# Quantum mechanics of one- and two-electron atoms The measurement of the Faraday then gives e.

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Mass-spectroscopy and N gives Mp , the proton mass. These non-atomic values agree with the more accurate ones, quoted above, to within approximately the experimental error the discrepancy is less than 1. Angularly dependent eigenfunctions and the angular momentum matrix. This equation can be separated in spherical polar coordinates.

We choose the nucleus as origin of a polar coordinate system with some arbitrary z be the coordinates of the electron. The left side is a function of r only. The equation on the right in 1. Furthermore, the real part of the spherical harmonic.

They are quantized; i. In view of 1. We could have saved ourselves the trouble of calculating the angularly dependent eigenfunctions explicitly, by making use of some general theorems about the angular momentum. These theorems state: 1. The components and the absolute value of the total angular momentum of an electron in any central force field are constants of the motion.

## Quantum Mechanics of One- and Two-Electron Atoms - Hans A. Bethe, Edwin E. Salpeter - Google книги

A quantum state is defined by specifying the magnitude and one of the components of the total angular momentum, i. Thus, by means of the generally valid formula 1. Before solving this equation, we shall calculate the matrix elements of the angular momenta kx and ky about the directions perendicular to the z -axis. By definition. The matrix elements which correspond to transitions between states of different orbital quantum numbers or of different radially dependent eigenfunctions are zero.

As is well known, the formulas 1. For even l , then, our wave functions are unaffected by the parity operation and we say the parity is even. For odd l , the wave functions change sign under the parity operation and the parity is odd. We shall now treat the differential equation satisfied by the radially dependent part of the eigenfunction. The last term in the above equation corresponds classically to the centrifugal force of the electron and increases with increasing angular momentum. Let us begin by assuming that the energy E is negative. Since the potential energy vanishes at infinity, a negative energy corresponds to a bound electron which possesses a positive kinetic energy only by virtue of the nuclear attraction.

First, we study the asymptotic behavior by neglecting the terms in r which are of lower power compared to the ones that are kept, then. Substitution of 2. The coefficient of each power of r must vanish in the above equation. Next, we require that f be a polynomial in r , i. Therefore, the breaking off of the series is necessary to assure that the eigenfunction be bounded at infinity. The energy does not depend on the other two quantum numbers m and l.

The independence of the magnetic quantum number m has its origin in the fact that all directions in space enter on equal terms. This holds for all atoms in the absence of an external field and is called directional degeneracy. The energy formula 2. The term scheme of hydrogen is shown in Fig. The spectral lines of the hydrogen spectrum the height of a line is proportional to its frequency, the number written next to a line is its wave length in ANGSTROM units. The wave numbers of many lines of the spectrum of the hydrogen atom can be measured to an accuracy of better than one part in a million.

This difference is due to correction terms allowing for the finite mass of the nucleus, the relativistic fine structure and hyperfine structure splitting of the energy levels and quantum electrodynamic level shifts all to be discussed later. These correction terms are small and well known.

Using this value for Ry, 2. After applying corrections for fine structure, etc. As pointed out in the Introduction, the atomic constants e , m and h , entering into the definition for Ry could also be obtained from experiments which do not require atomic theory for their analysis. Such a non-spectroscopic value for Ry would only be accurate to about one part in three thousand, but agrees with 2. We shall now deal with the radial eigenfunctions in greater detail.

Combining 2. Instead of evaluating 3. For this purpose we substitute the representation 3. We must now distinguish between two cases: left with the evaluation of the integral. A final integration with respect to 9 gives. The integral vanishes, as is evident from 3. By means of 3.

## Quantum Mechanics of One- And Two-Electron Atoms

The normalization integral 3. It is apparent that the maximum charge density recedes from the nucleus with increasing principal quantum number n. For eigenfunctions which have the same n , the one with lowest l has the largest amplitude in the neighborhood of the nucleus. The symbols K , etc. The charge distribution in the first few states of the hydrogen atom. The abscissa is r The curves are labelled with the corresponding nl values s-states in Fig.

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The differences between the various radial eigenfunctions become particularly apparent when we form the mean value of r , the distance between electron and nucleus, raised to various powers. The mean value rv is given by. Assigning particular values to the indices results in the explicit formulas.

This is readily explained. For positive powers the important contributions to the integral come from the large values of r for small r l.

### Quantum Mechanics of One- and Two-Electron Atoms (Dover Books on Physics)

The quantum mechanical analogue to BOHR orbits of large eccentricity small l is a large mean square deviation of the nucleus-electron separation:. As can be seen from 3. Aside from normalization, all eigenfunctions of a fixed orbital quantum number l and different principal quantum number n behave alike near the nucleus, provided n l.

This may be seen by neglecting l relative to n in formula 3. Then 2. Since the differential Eq. Its solution is. If the power series expansion for J is used in 3. First, we must get rid of the first derivative in Eq. Although 3. Thus, if we wish to evaluate the normalizing integral. In addition, we can make use of the fact that the cosine in. For large n the normalization 3. Friend Reviews. To see what your friends thought of this book, please sign up. Lists with This Book. Quantum mechanics of one- and two-electron atoms Quantum mechanics of one- and two-electron atoms Quantum mechanics of one- and two-electron atoms Quantum mechanics of one- and two-electron atoms Quantum mechanics of one- and two-electron atoms Quantum mechanics of one- and two-electron atoms

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