Wave functions squared

The hrst nontrivial radiative-recoil correction is of order a Za). We have already discussed the nonrecoil contribution of this order in Subsect. 3.3.2. Due to the wave function squared factor this correction naturally contained an explicit factor rrir/m). Below we will discuss radiative-recoil corrections of order a Za) with mass ratio dependence beyond this factor (nirfm). ... 

One could insist that the mass dependence in (5.9) is natural because the calculations leading to it are done without expansion over the mass ratio, and are therefore exact. On the other hand, the factor m / mM) symmetric with respect to the masses naturally arises in all apparently nonrecoil calculations just from the Schrodinger-Coulomb wave function squared. According to the tradition we preserve the coefficient before the logarithm squared term in the form given in (3.53). Then the new contribution contained in (5.9) has the form [12, 13]... 

In Eq. (15), 8(rik) is the Dirac delta function which, when integrated with the wave function, gives the value of the wave function at rik = 0. The two terms in Eq. (15) are in reality two limiting forms of the same interaction. The first term is the ordinary dipole-dipole interaction for two dipoles that are not too close to each other. It is the proper form of M S1 to be applied to p, d, and / electrons which are not found near the nucleus. For s electrons, which have a finite probability of being at the nucleus, the first term is clearly inappropriate, since it gives zero contribution at large values of rik and does not hold for small values of rik. From Dirac s relativistic theory of the electron, it is found (4) that the second term in Eq. (15) is the correct form for Si when the electron is close to the nucleus. Thus the contribution toJT S] from s electrons is proportional to the wave function squared at the site of the nucleus and the second term in Eq. (15) is often called the contact term in the hyperfine interaction. 

There is some justification for this interpretation that the wave function squared is proportional to the probability of finding an electron. With lightwaves,for example, while the wavelength provides the colour (more precisely the energy) of the wave, it is the amplitude squared that gives the brightness. 

Figure 9.5 Potential energy (dashed curve) and simphfied model potential (foil curve) for a - He atom in a tube of radius 0.5 rnn. The energies of the ground state and lowest azimuthally and radially excited states are shown as horizontal hnes along with the correspondmg wave functions (squares, circles and crosses, respectively). ( apte om e. [ ].)...

Mathematically, P describes the motion of an electron in an orbital. The modulus of the wave function squared, l P(r)P, is a direct measure of the probability of finding the electron at a particular location. The Schrodinger wave equation can be solved exactly for hydrogen. To apply it you must first transform it into polar coordinates (r,0,< )) and then solve using the method of separation of variables (described in, e.g., Kreyszig, 1999). 

The symbol H stands for the Hamiltonian operator, a set of mathanatical operations that represent the total energy (kinetic and potential) of the electron within the atom. The symbol E is the actual energy of the electron. The symbol tj/ is the wave function, a mathematical function that describes the wavelike nature of the electron. A plot of the wave function squared (i/r ) represents an orbital, a position probability distribution map of the electron. 

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