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American Journal of Physics

A real ‘‘thought’’ experiment for the hydrogen atom

I. E. McCarthy, E. Weigold

Published Feb 1, 1983• 10 citations
Publisher

Abstract

The absolute square of the normalized position-space wave function of the electron in a hydrogen atom is interpreted as the probability distribution for observations of the position of the electron. This is only a thought experiment, since the electron’s position cannot be observed. The first observation of the momentum distribution of the electron directly verifies the probability interpretation of the momentum-space wave function.

DOI
10.1119/1.13312

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The absolute square of the normalized position-space wave function of the electron in a hydrogen atom is interpreted as the probability distribution for observations of the position of the electron. This is only a thought experiment, since the electron’s position cannot be observed. The first observation of the momentum distribution of the electron directly verifies the probability interpretation of the momentum-space wave function.
The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which PPP, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates rrr, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits PPP and P+dPP+dPP+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.
The process of observation disturbs the thing being observed and that disturbance might possibly affect other observations. Heisenberg showed that simultaneous measurements of an electron's position and its momentum interfere with each other and thus induce essential uncertainties into those measurements. Quantum mechanics represents the behavior of electrons around nuclei in terms of mathematical constructs called wave functions, or orbitals, which contain all possible information about the electron. The square of the wave function gives the probability of finding the electron in a given region of space. For mathematical reasons, physicists and chemists almost always use the abstract concept of wavefunctions (orbitals), rather than that of electron probability clouds, to describe electrons. The distribution of probability within a probability cloud is not uniform. For some, the greatest probability is near, or even at, the nucleus
A few decades ago the electron was thought by some to be smeared out over space so that it could not be regarded as a discrete particle.\nThe distribution curve given by solutions of Schroedinger's wave equation is now widely interpreted as a probability of position, rather than as a true "picture" of a single electron. To even use the words "probability of position" implies that an individual electron must have a unique position in space and time. If then an electron has position in space and time the next logical step for curious theorists would seem to be to ask themselves what new models can be built up from that idea.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
The Bohr model is highly successful in explaining many features of the hydrogen atom. The ground state wave function does not have any angular dependence. The normalized hydrogen atom wave functions are directly accessible in Mathematica and can be readily verified as solutions of the Schrodinger equation. Get the hydrogen atom ground state wave function in terms of the Bohr radius a. The probability of finding the electron per volume at a given location is given by the square of the wave function. The electron has only a very tiny probability of being found inside the proton, whose size is about 1 fm.
The major problem that physicists were trying to solve was that of the wave nature of the electron. In 1926, the Austrian physicist Erwin Schrodinger presented an equation that described the wave behavior of the electron in the hydrogen atom. The physicist Max Born suggested instead that these waves were waves of probability; that is, they represented the probability of finding the electron in some region of space. In fact, an electron has a well-defined position only when it is being observed. Position is not an intrinsic property of an electron. The wave nature of electrons is the basis for the electron microscope, invented in 1932 by Ernst Ruska in Germany. Electron wavelengths are much shorter than those of visible light.