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For Problem 8.35, how do I prove, or perhaps verify, what it is they're asking for? 

8.69) and Table 8.1.
8.34 If you haven't already done so, do parts (a) and
(b) of Problem 8.33, and then do part (c), but for the
five spherical harmonics with/ 2.
r momentum
l expansion to
. (This shows
SECTIONS 8.7 and 8.8 (The Energy Levels of the
Hydrogen Atom and Hydrogenic Wave
Functions)
m mechanics
el.)
r the special
8.35 Prove that the degeneracy of the nth level in the hy-
drogen atom is n'; that is, verify the result (8.77). (But
be aware that this number gets doubled because of
the electron's spin, as we describe in Chapter 9.)
constant
d solution is
ow that this
acceptable).
r differential
mbination of
n, and prove
8.36
.It is known that a certain hydrogen atom has a defi-
nite value of l. (a) What does this statement tell
you
about the angular momentum? (b) What are the
allowed energies consistent with this information?
nstant.
8.37 The mean value (or expectation value) of 1/r for
any state is (1/r)
the 1s state of hydrogen. Comment. [Hint: See the
integrals in Appendix B.]
= J (1/r) P (r) dr. Find (1/r) for
r the case
ution. (Any
sin 0 is the
e complete
ependence
m = -1.
(a) It is known that a certain hydrogen atom has
8.38
2. How many different states are
n = 5 and m
consistent with this information? (b) Answer the
same question (in terms of n and m) for arbitrary
values ofn and m
leave it as
12
a maxi-
expand button
Transcribed Image Text:8.69) and Table 8.1. 8.34 If you haven't already done so, do parts (a) and (b) of Problem 8.33, and then do part (c), but for the five spherical harmonics with/ 2. r momentum l expansion to . (This shows SECTIONS 8.7 and 8.8 (The Energy Levels of the Hydrogen Atom and Hydrogenic Wave Functions) m mechanics el.) r the special 8.35 Prove that the degeneracy of the nth level in the hy- drogen atom is n'; that is, verify the result (8.77). (But be aware that this number gets doubled because of the electron's spin, as we describe in Chapter 9.) constant d solution is ow that this acceptable). r differential mbination of n, and prove 8.36 .It is known that a certain hydrogen atom has a defi- nite value of l. (a) What does this statement tell you about the angular momentum? (b) What are the allowed energies consistent with this information? nstant. 8.37 The mean value (or expectation value) of 1/r for any state is (1/r) the 1s state of hydrogen. Comment. [Hint: See the integrals in Appendix B.] = J (1/r) P (r) dr. Find (1/r) for r the case ution. (Any sin 0 is the e complete ependence m = -1. (a) It is known that a certain hydrogen atom has 8.38 2. How many different states are n = 5 and m consistent with this information? (b) Answer the same question (in terms of n and m) for arbitrary values ofn and m leave it as 12 a maxi-
Even when n and / are specified, there are still (21 +1) distinct
corresponding to the (2/ + 1) orientations m = 1, l - 1, - . . , -l. For s s
(=0), there is just one orientation; for p states ( = 1) there
(2 X 1)+1 = 3; for d states (I = 2) there are (2 x 2) + 1 = 5, and so
These numbers are shown in paren theses on the right of each horizontal be
Fig. 8.16. The total degeneracy of any level can be found by adding al
numbers for the level in question. For example, the n = 1 level is nondegee
ate; the n 2 level has degeneracy 4; the n =
1 = 0,1,., (n -1), and hence has degeneracy* (Problem 8.35)
single
momentum of a multielectron atom.
9
For the case n
(Problem 8.39)
3 level 9. The nth level
This wave functio
for the electron,
maximum at the
zero angular mon
state with / # 0.
classically: A cla
momentum is zer
I>0 has import
Chapter 10. It als
dent on the spat
1+3+5+ .. +(2n - 1) = n
(8.77
*Actually, the total degeneracy is twice this answer. This is because the electron has
other degree of freedom, called spin, which can be thought of as the angular m
tum due to its spinning on its own axis (much as the earth spins on its north-sou
This spin can have two possible orientations, and for each of the states describeu
there are really two states, one for each orientation of the spin. This will be discuss
Chapter 9.
energies of atom
radii.
expand button
Transcribed Image Text:Even when n and / are specified, there are still (21 +1) distinct corresponding to the (2/ + 1) orientations m = 1, l - 1, - . . , -l. For s s (=0), there is just one orientation; for p states ( = 1) there (2 X 1)+1 = 3; for d states (I = 2) there are (2 x 2) + 1 = 5, and so These numbers are shown in paren theses on the right of each horizontal be Fig. 8.16. The total degeneracy of any level can be found by adding al numbers for the level in question. For example, the n = 1 level is nondegee ate; the n 2 level has degeneracy 4; the n = 1 = 0,1,., (n -1), and hence has degeneracy* (Problem 8.35) single momentum of a multielectron atom. 9 For the case n (Problem 8.39) 3 level 9. The nth level This wave functio for the electron, maximum at the zero angular mon state with / # 0. classically: A cla momentum is zer I>0 has import Chapter 10. It als dent on the spat 1+3+5+ .. +(2n - 1) = n (8.77 *Actually, the total degeneracy is twice this answer. This is because the electron has other degree of freedom, called spin, which can be thought of as the angular m tum due to its spinning on its own axis (much as the earth spins on its north-sou This spin can have two possible orientations, and for each of the states describeu there are really two states, one for each orientation of the spin. This will be discuss Chapter 9. energies of atom radii.
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