This Quantum World/Implications and applications/Atomic hydrogen

Atomic hydrogenEdit

While de Broglie's theory of 1923 featured circular electron waves, Schrödinger's "wave mechanics" of 1926 features standing waves in three dimensions. Finding them means finding the solutions of the time-independent Schrödinger equation

 

with   the potential energy of a classical electron at a distance   from the proton. (Only when we come to the relativistic theory will we be able to shed the last vestige of classical thinking.)

 

In using this equation, we ignore (i) the influence of the electron on the proton, whose mass is some 1836 times larger than that of he electron, and (ii) the electron's spin. Since relativistic and spin effects on the measurable properties of atomic hydrogen are rather small, this non-relativistic approximation nevertheless gives excellent results.

For bound states the total energy   is negative, and the Schrödinger equation has a discrete set of solutions. As it turns out, the "allowed" values of   are precisely the values that Bohr obtained in 1913:

 

However, for each   there are now   linearly independent solutions. (If   are independent solutions, then none of them can be written as a linear combination   of the others.)

Solutions with different   correspond to different energies. What physical differences correspond to linearly independent solutions with the same  ?

Using polar coordinates, one finds that all solutions for a particular value   are linear combinations of solutions that have the form

 

  turns out to be another quantized variable, for   implies that   with   In addition,   has an upper bound, as we shall see in a moment.

Just as the factorization of   into   made it possible to obtain a  -independent Schrödinger equation, so the factorization of   into   makes it possible to obtain a  -independent Schrödinger equation. This contains another real parameter   over and above   whose "allowed" values are given by   with   an integer satisfying   The range of possible values for   is bounded by the inequality   The possible values of the principal quantum number   the angular momentum quantum number   and the so-called magnetic quantum number   thus are:

       
       
       
       
       
       
       

Each possible set of quantum numbers   defines a unique wave function   and together these make up a complete set of bound-state solutions ( ) of the Schrödinger equation with   The following images give an idea of the position probability distributions of the first three   states (not to scale). Below them are the probability densities plotted against   Observe that these states have   nodes, all of which are spherical, that is, surfaces of constant   (The nodes of a wave in three dimensions are two-dimensional surfaces. The nodes of a "probability wave" are the surfaces at which the sign of   changes and, consequently, the probability density   vanishes.)


 


Take another look at these images:



The letters s,p,d,f stand for l=0,1,2,3, respectively. (Before the quantum-mechanical origin of atomic spectral lines was understood, a distinction was made between "sharp," "principal," "diffuse," and "fundamental" lines. These terms were subsequently found to correspond to the first four values that   can take. From   onward the labels follows the alphabet: f,g,h...) Observe that these states display both spherical and conical nodes, the latter being surfaces of constant   (The "conical" node with   is a horizontal plane.) These states, too, have a total of   nodes,   of which are conical.

Because the "waviness" in   is contained in a phase factor   it does not show up in representations of   To make it visible, the phase can be encoded as color:

In chemistry it is customary to consider real superpositions of opposite   like   as in the following images, which are also valid solutions.

The total number of nodes is again   the total number of non-spherical nodes is again   but now there are   plane nodes containing the   axis and   conical nodes.

What is so special about the   axis? Absolutely nothing, for the wave functions   which are defined with respect to a different axis, make up another complete set of bound-state solutions. This means that every wave function   can be written as a linear combination of the functions   and vice versa.