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d V = 4 π r 2 d r .

The probability of finding the electron in the region r to r + d r (“at approximately r ”) is

P ( r ) d r = | ψ n 00 | 2 4 π r 2 d r .

Here P ( r ) is called the radial probability density function    (a probability per unit length). For an electron in the ground state of hydrogen, the probability of finding an electron in the region r to r + d r is

| ψ n 00 | 2 4 π r 2 d r = ( 4 / a 0 3 ) r 2 exp ( −2 r / a 0 ) d r ,

where a 0 = 0.5 angstroms. The radial probability density function P ( r ) is plotted in [link] . The area under the curve between any two radial positions, say r 1 and r 2 , gives the probability of finding the electron in that radial range. To find the most probable radial position, we set the first derivative of this function to zero ( d P / d r = 0 ) and solve for r . The most probable radial position is not equal to the average or expectation value of the radial position because | ψ n 00 | 2 is not symmetrical about its peak value.

A graph of the function P of r as a function of r is shown. It is zero at r = 0, rises to a maximum at r = a sub 0, then gradually decreases and goes asymptotically to zero at large r. The maximum is at the most probable radial position. The area of the region under the curve from r sub 1 to r sub 2 is shaded.
The radial probability density function for the ground state of hydrogen.

If the electron has orbital angular momentum ( l 0 ), then the wave functions representing the electron depend on the angles θ and ϕ ; that is, ψ n l m = ψ n l m ( r , θ , ϕ ). Atomic orbitals for three states with n = 2 and l = 1 are shown in [link] . An atomic orbital    is a region in space that encloses a certain percentage (usually 90%) of the electron probability. (Sometimes atomic orbitals are referred to as “clouds” of probability.) Notice that these distributions are pronounced in certain directions. This directionality is important to chemists when they analyze how atoms are bound together to form molecules.

This diagram illustrates the shapes of p orbitals. The orbitals are dumbbell shaped and oriented along the x, y, and z axes.
The probability density distributions for three states with n = 2 and l = 1 . The distributions are directed along the (a) x -axis, (b) y -axis, and (c) z -axis.

A slightly different representation of the wave function is given in [link] . In this case, light and dark regions indicate locations of relatively high and low probability, respectively. In contrast to the Bohr model of the hydrogen atom, the electron does not move around the proton nucleus in a well-defined path. Indeed, the uncertainty principle makes it impossible to know how the electron gets from one place to another.

The figure shows probability clouds for electrons in the n equals 1, 2 and 3, l equals 0, 1 and 2 states in a 3 by 3 grid. n=1, l=0 is a spherically symmetric distribution, brighter in the center and gradually fading with increasing radius, with no nodes. n=2, l=0 is a spherically symmetric distribution with a spherical, concentric node. The node appears as a black circle within the cloud. The cloud is brightest in the center, fading to black at the node, brightening again to outside the node (but not as bright as at the center of the cloud), then fading again at large r. n=2, l=1 has a planar node along the diameter of the cloud, appearing as a dark line across the distribution and indentations at the edges. The cloud is brightest near the center, above and below the node. n=3, l=0 is a spherically symmetric distribution with two spherical, concentric nodes. The nodes appear as concentric black circles within the cloud. The cloud is brightest in the center, fading to black at the first node, brightening again to a maximum brightness outside the node, fading to black at the second node brightening again, then fading again at large r. The local maxima (at the center, between the nodes, and outside the outer node) decrease in intensity. n=3, l=2 has both a concentric circular node and a planar node along the diameter, appearing as a circle in and line across the cloud. The cloud is brightest inside the circular node. A second local maximum brightness is seen within the lobes above and below the planar node. n=3, l=2 has two planar nodes, which appear as an X across the cloud. The quarters of the cloud thus defined are deeply indented at the edges, forming rounded lobes. The cloud is brightest near the center.
Probability clouds for the electron in the ground state and several excited states of hydrogen. The probability of finding the electron is indicated by the shade of color; the lighter the coloring, the greater the chance of finding the electron.

Summary

  • A hydrogen atom can be described in terms of its wave function, probability density, total energy, and orbital angular momentum.
  • The state of an electron in a hydrogen atom is specified by its quantum numbers ( n , l , m ).
  • In contrast to the Bohr model of the atom, the Schrödinger model makes predictions based on probability statements.
  • The quantum numbers of a hydrogen atom can be used to calculate important information about the atom.

Conceptual questions

Identify the physical significance of each of the quantum numbers of the hydrogen atom.

n (principal quantum number) total energy
l (orbital angular quantum number) total absolute magnitude of the orbital angular momentum
m (orbital angular projection quantum number) z -component of the orbital angular momentum

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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