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Chapter 5: Q9P (page 214)

(a) Suppose you put both electrons in a helium atom into the n=2state;

what would the energy of the emitted electron be?

(b) Describe (quantitatively) the spectrum of the helium ion,He+.

Short Answer

Expert verified

(a)The energy of the emitted electron is 2E1-4E1=-2E1=27.2eV

(b)Helium has one electron and it’s a hydrogenise ion with z=2 so the spectrum

is1/λ=4R(1/nf2-1/ni2)

Step by step solution

01

(a) The energy of the emitted electron

The energy of each electron isE=Z2E1/n2=4E1/4=E1=E1=-13.6eV,

so the total initial energy is2×-13.6eV=-27.2eV.

One electron drops to the ground state Z2E1/1=4E1, so the other is left

with 2E1-4E1=-2E1=27.2eV.

02

(b) The spectrum of the helium ion

(b) He+has one electron; it’s a hydrogenise ion with Z = 2, so the spectrum

is 1/λ=4R1/nf2-1/ni2, where R is the hydrogen Rydberg constant, and ni,nfare the

initial and final quantum numbers (1, 2, 3, . . . ).

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Most popular questions from this chapter

Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate ⟨(x1-x2)2⟩assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)

Suppose we use delta function wells, instead of spikes (i.e., switch the sign ofin Equation 5.57). Analyze this case, constructing the analog to Figure 5.6. this requires no new calculation, for the positive energy solutions (except that β is now negative; use β=-1.5 for the graph), but you do need to work out the negative energy solutions (letk≡-2mE/handZ≡-ka,forE<0) and , for). How many states are there in the first allowed band?

a) Hund’s first rule says that, consistent with the Pauli principle, the state with the highest total spin (S) will have the lowest energy. What would this predict in the case of the excited states of helium?

(b) Hund’s second rule says that, for a given spin, the state with the highest total orbital angular momentum (L) , consistent with overall antisymmetrization, will have the lowest energy. Why doesn’t carbon haveL=2? Note that the “top of the ladder”(ML=L)is symmetric.

(c) Hund’s third rule says that if a subshell(n,l)is no more than half filled,
then the lowest energy level hasJ=lL-SI; if it is more than half filled, thenJ=L+Shas the lowest energy. Use this to resolve the boron ambiguity inProblem 5.12(b).

(d) Use Hund’s rules, together with the fact that a symmetric spin state must go with an antisymmetric position state (and vice versa) to resolve the carbon and nitrogen ambiguities in Problem 5.12(b). Hint: Always go to the “top of the ladder” to figure out the symmetry of a state.

Obtain equation 5.76 by induction. The combinatorial question is this: How many different ways you can put N identical balls into d baskets (never mind the subscriptfor this problem). You could stick all of them into the third basket, or all but one in the second basket and one in the fifth, or two in the first and three in the third and all the rest in the seventh, etc. Work it out explicitly for the casesN=1,N=2,N=3andN=4; by that stage you should be able to deduce the general formula.

Show that most of the energies determined by Equation 5.64are doubly degenerate. What are the exceptional cases? Hint: Try it for N=1,2,3,4.... , to see how it goes. What are the possible values of cos(ka)in each case?
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