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# Chemistry 120A Problem Set 6 (due March 17, 2010) 1. Problems 6-43, 6-44 and 6-45 in McQuarrie and Simon. These problems lead you through the...

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Chemistry 120A Problem Set 6 (due March 17, 2010) 1. Problems 6-43, 6-44 and 6-45 in McQuarrie and Simon. These problems lead you through the textbook’s treatment of orbital magnetization in the hydrogen atom. You might ﬁnd them helpful as background for Problem 2 below. 2. The orbital motion of the electron in a hydrogen atom produces a magnetic moment in the z -direction, μ z = - ( e/ 2 m e c ) L z , where - e and m e are, respectively, the charge and mass of an electron, c is the speed of light, and L z is the diﬀerential operator giving the z -component of the angular momentum. The energy of a magnetic moment in a static magnetic ﬁeld, ~ B = B ˆ z , is E mag = - μ z B . Use ﬁrst order perturbation theory for the energy and calculate the expectation values for the energy when the hydrogen atom is in the states ( n,‘,m )=(2,1,1), (2,1,0), and (2,1,-1). Compare your results with the magnetic energies discussed in lecture. 3. The instantaneous dipole moment of a hydrogen atom is = - e~ r , where ~ r is the position of the electron relative to that of the nucleus. The energy of a dipole in an electric ﬁeld, ~ E , is E elec = - · ~ E . With this formula in mind, you will consider the Hamiltonian for a hydrogen atom perturbed by an electric ﬁeld and use ﬁrst order degenerate perturbation theory to estimate the energy levels of this atom. (a) With the unperturbed hydrogen stationary states | ψ n,‘,m i for ( n,‘,m ) = (1,0,0), (2,0,0), (2,1,-1), (2,1,0) and (2,1,1) determine the corresponding elements of the 5 × 5 Hamiltonian matrix for the hydrogen atom in an electric ﬁeld. Take the z -axis to be the direction of the electric ﬁeld. (Hint: Most of these matrix elements are zero, and not much work is required to see why. Do or consider doing your θ and φ integrations ﬁrst! In this way, you can quickly identify the few integrals that actually need to be evaluated explicitly.) (b) Use ﬁrst order degenerate perturbation theory to calculate the perturbed en- ergy eigenvalues for the ﬁve stationary states of lowest energy. (c) The splitting of energy levels by a static electric ﬁeld is called the Stark eﬀect. Draw an energy diagram showing the perturbed energies relative to the two lowest unperturbed energy levels. Indicate how the Stark eﬀect energy level splittings depend upon the size of the electric ﬁeld, E . 1
(d) How would your estimates of these ﬁve perturbed energy levels change if you expanded your basis set to include n = 3 states? Explain. 4. A system in a stationary state at any point in time will remain in that station- ary state for all time unless the system is perturbed, i.e., unless the Hamiltonian changes. According to ﬁrst-order time-dependent perturbation theory, the appli- cation of a perturbation causes transitions between stationary states i and j if h ψ i |H 1 | ψ j i 6 = 0, where H 1 is the perturbation Hamiltonian. Though we have not yet derived it, you should ﬁnd this statement reasonable because you know, from Schr¨ odinger’s time-dependent equation, that the operation of the Hamiltonian on a state is a measure of that state’s rate of change. Accepting the statement as true, consider the following: The observed spectrum of hydrogen is a consequence of transitions between dif- ferent stationary states. The perturbation causing these transitions is the energy of interaction between the hydrogen atoms and the electric ﬁeld associated with light. As noted in the previous problem, this interaction energy is - · ~ E , hence the perturbation referred to in the the previous paragraph is H 1 = - μ · E = e~ r · ~ E . As such, we expect transitions occur between stationary states i and j only when h ψ i | e~ r · ~ E | ψ j i 6 = 0 . By studying the integrals associated with this object, taking the z -axis to be the direction of the electric ﬁeld, it is possible to show that transitions between hydrogen states ( n,‘,m ) and ( n 0 ,‘ 0 ,m 0 ) occur only when Δ = 0 - = ± 1 , and Δ m = m 0 - m = 0 , ± 1 , and there is no restriction on Δ n = n 0 - n . Transitions that obey these rules are said to be “allowed” transitions; these conditions for allowed transitions are called “selection rules.” (a) By considering the appropriate integrals, show that transitions between 2s and 1s states are not allowed; i.e., show that the matrix elements for these transitions are zero. (b) By considering the appropriate integrals, show that 3p to 1s and 2p to 1s transitions are allowed; i.e., show that the matrix elements for these transitions are not zero. 2
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