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Chapter 8: Q71E (page 344)
Repeat example 8.6 but assume that the upper state is the rather than the
The and states are both split into two levels.
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In its ground state, carbon's 2pelectrons interact to produce . Given Hund's rule. what does this say about the total orbital angular momentum of these electrons?
Exercise 44 gives an antisymmetric multiparticle state for two particles in a box with opposite spins. Another antisymmetric state with spins opposite and the same quantum numbers is
Refer to these states as 1 and 11. We have tended to characterize exchange symmetry as to whether the state's sign changes when we swap particle labels. but we could achieve the same result by instead swapping the particles' stares, specifically theandin equation (8-22). In this exercise. we look at swapping only parts of the state-spatial or spin.
(a) What is the exchange symmetric-symmetric (unchanged). antisymmetric (switching sign). or neither-of multiparticle states 1 and Itwith respect to swapping spatial states alone?
(b) Answer the same question. but with respect to swapping spin states/arrows alone.
(c) Show that the algebraic sum of states I and II may be written
Where the left arrow in any couple represents the spin of particle 1 and the right arrow that of particle?
(d) Answer the same questions as in parts (a) and (b), but for this algebraic sum.
(e) ls the sum of states I and 11 still antisymmetric if we swap the particles' total-spatial plus spin-states?
(f) if the two particles repel each other, would any of the three multiparticle states-l. II. and the sum-be preferred?
Explain.
Your friends ask: “Why is there an exclusion principle?” Explain in the simplest terms.
Consider potassium. As a rough approximation assume that each of itselectron s orbits 19 pro. tons and half an electron-that is, on average, half its fellowelectron. Assume that each of itselectrons orbits 19 protons, two Is electrons. and half of the seven otherelectrons. Continue the process, assuming that electrons at eachorbit a correspondingly reduced positive charge. (At each, an electron also orbits some of the electron clouds of higher. but we ignore this in our rough approximation.)
(a) Calculate in terms ofthe orbit radii of hydrogenlike atoms of these effective Z,
(b) The radius of potassium is often quoted at around. In view of this, are yourthroughradii reasonable?
(c) About how many more protons would have to be "unscreened" to theelectron to agree with the quoted radius of potassium? Considering the shape of its orbit, should potassium'selectron orbit entirely outside all the lower-electrons?
Exercise 45 refers to state I and II and put their algebraic sum in a simple form. (a) Determine algebraic difference of state I and state II.
(b) Determine whether after swapping spatial state and spin state separately, the algebraic difference of state I and state II is symmetric, antisymmetric or neither, and to check whether the algebraic difference becomes antisymmetric after swapping spatial and spin states both.
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