SOLUTIONSProblems Exam3Spring 2008
1.
Suppose that a particle in the onedimensional infinite well obeyed a rule that the
quantum number n could change by only one unit as the particle make transitions among
the excited states. Show that the photons emitted in those transitions have energies 3E
0
, 5
E
0
, 7 E
0
… (2n1) E
0
… where E
0
is the groundstate energy.
Solution: Consider “L” the width of the infinite well and “m” the mass of the particle.
The energy of the particle confined in the one dimensional infinite well is quantized:
2
2
2
2
2
2
±
²
³
´
µ
¶
=
=
L
n
m
m
k
E
n
n
·
h
h
.
The energy of the ground state (n=1) is
0
2
2
1
2
E
L
m
E
=
±
²
³
´
µ
¶
=
·
h
,
the energy of the n=2 state is
0
2
2
2
4
2
2
E
L
m
E
=
±
²
³
´
µ
¶
=
·
h
, of the n=3 state is
0
3
9
E
E
=
. In
general the energy of the “n” state is
0
2
E
n
E
n
=
. Since the quantum number n can change
by only one unit the energy of the photon emitted in these transitions is given by:
0
0
2
2
1
)
1
2
(
)
)
1
(
(
E
n
E
n
n
E
E
n
n
±
=
²
±
±
=
±
±
, n has to be bigger than 1:
For example for n=2,
0
0
1
2
3
)
1
2
2
(
E
E
E
E
=
±
²
=
±
and for n=3
0
2
3
5
E
E
E
=
±
etc.
2.
A particle is trapped in a onedimensional region of dimension
L
. Show that the
probability is 1/
n
to find the particle between
x=0
and
x=L/n
in the state of quantum
number
n
.
Solution: The wave function of a particle trapped in a one dimensional region of length L
is given by
±
²
³
´
µ
¶
·
=
L
x
n
A
x
n
¸
¹
sin
)
(
. This wave function satisfies the Schrödinger equation
and the boundary conditions
0
)
(
)
0
(
=
=
L
n
n
±
±
. The constant A can be found from the
normalization condition: the probability to find the particle somewhere in confining
region of length L is one.
1
2
/
2
)
(
sin
)
/
(
sin

)
(

2
2
0
2
2
0
2
2
0
2
=
=
=
=
=
±
±
±
L
A
n
n
L
A
dy
y
n
L
A
dx
L
x
n
A
dx
x
n
L
L
n
²
²
²
²
³
²
.
Therefore
L
A
2
=
and
±
²
³
´
µ
¶
·
=
L
x
n
L
x
n
¸
¹
sin
2
)
(
.
The probability to find the particle (in the state of quantum number n) located somewhere
between x=0 and x=L/n is:
n
n
L
L
dy
y
n
L
A
dx
L
x
n
A
dx
x
n
L
n
L
n
1
2
2
)
(
sin
)
/
(
sin

)
(

0
2
2
/
0
2
2
/
0
2
=
=
=
=
±
±
±
²
²
²
²
³
²
.
3.
According to the correspondence principle, quantum theory should give the same
results as classical physics in the limit of large quantum numbers. Show that as the
quantum number
±
²
n
, the probability of finding the trapped particle in a one
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dimensional infinite potential well between
x
and
x+
_
x
is
_
x/L
and so is independent of
x
,
which is the classical expectation.
Solution. The wave function of a particle confined in a one dimensional infinite potential
well is given by
±
²
³
´
µ
¶
·
=
L
x
n
L
x
n
¸
¹
sin
2
)
(
. Consider a region of length _a along the x axis
between x
1
=a and x
2
=a+_a.
The probability to find the particle in this region is given by:
L
a
a
n
L
a
n
L
a
a
n
L
a
n
a
a
a
a
a
a
n
y
y
n
dy
y
L
n
L
dx
L
x
n
L
dx
x
/
)
(
/
/
)
(
/
2
2
2
2
2
sin
2
2
)
(
sin
2
)
/
(
sin
2

)
(

±
+
±
+
±
+
±
+
²
³
´
µ
¶
·
¸
=
=
=
¹
¹
¹
º
º
º
º
º
º
º
»
±
²
³
´
µ
¶
+
·
¸
+
·
¸
+
=
¹
¸
+
2
)
/
2
sin(
2
)
/
)
(
2
sin(
)
(
1

)
(

2
L
a
n
L
a
n
L
a
a
n
L
a
a
n
n
dx
x
a
a
a
n
º
º
º
º
º
»
(
)
)
/
)
(
2
sin(
)
/
2
sin(
2
1
)
(

)
(

2
L
a
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 Spring '08
 Alarcon
 Physics, Photon, Particle, wave function, OX

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