41.2.1 Radial Wave Functions

In practice, the probability of finding the electron at a certain point in space is often less useful than the probability of finding the electron at a certain distance from the nucleus. That is, what is the probability that the electron is to be found within the small range of distances $δ r$ at the distance r?

It turns out that the solutions to the three-dimensional Schrödinger equation, the wave functions $ψ (x, y, z)$ , can be written in a form that focuses on the electron’s radial distance $r$ from the proton. The portion of the wave function that depends only on $r$ is called the radial wave function. These functions, which depend on the quantum numbers $n$ and $l$ , are designated $R_{n l} (r)$ . The first three radial wave functions are

\begin{array}{l} R_{1 s} (r) = \frac{1}{\sqrt{π a_{B}^{3}}} e^{- r / a_{B}} \\ R_{2 s} (r) = \frac{1}{\sqrt{8 π a_{B}^{3}}} (1 - \frac{r}{2 a_{B}}) e^{- r /2 a_{B}} \\ R_{2 p} (r) = \frac{1}{\sqrt{24 π a_{B}^{3}}} (\frac{r}{2 a_{B}}) e^{- r /2 a_{B}} \end{array}

(41.7)

where $a_{B}$ is the Bohr radius.

The radial wave functions may seem mysterious, because we haven’t shown where they come from, but they are essentially the same as the one-dimensional wave functions $ψ (x)$ you learned to work with in Chapter 40. In fact, these radial wave functions are mathematically similar to the one-dimensional wave functions of the simple harmonic oscillator. One important difference, however, is that $r$ ranges from 0 to $\infty$ . For one-dimensional wave functions, $x$ ranged from $- \infty$ to $\infty$ .

NOTE Don’t be confused by the notation. $R$ is not a radius but, like $ψ$ , is the symbol for a wave function, the radial wave function. It is a function of the distance $r$ from the proton.

FIGURE 41.5 shows the radial wave functions for the $1 s$ and $2 s$ states. Notice that the radial wave function is nonzero at $r = 0$ , the position of the nucleus. This is surprising, but it is consistent with our observation in Figure 41.4 that the $1 s$ and $2 s$ electrons have a strong probability of being found at the origin.

The figure illustrates the 1 s and 2 s radial wave functions of hydrogen.

FIGURE 41.5 Full Alternative Text

Our purpose for introducing the radial wave functions was to determine the probability of finding the electron a certain distance from the nucleus. FIGURE 41.6 shows a shell of radius $r$ and thickness $δ r$ centered on the nucleus. The probability of finding the electron at distance $r$ from the nucleus is equivalent to the probability that the electron is located somewhere within this shell. The volume of a thin shell is its surface area multiplied by its thickness $δ r$ . The surface area of a sphere is $4 π r^{2}$ , so the volume of this thin shell is

δ V = 4 π r^{2} δ r

The figure illustrates the radial probability density.

FIGURE 41.6 Full Alternative Text

Just as $| ψ (x) |^{2}$ is the probability in one dimension of finding a particle within an interval $δ x$ , the probability of locating the electron within this spherical shell can be written in terms of the radial wave function $R_{n l} (r)$ as

Prob(in δ r at r) = | R_{n l} (r) |^{2} δ V = 4 π r^{2} | R_{n l} (r) |^{2} δ r

(41.8)

If we define the radial probability density $P_{r} (r)$ for state $n l$ as

P_{r} (r) = 4 π r^{2} | R_{n l} (r) |^{2}

(41.9)

then, exactly analogous to the one-dimensional quantum mechanics of Chapter 40, we can write the probability of finding the electron within a small interval $δ r$ at distance $r$ as

Prob(in δ r at r) = P_{r} (r) δ r

(41.10)

The radial probability density tells us the relative likelihood of finding the electron at distance $r$ from the nucleus. The volume factor $4 π r^{2}$ reflects the fact that more space is available in a shell of larger $r$ , and this additional space increases the probability of finding the electron at that distance.

The probability of finding the electron between $r_{min}$ and $r_{max}$ is

Prob (r_{min} \leq r \leq r_{max}) = \int_{r_{min}}^{r_{max}} P_{r} (r) d r = 4 π \int_{r_{min}}^{r_{max}} r^{2} {| R_{n l} (r) |}^{2} d r

(41.11)

The electron must be somewhere between $r = 0$ and $r = \infty$ , so the integral of $P_{r} (r)$ between 0 and $\infty$ must equal 1. This normalization condition was used to determine the constants in front of the radial wave functions of Equations 41.7.

FIGURE 41.7 shows the radial probability densities for the $n = 1, 2$ , and 3 states of the hydrogen atom. You can see that the $1 s$ , $2 p$ , and $3 d$ states, with maxima at $a_{B}, 4 a_{B}$ , and $9 a_{B}$ , respectively, are following the pattern $r_{peak} = n^{2} a_{B}$ of the radii of the orbits in the Bohr hydrogen atom. There we simply bent a one-dimensional de Broglie wave into a circle of that radius. Now we have a three-dimensional wave function for which the electron is most likely to be this distance from the nucleus, although it could be found at other values of $r$ . Quantum mechanics reproduces some aspects of the Bohr hydrogen atom.

The figure consists of three parts that illustrate the radial probability densities for n equals 1, 2, and 3 using a probability versus radius graph.

FIGURE 41.7 Full Alternative Text

41.2.1 Radial Wave Functions

FIGURE 41.5 The $1 s and 2 s$ radial wave functions of hydrogen.

FIGURE 41.6 The radial probability density gives the probability of finding the electron in a spherical shell of thickness $δ r$ at radius $r$ .

FIGURE 41.7 The radial probability densities for $n = 1, 2$ , and $3$ .