Using the time-independent Schrödinger equation with the potential energy term V = –e2/r, where e is the absolute value of the charge both of the electron and of the proton, we again find that bound states exist only for specific values of the total energy E. These are exactly the values that Bohr had obtained via his 1913 postulate.
Just as factorizing ψ(x,y,z,t) into Ψ(x,y,z) and [1:−Et/ℏ] led to a time-independent Schrödinger equation and a discrete set of values En, so factorizing Ψ(r,φ,θ) — which is Ψ(x,y,z) in polar coordinates — into ψ(r,θ) and [1;Lzφ/ℏ] leads to a φ-independent Schrödinger equation and a discrete set of values Lz.
The φ-independent Schrödinger equation contains a real parameter whose possible values are given by L(L+1)ℏ2, where L is an integer satisfying the condition 0 ≤ L ≤ n-1. The possible values of Lz are integers satisfying the inequality |Lz| ≤ L. The possible combinations of the quantum numbers n, L, and Lz are thus
n = 1xxxL = 0xxxLz = 0
n = 2xxxL = 0xxxLz = 0
n = 2xxxL = 1xxxLz = –1, 0, +1
n = 3xxxL = 0xxxLz = 0
n = 3xxxL = 1xxxLz = –1, 0, +1
n = 3xxxL = 2xxxLz = –2, –1, 0, +1, +2
All of these states are stationary. n is known as the principal quantum number, L as the angular momentum (or orbital, or azimuthal) quantum number, and Lz as the magnetic quantum number (hence the letter m is often used instead).
States with L = 0, 1, 2, 3 were originally labeled s, p, d, f — for “sharp,” “principal,” “diffuse,” and “fundamental,” respectively. The purpose of these letters was to characterize spectral lines. States with higher L follow the alphabet (g, h, …). Figure 2.6.2 maps the radial dependencies of the first three s states, which are spherically symmetric. The plots can be identified by the number N of their nodes (N = n-1).
Figures 2.6.3 and 2.6.4 plot the position probability distributions defined by some non-spherical stationary states with m = 0. Figure 2.6.3 emphasizes the fuzziness of these orbitals at the expense of their rotational symmetry. By plotting surfaces of constant probability, Figure 2.6.4 emphasizes their 3-dimensional shape at the expense of their fuzziness.
It must be stressed that what we see in these images is neither the nucleus nor the electron but the fuzzy position of the electron relative to the nucleus. Nor do we see this fuzzy position “as it is.” What we see is the plot of a position probability distribution. This is defined by outcomes of three measurements, determining the values of n, L, and Lz, and it defines a fuzzy position by determining the probabilities of the possible outcomes of a subsequent measurement of the position of the electron relative to the nucleus.
Here is how such a probability can be calculated. Imagine a small region V of space in the vicinity of the nucleus — so small that the probability density ρ (probability per unit volume) inside it can be considered constant. The probability of finding the electron inside V (if the appropriate measurement is made) is the product ρV. If the gray inside V is a lighter shade, this probability is lower; if it’s a darker shade, this probability is higher. To calculate the probability associated with a larger region, divide it into sufficiently many sufficiently small regions and add up the probabilities associated with them.
Since the dependence on φ is contained in the factor [1;Lzφ/ℏ], it cannot be seen in plots of |Ψ(r,φ,θ)|2. To make this dependence visible, it is customary to replace the complex number [1;Lzφ/ℏ] by its real part, as has been done in Figure 2.6.5.