Abstract
The hydrogen atom is a concrete example of the general case of a two-body problem, in which two interacting bodies are considered without any external forces. The total energy of this system is composed of the kinetic energies of the two bodies and their potential energy, i.e. the energy of interaction V between them:
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Notes
- 1.
The relativistic case is treated in Appendix F, Vol. 2.
- 2.
Hence, the interaction of the two bodies does not depend on their absolute position in space, but only on their relative positions w.r.t. each other.
- 3.
If one of the bodies has a much greater mass than the other (\(m_{1}\gg m_{2}\), e.g. hydrogen atom), we have \(\mu \approx m_{2}\); if the two masses are equal (\(m_{1}=m_{2}\), e.g. positronium), we have \(\mu =m_{1}/2\).
- 4.
An even more compact notation (which we however will not need in the following) can be formulated using the radial momentum
$$\begin{aligned} p_{r}= \frac{\hbar }{i}\frac{1}{r}\frac{\partial }{\partial r}r=\frac{\hbar }{i} \left( \frac{\partial }{\partial r}+\frac{1}{r}\right) , \end{aligned}$$namely
$$\begin{aligned} \mathbf {p}^{2}=p_{r}^{2}+\frac{{\mathbf {l}}^{2}}{r^{2}}\ \ \mathrm{or}\ \ {\varvec{\nabla }}_{\mathbf {r}}^{2}=-\frac{p_{r}^{2}}{\hbar ^{2}}-\frac{{\mathbf {l}}^{2} }{\hbar ^{2}r^{2}}. \end{aligned}$$It holds that \(\left[ r, p_{r}\right] =i\hbar \).
- 5.
The dependence on \(\mu \) and on the details of the potential is not noted in general.
- 6.
We have \(\int Y_{l}^{m*}\left( \vartheta ,\varphi \right) Y_{L}^{M}\left( \vartheta ,\varphi \right) \mathrm{d}\Omega =\delta _{Ll}\delta _{Mm}\).
- 7.
Reductions of this kind make life considerably easier, both in theoretical and in computational terms.
- 8.
The irregular solution behaves at the origin as \(\sim \!r^{-l}\).
- 9.
We repeat the remark of Chap. 1, Vol. 1, that the term V in the SEq, although in fact the potential energy, is usually called just ‘potential’. In electrodynamics, \(V(r)= \frac{1}{4\pi \varepsilon _{0}}\frac{q}{r}\) is the potential of a point charge q. Since in quantum mechanics V(r) denotes the potential energy, there must be two charges.
- 10.
Somewhat misleadingly sometimes also called the radius of the hydrogen atom (actually the hydrogen atom does not have a well-defined radius).
- 11.
Another possibility to find a solution would be e.g. to look up a book on ‘special functions’, such as Abramowitz, and to convince oneself that the solution can be formulated in terms of special functions, here the Laguerre polynomials; see below. An entirely different option is the algebraic approach, keyword Lenz vector (see Appendix G, Vol. 2).
- 12.
The convention that the same symbol R is used for the radial function and for the Rydberg constant may not be very clever didactically, but it is well established.
- 13.
Because of the dependence on the reduced mass, the Rydberg constant of positronium has only half the value of that for hydrogen. Also, for ‘normal’ and heavy hydrogen, the Rydberg constants differ due to the dependence on the masses. This allows one to determine spectroscopically the proportions of the two isotopes. Incidentally, the notation
$$\begin{aligned} R_{\infty }=\frac{m_{e}e^{4}}{2\hbar ^{2}\left( 4\pi \varepsilon _{0}\right) ^{2}} \end{aligned}$$is also common; it refers to an infinite nuclear mass.
- 14.
The reason is that a further conserved quantity exists: the Lenz vector (see Appendix G, Vol. 2).
- 15.
If one describes the interaction more realistically, the term diagram changes in subtle ways (cf. Chap. 19).
- 16.
If they are not degenerate, it is sufficient to specify the energy eigenvalue.
- 17.
And the spin quantum number \(s_{z}\), if the appropriate operators are incorporated into the Hamiltonian, see Chap. 19.
- 18.
If two operators A and B commute with a third operator C, that does not necessarily mean \(\left[ A, B\right] =0\). An example of this behavior, called contextuality, is provided by the components of the angular-momentum operator \(\left[ l_{x},\mathbf {l}^{2}\right] =0\), \(\left[ l_{y},\mathbf {l}^{2}\right] =0\), \(\left[ l_{x}, l_{y}\right] \ne 0\). Thus, for a CSCO, observables are required which commute pairwise and with H.
Contextuality means that the result of a measurement depends on other measurements performed at the same time; see also Chap. 27.
- 19.
See also the remarks on the Galilean transformation in Appendix L, Vol. 2 and the discussion of relativistic topics in Appendix U, Vol. 1, and Appendix W, Vol. 2.
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Pade, J. (2018). The Hydrogen Atom. In: Quantum Mechanics for Pedestrians 2. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00467-5_17
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DOI: https://doi.org/10.1007/978-3-030-00467-5_17
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