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We now have all the mathematical machinery to quantum mechanically analyze the simplest real physical system, the hydrogen atom. Before we do this, we examine two questions. What do we expect from the hydrogen atom based on classical mechanics? And why was quantum mechanics such a great triumph of human thought in the beginning of the 20th century? A simplified picture of a hydrogen atom is shown in figure 1.7. An electron of charge —e at distance R is revolving around its nucleus of charge +e.

The coulombic force must act as a centrifugal force. Hence,

where со = 2nf is the angular rate of rotation with / being the frequency. There is nothing in equation 1.34 to suggest that the radius R is quantized and consequently the energy is quantized. However, ample spectroscopic evidence existed before the discovery (or creation) of quantum mechanics to suggest that the energies were quantized. Spectroscopists found specific discrete frequencies coming out of the hydrogen atom and other atoms not a continuous range of frequencies, suggesting that discrete energy levels exist in the atoms. Furthermore, as is usually taught in advanced electromagnetism courses, an accelerating charge, like the electron in hydrogen, emits radiation so its orbit will not be circular but a spiral falling ultimately in the nucleus, something which clearly does not happen as hydrogen is a stable atom.

A classical picture of an electron (e) circulating around a proton (p) at a radius R with the centrifugal force m(i)R acting on it

FIGURE 1.7 A classical picture of an electron (e) circulating around a proton (p) at a radius R with the centrifugal force m(i)2R acting on it.

The Coulomb potential seen by an electron inside a hydrogen atom along two opposite directions

FIGURE 1.8 The Coulomb potential seen by an electron inside a hydrogen atom along two opposite directions.

We now begin our quantum mechanical analysis of atoms which will resolve all the above contradictions. We begin by writing down the classical equation of the energy of an electron in the field of the nucleus. The potential energy is due to the coulombic attraction that holds the electron close to the nucleus. The latter is shown in figure 1.8. Note that the radial distance r from the nucleus is always positive but what in figure 1.8 is meant by a negative r is the distance r in the opposite (-180°) direction from the one we have chosen to measure r initially. The reason we have chosen to present the coulombic attraction graphically this way is because it should be clear from figure 1.8 that the electron is again in a quantum well, albeit of a more complicated nature this time.

Now the electron in the hydrogen atom has kinetic and potential energy. Hence,

To construct the hamiltonian we have to apply the following rules of quantum mechanics: E —» H, p —> -ihV and r —» r.

Therefore, we get

for the hamiltonian and for the Schroedinger equation.

Unfortunately, this hamiltonian is in mixed cartesian and spherical coordinates. We therefore have either to express the operator V2 in spherical coordinates or substitute y]x2 + y2 +z2 for r. The former choice is more convenient, so we proceed that way. Then the Schroedinger equation becomes

This is an awkward looking partial differential equation that can, however, be decomposed into 3 separate ordinary differential equations just like we did in the 3-dimensional rectangular quantum box. The procedure is far from as straightforward as the simple rectangular well, so we do not find it useful to give this procedure here as it only entails pure mathematical steps of the separation of variables method. So we give the final results directly which have all the physical significance embedded in the equations. So for the wavefunctions we have

We now analyze all the symbols in 1.38.

The n, l, mi are quantum numbers labelling the wavefunctions just as in the 3-dimensional rectangular quantum well of the previous section we had 3 quantum numbers labelling the wavefunctions. The very simple reason why discrete states appear in the quantum version of the hydrogen atom and not in the classical is that the Schroedinger equation is an eigenvalue equation whereas the classical equations of motion do not have this property. All the mathematical steps we have omitted in going from equation 1.36 to the solution 1.38 for vFn>i>m( have this common theme. The quantum numbers n, /, mt are given by the following formulae:

The eigenvalues corresponding to any*P are given by the equation

where h is Planck’s constant (not h bar). As can be seen, the energies are negative because we have taken as the zero of energy the configuration where the electron is infinitely away from the nucleus, i.e. ionized. This has been implicitly assumed in the form of the coulombic potential in equation 1.36a. Furthermore, it is also evident that the eigenenergies depend only on the quantum number n and for this reason it is called the principal quantum number. Therefore, eigenfunctions with the same quantum number n but different / and m, belong to the same energy. This is called degeneracy. We have already encountered this concept in connection with spin in discussing the Fermi-Dirac statistics. We remind the reader that the electron has an inherent spin or magnetic moment that can take only two values, “up” or “down”. In the absence of a magnetic field, the “up” and “down” spin electrons have the same energy. Therefore, each can accommodate two electrons.

