Chapter 20 The Hydrogen Atom. Physical Chemistry 2 nd Edition. Thomas Engel, Philip Reid. Objectives. Solve the Schrödinger equation for the motion of an electron in a spherically symmetric Coulomb potential.
where e = electron charge me = electron massε0 = permittivity of free space
where (Bohr radius)
Normalize the functions in three-dimensional
In general, a wave function is normalized by multiplying it by a constant N defined by
. In three-dimensional spherical coordinates, it is
The normalization integral
For the first function,
We use the standard integral
Integrating over the angles , we obtain
Evaluating the integral over r,
For the second function,
This simplifies to
Integrating over the angles using the result , we obtain
Using the same standard integral as in the first part of the problem,
a. Consider an excited state of the H atom with the electron in the 2s orbital.Is the wave function that describes this state,an eigenfunction of the kinetic energy? Of the potential energy?
b. Calculate the average values of the kinetic and
potential energies for an atom described by this wave function.
a. We know that this function is an eigenfunction of the
total energy operator because it is a solution of the
Schrödinger equation. You can convince yourself that
the total energy operator does not commute with either
the kinetic energy operator or the potential energy
operator by extending the discussion of Example
Problem 20.1. Therefore, this wave function cannot
be an eigenfunction of either of these operators.
b. The average value of the kinetic energy is given by
We use the standard integral,
Using the relationship
The average potential energy is given by
We see that
The relationship of the kinetic and potential energies is
a specific example of the virial theorem and holds for
any system in which the potential is Coulombic.
Locate the nodal surfaces in
The radial part of the equations is zero for finite values of for .
This occurs at .
a. At what point does the probability density for the
electron in a 2s orbital have its maximum value?
b. Assume that the nuclear diameter for H is 2 × 10-15 m. Using this assumption, calculate the total
probability of finding the electron in the nucleus if it
occupies the 2s orbital.
a. The point at which and , therefore, has its greatest value is found from the wave function:
which has its maximum value at r=0, or at the nucleus
b. The result obtained in part (a) seems unphysical, but is a consequence of wave-particle duality in describing electrons. It is really only a problem if the total probability of finding the electron within the nucleus is significant. This probability is given by
Because , we can evaluate the integrand by assuming that is constant over the interval
Because this probability is vanishingly small, even though the wave function has its maximum amplitude at the nucleus, the probability of finding the electron in the nucleus is essentially zero.
Calculate the maxima in the radial probability distribution for the 2s orbital. What is the most probable distance from the nucleus for an electron in this orbital? Are there subsidiary maxima?
The radial distribution function is
To find the maxima, we plot P(r) and
versus and look for the nodes in this function.
These functions are plotted as a function of in the following figure:
The resulting radial distribution function only depends on r, and not on . Therefore, we can display P(r)dr versus r in a graph as shown