THE HYDROGEN ATOM
Let’s continue the discussion of the hydrogen atom that we began in Chapter 40. In the Bohr model, electrons moved in circular rbits like Newtonian particles, but with quantized values of angular momentum, While this model gave the correct energy levels’of the hydrogen atom, deduced from spectra, it had many conceptual difficulties. It mixed classical physics with new and seemingly contradictory concepts. It provided no insight into the process by which photons are emitted and absorbed. It could not be generalized to atoms with more than one electron. It predicted the wrong magnetic properties for the hydrogen atom. And perhaps most important, its picture of the electron as a point particle was inconsistent with the more general view we developed in Chapters 41 and 42.
Now let’s apply the Schrodinger equation to the hydrogen atom. As we discussed in Section 40-6, we include the motion of the nucleus b, simply replacing the electron mass m with the reduced mass m,. We discussed the three-dimensional version of the Schrodinger equation in Section 42-7. The hydrogen-atom problem is best formulated in spherical coordinates (r, 9, ~), shown in Fig. 42-22; the potential energy is then simply the Schrodinger equation with this potential-energy function can be solved exactly; the solutions are combinations of familiar function.
A lot of detail, we can describe the most important features of the pro the results. Solutions are obtained by a method called separation of variables. in which the wave function yt (r, 9,1/1) as a product of three functions, each one a function one of the three coordinates:
The function R(r) depends only on r, e(9) depends only on 9, and <1>(1/1) depends. When we substitute Eq. (43-2) into the Schrodinger equation, we get three equations, each containing only one of the coordinates. This is an enormous cation; it reduces the problem of solving a fairly complex partial differential .on with three independent variables to the much simpler problem of olving three separate ordinary differential equations with one independent variable each. TIle physically acceptable solutions of these three equations are determined by boundar» conditions. The radial function R(r) must approach zero at large r, because we describing bound states of the electron that are localized near the nucleus. This is lIDlt.lq~lS to the requirement that the harmonic oscillator wave functions (Section approach zero at large x. The angular functions e( 9) and <1>(1/1) must be periexample, (r, 9, 1/1) and (r, 9,1/1 +2n) describe the same point, so <1>(1/1+2n) must .). Also, the angular functions .must be finite for all relevant values of the For example, there are solutions of the e equation that become infinite at 9 = 0 = r, these are unacceptable, since yt(r, 9, 1/1) must be nonreturnable. radial functions R(r) turn out to be an exponential function e-ur (where a is stippled by a polynomial in r. The functions 9(9) are polynomials containing ‘I’Z:;:;.or;:s powers of sin 9 and cos 9, and the functions cI>(1/I) are simply proportional to 15 an integer that may be positive, zero, or negative. [10 of finding solutions that satisfy the boundary conditions, we. also find = Nadine,g energy levels. Their energies, which we denote by En (n = I, 2, 3, … ), identical to those from the Bohr model, as given by Eq. (40-16), with Q:~e:!i=::a:~ ~S1 mass m replaced by the reduced mass m; Rewriting that equation.