2ObjectivesSolve the Schrödinger equation for the motion of an electron in a spherically symmetric Coulomb potential.Emphasize the similarities and differences between quantum mechanical and classical models.Comparison is made between the quantum mechanical picture of the hydrogen atom.
3Outline Formulating the Schrödinger Equation Solving the SchrödingerEquation for the Hydrogen AtomEigenvalues and Eigenfunctions for the Total EnergyThe Hydrogen Atom OrbitalsThe Radial Probability Distribution FunctionThe Validity of the Shell Model of an Atom
420.1 Formulating the Schrödinger Equation Hydrogen atom as made up of an electron moving about a proton located at the origin of the coordinate system.The two particles attract one another and the interaction potential is given by a simple Coulomb potential:where e = electron charge me = electron mass ε0 = permittivity of free space
520.1 Formulating the Schrödinger Equation As the potential is spherically symmetrical, we choose spherical polar coordinates to formulate the Schrödinger equation.
620.2 Solving the Schrödinger Equation for the Hydrogen Atom Separation of variablesThus the differential equation for R(r) is obtained.The second term can be viewed as an effective potential.
720.2 Solving the Schrödinger Equation for the Hydrogen Atom Each of the terms that contribute to Veff (r) and their sums can be graphed as a function of distance.
820.3 Eigenvalues and Eigenfunctions for the Total Energy Note that the energy, E, only appears in the radial equation and not in the angular equation.R(r) can be well behaved at large values of r [R(r) → 0 as r → ∞].where (Bohr radius)The definition leads to
920.3 Eigenvalues and Eigenfunctions for the Total Energy The other two quantum numbers are l and ml, which arise from the angular coordinates.Their relationship is given by
1020.3 Eigenvalues and Eigenfunctions for the Total Energy The quantum numbers associated with the wave functions are
1120.3 Eigenvalues and Eigenfunctions for the Total Energy The quantum numbers associated with the wave functions are
12Example 20.1 Normalize the functions in three-dimensional spherical coordinates.
13SolutionIn general, a wave function is normalized by multiplying it by a constant N defined by. In three-dimensional spherical coordinates, it isThe normalization integralFor the first function,
14Solution We use the standard integral Integrating over the angles , we obtainEvaluating the integral over r,
15Solution For the second function, This simplifies to Integrating over the angles using the result , we obtain
16SolutionUsing the same standard integral as in the first part of the problem,
1720.3 Eigenvalues and Eigenfunctions for the Total Energy The angular part of each hydrogen atom total energy eigenfunctions is a spherical harmonic function.
18Example 20.2a. 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 andpotential energies for an atom described by this wave function.
19Solution a. We know that this function is an eigenfunction of the total energy operator because it is a solution of theSchrödinger equation. You can convince yourself thatthe total energy operator does not commute with eitherthe kinetic energy operator or the potential energyoperator by extending the discussion of ExampleProblem Therefore, this wave function cannotbe an eigenfunction of either of these operators.
20Solutionb. The average value of the kinetic energy is given by
21SolutionWe use the standard integral,Using the relationship
22SolutionThe average potential energy is given by
23SolutionWe see thatThe relationship of the kinetic and potential energies isa specific example of the virial theorem and holds forany system in which the potential is Coulombic.
2420.3 Eigenvalues and Eigenfunctions for the Total Energy The radial distribution function is used to extract information from the H atom orbitals.We first look at the ground-state (lowest energy state) wave function for the hydrogen atom,We need a four-dimensional space to plot as a function of all its variables.
2520.4 The Hydrogen Atom Orbitals Since such a space is not readily available, the number of variables is reduced.It is reduced by evaluating in one of the x–y, x–z, or y–z planes by setting the third coordinate equal to zero.r are spherical nodal surfaces rather than nodal points (one-dimensional) potentials.
26Example 20.3 Locate the nodal surfaces in Solution: The radial part of the equations is zero for finite values of forThis occurs at
2720.5 The Radial Probability Distribution Function 3D perspective plots of the square of the wave functions for the orbitals is indicated.
28Example 20.4 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 × m. Using this assumption, calculate the totalprobability of finding the electron in the nucleus if itoccupies the 2s orbital.
29Solutiona. 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
30Solutionb. 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
31SolutionBecause , we can evaluate the integrand by assuming that is constant over the interval
32SolutionBecause 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.
3320.5 The Radial Probability Distribution Function It is most meaningful for the s orbitals whose amplitudes are independent of the angular coordinates.The radial distribution P(r) is the probability function of choice to determine the most likely radius to find the electron for a given orbital
34Example 20.6Calculate 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?
35Solution The radial distribution function is To find the maxima, we plot P(r) andversus and look for the nodes in this function.
36SolutionThese functions are plotted as a function of in the following figure:
37SolutionThe 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
3820.6 The Validity of the Shell Model of an Atom The idea of wave-particle duality is that waves are not sharply localized.