The Bound-State Hydrogen-Atom Wave Functions

6 THE BOUND-STATE HYDROGEN-ATOM WAVE FUNCTIONS 

The Radial Factor. Using (6.93), we have for the recursion relation (6.86) 

How many nodes does R r) have The radial function is zero at r = oo, at r = 0 for / 7 0, and at values of r that make A/(r) vanish. A/(r) is a polynomial of degree n - I - 1, and it can be shown that the roots of A/(r) = 0 are all real and positive. Thus, aside from the origin and infinity, there are n — / — 1 nodes in R(r). The nodes of the spherical harmonics are discussed in Problem 6.35. 

Ground-State Wave Function and Energy. For the ground state of the hydrogenlike atom, we have = 1, / = 0, and m = 0. The radial factor (6.100) is 

Multiplying by Yq = 1/(47t), we have as the ground-state wave function 

6 The Bound-State Hydrogen-Atom Wave Functions 135 

The radial equation for the hydrogen atom can also be solved by the use of ladder operators (also known as factorization), see Z. W. Salsburg, Am. J. Phys., 33, 36 (1965). 

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Section 6.6 The Bound-State Hydrogen-Atom Wave Functions 147 since the spherical harmonics are normalized ... 

Since the interaction (4.304) is central, the associate wave equation may be separated in spherical polar coordinates to produce the normalized radial function. For the bound states hydrogenic atoms in the case of an infinitely heavy nucleus it looks like (Bransden Joachain, 1983) ... 

It divides into two categories. For E < 0 we obtain an infinite discrete sequence of bounded states for > 0 we obtain a continuous spectrum. Due to the similarity of (6.1.22) to the Z = 0 radial Hamiltonian of the hydrogen atom, the wave functions and energies can be computed analytically. For the bound state wave functions we obtain... 

We now consider the bound states of the hydrogen atom, with < 0. In this case, the quantity in parentheses in (6.66) is positive. Since we want the wave functions to remain finite as r goes to infinity, we prefer the minus sign in (6.66), and in order to get a two-term recursion relation, we make the substitution... 

Degeneracy. Are the hydrogen-atom energy levels degenerate For the bound states, the energy (6.94) depends only on n. However, the wave function (6.61) depends on all three quantum numbers n, I, and m, whose allowed values are [Eqs. (6.91), (6.92), (5.108), and (5.109)]... 

How about using the hydrogen-atom bound-state wave functions to expand an arbitrary function /(r, 6,4>)1 The answer is that these functions do not form a complete set, and we cannot expand / using them. To have a complete set, we must use all the eigenfunctions of a particular Hermitian operator. In addition to the bound-state... 

We denoted the hydrogen-atom bound-state wave functions by three subscripts that give the values of n, I, and m. In an alternative notation, the value of / is indicated by a letter ... 

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