Problem Set #3 Answers
Note: Relevant Chapter in Mortimer is Chapter 16.
Linear Momentum; Orbital Angular Momentum
1. Calculate the average linear momentum of a particle described by the wave function , where .
2. Given that , derive an expression for , the orbital angular momentum operator.
3. Show that the wave function is an eigenfunction of . What is the eigenvalue? Hint:
The eigenvalue is .
4. What is the orbital angular momentum for electrons in 3s, 3p, and 3d orbitals (expressed in terms of )? How many radial and angular nodes do each of these orbitals have?
For the 3s orbital, l = 0.
The angular momentum is zero.
The number of angular nodes equals l, so there are no angular nodes.
The number of total nodes is n - 1 = 3 - 1 = 2. Therefore there are 2 radial nodes.
For the 3p orbital, l = 1.
The number of angular nodes equals l, so there is one angular node.
The number of total nodes is n - 1 = 3 - 1 = 2. Therefore there is one radial node.
For the 3d orbital, l = 2.
The number of angular nodes equals l, so there are 2 angular nodes.
The number of total nodes is n - 1 = 3 - 1 = 2. Therefore there are no radial nodes.
5. For a 2p electron in a hydrogen-like atom, what is the magnitude of the orbital angular momentum and what are the possible values of Lz? Express all answers in terms of .
Laguerre Polynomials; Radial Factors
6. The radial wave function for the 2s orbital of the hydrogen atom can be written as where . Find the location of the radial nodes in this orbital in terms of a0.
The radial nodes occur when the radial probability function equals zero.
At this point, and therefore . For a 2s orbital, n = 2; and for a hydrogen atom Z = 1. Therefore, we obtain the following:
7. The eigenfunction for a 1s electron of a hydrogen-like atom is given by , where k is a constant, ao is the radius of the first Bohr orbit for hydrogen. Show that the radius at which there is a maximum probability of finding a 1s electron (in any direction) is just .
The probability of finding an electron in unit element of volume at distance r is given by . The probability of finding the electron at distance r, irrespective of direction, is given by , which can be written as , where is a constant.
The radius at which P is maximum occurs when . We obtain the following:
Hamiltonians and the Schrodinger Equation
8. Give an expression for the Hamiltonian of a two-electron atom. Explain what each term represents and which of the terms is the most difficult to evaluate. Hint: Ignore any magnetic interaction between the spin and the orbital motions of the electrons (i.e. the spin-orbit coupling term, which is small for light atoms).
The Hamiltonian is as follows
These terms represent, in the order given,
The last term contains r12 which is very complicated mathematically.
Normalization of Hydrogenlike Orbitals
9. The radial wave function for the 1s orbital of a hydrogen atom is . Find the normalization constant A. Hint:
Orbitals and Quantum Numbers
10. A hydrogen-like wave function is of the form . Explain what the values of n, l, and m are.
The radial portion of a hydrogen-like wave function is a simple power of r (not a polynomial in r) when l has the maximum possible value, and the power is then (n-1). Thus, in this case n = 5, and l = 4. The function is a simple power of if m has the maxium value for that l, and the power is then l. Thus in this case m = 4. This is further confirmed by the function in .
11. The hydrogen-like wave functions for n = 2 are
where ao is the radius for the first Bohr orbital for hydrogen, and . For each of the above wave functions, explain if the wave function is the function. Give the l and m quantum number values for each of the above wave functions.
is the only spherically symmetric function, and thus must be which has l = m = 0. Next, is symmetric about the z-axis, and thus must be with (l = 1 and m = 0). Then and must be for l = 1 and m = ± 1. Further, has a maximum at , so must be and must be .
12. The following figure shows a plot of the radial wave function R and a polar plot of the wave function, in the xy plane, for a particular hydrogen-like orbital. Explain what the values of n, l, and m must be, or at least what can definitely be said about them.
The R function has no nodes. Therefore, n - l - 1 = 0 so that n = l +1. The function must be because there is no maximum at ; in addition, there is a maximum at indicating that m = 2. If the maximum extension is in the xy plane, then the l value is the maximum possible for the m value, or l = 2. Then, n = 3.