Hydrogen has one electron in it, so that it will go to the lowest energy with n = 1. For this state, l can take only the value l = 0 and hence w, can only be mt = 0. The spin, however, is undetermined so we do not know in what spin state the electron is.

We move to the examination of the wavefunctions in greater detail. As can be seen from equation 1.38, the wavefunctions are a product of a function R„i{r), which is a function of only the radial distance r, and a function Y/m(0,R„i(r). The function Ylm (0,VF|2. The Ylm are called spherical harmonics. The Ylm for various / and mt are shown in figure 1.9. For the l = 0 as in the hydrogen case, they give a spherically symmetric distribution in space.

The R„i in particular are a product of a polynomial and a decaying exponential The spatial distribution of the Yj orbitals

FIGURE 1.9 The spatial distribution of the Yjm orbitals. For historical reasons the values for /= 1, 3 are denoted by the letters s, p, d respectively while the subscripts of s, p, d are notations for the third quantum number.

The units of (3 are obviously 1/distance so it is necessary at this point to define a characteristic length useful for atomic distances. The obvious way to do it is through equation 1.40 for the energy levels. We observe that E„ can be written as


The value of (3 can be written in terms of a0. Calculations following the basic differential equation 1.37 (they are by no means short) show that

Given that the extent in space of the hydrogen atom is determined by the decaying exponential in 1.41, we deduce the obvious result that the orbits (orbitals to be precise, as they only have a probabilistic nature) of states of higher energy lie further away from the nucleus. The extent of the atoms is generally a few Bohr radii. Equations 1.41-1.43 show the quantization in both space and energy of the hydrogen atom. Table 1.1 shows the R„i(r) for n = 1,2.

The reason why we have spent so much time on the hydrogen atom is because the holy grail of atomic physics (the atoms of the periodic table) is only a small step away. In what way will the Schroedinger equation of any atom of the periodic table be different from equation 1.37? Only the potential energy V(r) will be different and include not only the coulombic attraction from the nucleus of atomic number Z - but also the coulombic repulsion between any electron and the remaining Z-l electrons.

Hence we can write

TABLE 1.1 Radial part of the hydrogen wavefunctions for n = 1 and n = 2










Obviously V(r) in equation 1.44 will be a complicated expression of not only r but also of the r, of the remaining electrons. However, if we decide that every electron moves in the average field of the other electrons and if we further assume that this is, to a first approximation, spherically symmetric, we can write

where approximations Vefpr (r) to Vee (r) can be found. How then would the wavefunctions of this hamiltonian look like? Answer: they are formally exactly the same as those of equation 1.38. The R„i(r) of course are not given by simple algebraic expressions as those of table 1.1 and can only be found numerically by solving by computer the corresponding differential equation for the R„i(r) but the angular part is given by the same functions, the spherical harmonics Yjm (0,cp). All these simplifications are a consequence of the spherical average we have taken for Vee(r).

As far as the eigenvalues are concerned those are not only a function of the first quantum number n but also a function of the second quantum number /, but not of the third one mh i.e. they have subscripts as For historical reasons the values l = 0,1,2,3 are not given numerically but instead are designated by the letters s, p, d,f respectively. The electrons will fill in these states as described in section 1.5. However, to each eigenvalue E„i we can have as many electrons as there are different mi values and different spins. Hence the number of electrons in a given state E„i, called an orbital, is therefore N = 2(21 + 1) where the 2 comes from spin and the (2/+1) from the possible values of m,. As we have stated before, this is the degeneracy of the atomic state or orbital. The atomic states are symbolized as, for example, Is1 meaning the hydrogen state with n = 1 and l = 0 with one electron in it or 2p4 meaning a state with n = 2 and l = 1 with 4 electrons in it and so on. The atoms themselves are designated of course by the atomic number Z. The usual rule of high school chemistry, where a simple atom has to have 8 electrons in its outer shell to complete bonding, refers to atoms with s and p states in their outer shell. Note that 8 electrons are required to completely fill these states or orbitals. Figure 1.10 gives schematically the various atomic energies for an atom with a maximum quantum number n = 4.

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