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Chapter 2
2.1 Sketch_______________________________________
2.2 Sketch_______________________________________
2.3 Sketch_______________________________________
2.4
From Problem 2.2, phase
= constant Then
From Problem 2.3, phase
= constant Then
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2.5
Gold: eV J
So,
cm or m Cesium: eV
J So,
cm or m_______________________________________
2.6
(a)
kg-m/s
m/s
or cm/s
(b)
kg-m/s
m/s
or cm/s (c) Yes_______________________________________
2.7(a) (i)
kg-m/s
m
or
(ii)
kg-m/s
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m
or
(iii)
kg-m/s
m
or
(b)
kg-m/s
m
or
_______________________________________
2.8
eV Now
or kg-m/s Now
m
or
_______________________________________
2.9
Now
and
Set and Then
which yields
J keV_______________________________________
2.10
(a)
kg-m/s
m/s
or cm/s
J
or eV
(b)
J
or eV
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kg-m/s
m
or
_______________________________________
2.11(a)
J Now
V kV(b)
kg-m/s Then
m
or
_______________________________________
2.12
kg-m/s_______________________________________
2.13(a) (i)
kg-
m/s
(ii)
Now
kg-m/sso
J
or eV
(b) (i) kg-m/s (ii)
kg-m/s
J
or
eV_______________________________________
2.14
kg-m/s
m/s_______________________________________
2.15(a)
s
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(b)
kg-m/s_______________________________________
2.16(a) If and are solutions
to Schrodinger's wave equation, then
and
Adding the two equations, we obtain
which is Schrodinger's wave equation. So is also a solution.
(b) If were a solution toSchrodinger's wave equation, then we could write
which can be written as
Dividing by , we find
Since is a solution, then
Subtracting these last two equations, we have
Since is also a solution, we have
Subtracting these last two equations, we obtain
This equation is not necessarily valid, which means that is, in general, not a solution to Schrodinger's wave equation._______________________________________
2.17
so
or
_______________________________________
2.18
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or _______________________________________
2.19
Note that
Function has been normalized.(a) Now
or
which yields (b)
or
which yields (c)
which yields
_______________________________________
2.20
(a)
or
(b)
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or
(c)
or _______________________________________
2.21
(a)
or
(b)
or
(c)
or _______________________________________
2.22
(a) (i) m/s
or cm/s
m
or
(ii) kg-m/s
J
or eV
(b) (i) m/s
or cm/s
m
or
(ii) kg-m/s eV
_______________________________________
2.23(a)
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(b)
so m/scm/sFor electron traveling in direction, cm/s
kg-m/s
m
m
or rad/s_______________________________________
2.24(a) kg-m/s
m
m
rad/s(b) kg-m/s
m
m
rad/s
_______________________________________
2.25
J or
or eV Then eV
eV
eV_______________________________________
2.26(a)
Jor
eV
Then eV eV eV
(b)
J
mor nm
_______________________________________
2.27
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(a)
or (b) mJ(c) No
_______________________________________
2.28 For a neutron and :
J or
eV
For an electron in the same potential well:
J or
eV
_______________________________________
2.29 Schrodinger's time-independent wave equation
We know that
for and
We have
for
so in this region
The solution is of the form
where
Boundary conditions:
at
First mode solution: where
Second mode solution: where
Third mode solution: where
Fourth mode solution: where
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2.30 The 3-D time-independent wave equation in cartesian coordinates for is:
Use separation of variables, so let Substituting into the wave equation, we obtain
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Dividing by and letting ,
we find
(1)
We may set
Solution is of the form
Boundary conditions:
and
where Similarly, let
and
Applying the boundary conditions, we find
,
,
From Equation (1) above, we have or
so that
_______________________________________
2.31 (a)
Solution is of the form:
We find
Substituting into the original equation, we find:
(1)
From the boundary conditions,
, where
So ,
Also , where
So ,
Substituting into Eq. (1) above
(b)Energy is quantized - similar to 1-D result. There can be more than one quantum state per given energy - different than 1-D result.
_______________________________________
2.32(a) Derivation of energy levels exactly the
same as in the text
(b)
For Then
(i) For
J
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or
eV
(ii) For cm
J or
eV
_______________________________________
2.33(a) For region II,
General form of the solution is
where
Term with represents incident wave and term with represents reflected wave. Region I,
General form of the solution is
where
Term involving represents the transmitted wave and the term involving represents reflected wave: but if a particle is
transmitted into region I, it will not be reflected so that . Then
(b) Boundary conditions:(1)
(2)
Applying the boundary conditions to the solutions, we find
Combining these two equations, we find
The reflection coefficient is
The transmission coefficient is
_______________________________________
2.34
where
m
(a) For m
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(b) For m
(c) For m
_______________________________________
2.35
where
or m(a) For m
(b) For m
(c) , where is the density of transmitted electrons. eV J
m/scm/s
electrons/cm
Density of incident electrons,
cm
_______________________________________
2.36
(a) For
or m Then
or
(b) For
=
or m Then
or _______________________________________
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2.37
where
m (a)
(b)
or m_______________________________________
2.38 Region I , ; Region II , Region III ,
(a) Region I:
(incident) (reflected)
where
Region II:
where
Region III:
(b) In Region III, the term represents a reflected wave. However, once a particle is transmitted into Region III, there will not be a reflected wave so that . (c) Boundary conditions: At :
At :
The transmission coefficient is defined as
so from the boundary conditions, we want to solve for in terms of . Solving for in terms of , we find
We then find
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We have
If we assume that , then will be large so that We can then write
which becomes
Substituting the expressions for and , we find
and
Then
Finally,
_____________________________________
2.39 Region I:
incident reflected where
Region II:
transmitted reflected where
Region III:
transmitted where
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There is no reflected wave in Region III. The transmission coefficient is defined as:
From the boundary conditions, solve for in terms of . The boundary conditions are: At :
At :
But
Then, eliminating , , from the boundary condition equations, we find
_______________________________________
2.40(a) Region I: Since , we can write
Region II: , so
Region III: The general solutions can be written, keeping in mind that must remain finite for , as
where
and
(b) Boundary conditions At :
At :
or (c)
and since , then
From , we can write
or
This equation can be written as
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or
This last equation is valid only for specific values of the total energy . The energy levels are quantized.
_______________________________________
2.41
(J)
(eV)
or
(eV)
eV eV eV eV_______________________________________
2.42 We have
and
or
To find the maximum probability
which gives
or is the radius that gives the greatest probability._______________________________________
2.43 is independent of and , so the wave equation in spherical coordinates reduces to
where
For
Then
so
We then obtain
Substituting into the wave equation, we have
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where
Then the above equation becomes
or
which gives 0 = 0 and shows that is indeed a solution to the wave equation._______________________________________
2.44 All elements are from the Group I column of the periodic table. All have one valence electron in the outer shell._______________________________________