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- approximation) k V+y2 = absolute value (or modulus) of z OME INTEGRALS r-iy=complex conjugate of z gerals of the form x"e Av? dx P- A is a positive number, occur positive integer, their value can be found from the following: frequently in several branches of physics. When n ere IGONOMETRIC RELATIONS T lo 4A TT 12= 21 1 I3 14 2A2 1613 (Euler's relation) cos + i sin 0 e TT 8 A +e cos 8 = 2 dIn-2 sin 6 = ice that the integral e Ax dx equals 21, when n is even, but is zero if n is odd. 2i Another common integral is the indefinite integral fre. Jn "e"x/b dx hat is analytic near the point z = a can be expanded in a Taylor is a small integer, this is easily evaluated by parts, for example, J4-(b +bx)exb J2=(2b +2bx + bx)eb 4-6e1 (z- a) (a)( - a) + a) (Taylor's series) 2! general 6 t special cases of this expansion are 22 ite in particular, that 2! 3! reb dx 2b xelb dx = b2 In(1 + z) z 3 aso that cos z 1 2! 4! νTο Vxelb dx 2 sin z z 31 5! n 1) n(n - 1(n- 2) + + + 1 1 1.
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- approximation) k V+y2 = absolute value (or modulus) of z OME INTEGRALS r-iy=complex conjugate of z gerals of the form x"e Av? dx P- A is a positive number, occur positive integer, their value can be found from the following: frequently in several branches of physics. When n ere IGONOMETRIC RELATIONS T lo 4A TT 12= 21 1 I3 14 2A2 1613 (Euler's relation) cos + i sin 0 e TT 8 A +e cos 8 = 2 dIn-2 sin 6 = ice that the integral e Ax dx equals 21, when n is even, but is zero if n is odd. 2i Another common integral is the indefinite integral fre. Jn "e"x/b dx hat is analytic near the point z = a can be expanded in a Taylor is a small integer, this is easily evaluated by parts, for example, J4-(b +bx)exb J2=(2b +2bx + bx)eb 4-6e1 (z- a) (a)( - a) + a) (Taylor's series) 2! general 6 t special cases of this expansion are 22 ite in particular, that 2! 3! reb dx 2b xelb dx = b2 In(1 + z) z 3 aso that cos z 1 2! 4! νTο Vxelb dx 2 sin z z 31 5! n 1) n(n - 1(n- 2) + + + 1 1 1.
Related questions
Question
For Problem 8.37, how do I find <1/r> within the integral? I think that the exponent function inside of P(r) is actually troublesome to finding what I need to find; however, I am not certain of what's really the correct procedure here.
![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-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81ed2a7d-ef58-4ac0-a29f-af1e2992bb8e%2Fd5249d57-927c-4080-ae3e-92dac4a2ae37%2F7rpa91i.jpeg&w=3840&q=75)
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-
Transcribed Image Text:approximation)
k V+y2 = absolute value (or modulus) of z
OME INTEGRALS
r-iy=complex conjugate of z
gerals of the form
x"e Av?
dx
P-
A is a positive number, occur
positive integer, their value can be found from the following:
frequently in several branches of physics. When n
ere
IGONOMETRIC RELATIONS
T
lo 4A
TT
12=
21
1
I3
14
2A2
1613
(Euler's relation)
cos
+ i sin 0
e
TT
8 A
+e
cos 8 =
2
dIn-2
sin 6 =
ice that the integral e Ax dx equals 21, when n is even, but is zero if n is odd.
2i
Another common integral is the indefinite integral
fre.
Jn
"e"x/b dx
hat is analytic near the point z = a can be expanded in a Taylor
is a small integer, this is easily evaluated by parts, for example,
J4-(b +bx)exb
J2=(2b +2bx + bx)eb
4-6e1
(z- a)
(a)( - a) + a)
(Taylor's series)
2!
general
6
t special cases of this expansion are
22
ite in particular, that
2!
3!
reb dx 2b
xelb dx = b2
In(1 + z) z
3
aso that
cos z 1
2!
4!
νTο
Vxelb dx
2
sin z z
31
5!
n 1)
n(n - 1(n- 2)
+
+
+
1
1
1.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
