Elements of Modern Physics
96
which means that operation by L ± produces eigenstates with eigenvalues
(m ± 1) , or annihilates that state if states with eigenvalues (m ± 1) do not
exist. Hence L± are called raising and lowering operators.
It may also be noted that for potentials which are functions of the radial
distance r only,
[H, L] = 0
(3.166)
so that it is possible to obtain eigenstates of H which are simultaneously
eigenstates of L
2
and L z , or L
2
and L x , or L
2
and L y .
Example 3
Consider the tunnelling of particles across a barrier potential of height V and
with d. Imposing the conditions of continuity of the wave function in Eq. (3.94),
and its derivative at x = 0 and x = d,
a + + a – = b + + b –
a + – a_ =
i
p
α (b + – b_)
b + e
–
α
d
+ b – eα
d
= c + e
ipd
(3.167)
b + e
–
α
d
– b_ eα
d
=
ipd
ip c e
−
+
α
(3.167)
Expressing c + in terms of a + , we get
a
c
+
+
=
1
1
1
1
1
4
ipd
d
d
ip
ip
i
i
e
e
e
p
p
α
− α
α
α
+
−
+ −
+
α
α
(3.168)
so that the transmission coefficient T is given by
2
a
c
+
+
=
1
T
=
(
)
(
)
2
1 2
16
d
d
d
d
p
e
e
i
e
e
p
α
− α
α
− α
α
+
+
−
−
α
=
(
)
2
2
1
1 16
d
d
p
e
e
p
α
− α
α
+
+
−
α
(3.169)
For a very broad barrier, i.e. for large d, this leads to
T ≈
(
)
2 2
2
2
2
16
d
p e
p
− α
α
α +
(3.170)
96
which means that operation by L ± produces eigenstates with eigenvalues
(m ± 1) , or annihilates that state if states with eigenvalues (m ± 1) do not
exist. Hence L± are called raising and lowering operators.
It may also be noted that for potentials which are functions of the radial
distance r only,
[H, L] = 0
(3.166)
so that it is possible to obtain eigenstates of H which are simultaneously
eigenstates of L
2
and L z , or L
2
and L x , or L
2
and L y .
Example 3
Consider the tunnelling of particles across a barrier potential of height V and
with d. Imposing the conditions of continuity of the wave function in Eq. (3.94),
and its derivative at x = 0 and x = d,
a + + a – = b + + b –
a + – a_ =
i
p
α (b + – b_)
b + e
–
α
d
+ b – eα
d
= c + e
ipd
(3.167)
b + e
–
α
d
– b_ eα
d
=
ipd
ip c e
−
+
α
(3.167)
Expressing c + in terms of a + , we get
a
c
+
+
=
1
1
1
1
1
4
ipd
d
d
ip
ip
i
i
e
e
e
p
p
α
− α
α
α
+
−
+ −
+
α
α
(3.168)
so that the transmission coefficient T is given by
2
a
c
+
+
=
1
T
=
(
)
(
)
2
1 2
16
d
d
d
d
p
e
e
i
e
e
p
α
− α
α
− α
α
+
+
−
−
α
=
(
)
2
2
1
1 16
d
d
p
e
e
p
α
− α
α
+
+
−
α
(3.169)
For a very broad barrier, i.e. for large d, this leads to
T ≈
(
)
2 2
2
2
2
16
d
p e
p
− α
α
α +
(3.170)