8.1 The hydrogen atom  (Page 2/11)

The factor $r\text{sin}\theta$ is the magnitude of a vector formed by the projection of the polar vector onto the xy -plane. Also, the coordinates of x and y are obtained by projecting this vector onto the x - and y -axes, respectively. The inverse transformation gives

Schrödinger’s wave equation for the hydrogen atom in spherical coordinates is discussed in more advanced courses in modern physics, so we do not consider it in detail here. However, due to the spherical symmetry of U ( r ), this equation reduces to three simpler equations: one for each of the three coordinates $(r,\theta ,\text{and}\varphi ).$ Solutions to the time-independent wave function are written as a product of three functions:

where R is the radial function dependent on the radial coordinate r only; $\text{Θ}$ is the polar function dependent on the polar coordinate $\theta$ only; and $\text{Φ}$ is the phi function of $\varphi$ only. Valid solutions to Schrödinger’s equation $\psi (r,\theta ,\varphi )$ are labeled by the quantum numbers n , l , and m .

(The reasons for these names will be explained in the next section.) The radial function R depends only on n and l ; the polar function $\text{Θ}$ depends only on l and m ; and the phi function $\text{Φ}$ depends only on m . The dependence of each function on quantum numbers is indicated with subscripts:

Not all sets of quantum numbers ( n , l , m ) are possible. For example, the orbital angular quantum number l can never be greater or equal to the principal quantum number $n\left(l<n\right)$ . Specifically, we have

Notice that for the ground state, $n=1$ , $l=0$ , and $m=0$ . In other words, there is only one quantum state with the wave function for $n=1$ , and it is ${\psi }_{100}$ . However, for $n=2$ , we have

Therefore, the allowed states for the $n=2$ state are ${\psi }_{200}$ , ${\psi }_{21-1},{\psi }_{210}$ , and ${\psi }_{211}$ . Example wave functions for the hydrogen atom are given in [link] . Note that some of these expressions contain the letter i , which represents $\sqrt{\mathrm{-1}}$ . When probabilities are calculated, these complex numbers do not appear in the final answer.

Physical significance of the quantum numbers

Each of the three quantum numbers of the hydrogen atom ( n , l , m ) is associated with a different physical quantity. The principal quantum number     n is associated with the total energy of the electron, $E_n$ . According to Schrödinger’s equation:

where $E_0=\mathrm{-13.6}\text{eV}\text{.}$ Notice that this expression is identical to that of Bohr’s model. As in the Bohr model, the electron in a particular state of energy does not radiate.