Quantum Theory (Cambridge University Press, Cambridge, 2002)
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Tamvakis, K. (2019). Atoms. In: Basic Quantum Mechanics. Undergraduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-22777-7_15
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DOI: https://doi.org/10.1007/978-3-030-22777-7_15
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