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Chapter 6: Q16P (page 276)
Question: Evaluate the following commutators :
a)
b)
c)role="math" localid="1658226147021"
d)
e)
f)
Hint: L and S satisfy the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134 ), but they commute with each other.
a)
b)
c) 0
d) 0
e) 0
f) 0
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Question: Sometimes it is possible to solve Equation 6.10 directly, without having to expand in terms of the unperturbed wave functions (Equation 6.11). Here are two particularly nice examples.
(a) Stark effect in the ground state of hydrogen.
(i) Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field (the Stark effect-see Problem 6.36). Hint: Try a solution of the form
your problem is to find the constants , and C that solve Equation 6.10.
(ii) Use Equation to determine the second-order correction to the ground state energy (the first-order correction is zero, as you found in Problem 6.36(a)). Answer: .
(b) If the proton had an electric dipole moment p the potential energy of the electron in hydrogen would be perturbed in the amount
(i) Solve Equation 6.10 for the first-order correction to the ground state wave function.
(ii) Show that the total electric dipole moment of the atom is (surprisingly) zero, to this order.
(iii) Use Equation 6.14 to determine the second-order correction to the ground state energy. What is the first-order correction?
Starting with Equation 6.80, and using Equations 6.57, 6.61, 6.64, and 6.81, derive Equation 6.82.
Question: In a crystal, the electric field of neighbouring ions perturbs the energy levels of an atom. As a crude model, imagine that a hydrogen atom is surrounded by three pairs of point charges, as shown in Figure 6.15. (Spin is irrelevant to this problem, so ignore it.)
(a) Assuming that show that
where
(b) Find the lowest-order correction to the ground state energy.
(c) Calculate the first-order corrections to the energy of the first excited states Into how many levels does this four-fold degenerate system split,
(i) in the case of cubic symmetry, (ii) in the case of tetragonal symmetry, (iii) in the general case of orthorhombic symmetry (all three different)?
Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential energy is shifted by an amount.(a) Show that there is no first-order change in the energy levels, and calculate the second-order correction. Hint: See Problem 3.33.
(b) The Schr枚dinger equation can be solved directly in this case, by a change of variables. Find the exact energies, and show that they are consistent with the perturbation theory approximation.
If I=0, then j=s,, and the "good" states are the same for weak and strong fields. Determine(from Equation) and the fine structure energies (Equation 6.67), and write down the general result for the I=O Zeeman Effect - regardless of the strength of the field. Show that the strong field formula (Equation 6.82) reproduces this result, provided that we interpret the indeterminate term in square brackets as.
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