Chattaraj P.K. (Ed.) Quantum Trajectories
Подождите немного. Документ загружается.
24 Quantum Trajectories
The ratio w = ψ
D
/ψ, where ψ
D
and ψ are two real linearly independent solutions
of Equation 2.22 is, in deep analogy with uniformization theory, the trivializing map
transforming any W to W
0
≡ 0 [1–6,14]. This formulation extends to higher dimen-
sions and to the relativistic case as well [1–7].
Let q
−/+
be the lowest/highest q for which W(q) changes sign. Then we
have Refs [1–6]
Theorem 2.2
If
V (q) −E ≥
P
2
−
> 0, q<q
−
,
P
2
+
> 0, q>q
+
,
(2.23)
then w is a local self-homeomorphism of
ˆ
R = R ∪ {∞} if and only if Equation 2.22
has an L
2
(R) solution.
The crucial consequence is that since the QSHJE is defined if and only if w is a local
self-homeomorphism of
ˆ
R, it follows that the QSHJE by itself implies energy quan-
tization. We stress that this result is obtained without any probabilistic interpretation
of the wavefunction.
2.3 PROOF OF THEOREM 1
The main steps in proving Theorem 1 are two lemmas [1–6]. Let us start by observing
that if the cocycle condition Equation 2.20 is satisfied by (f (q);q), then this is still
satisfied by adding a coboundary term
(f (q);q) −→ (f (q);q) +(∂
q
f )
2
G(f (q)) −G(q). (2.24)
Since (Aq;q) evaluated at q = 0 is independent of A, we have
0 = (q;q) = (q;q)
|q=0
= (Aq;q)
|q=0
. (2.25)
Therefore, if both (f (q);q) and Equation 2.24 satisfy Equation 2.20, then G(0) = 0,
which is the unique condition that G should satisfy. We now use Equation 2.11 to fix
the ambiguity Equation 2.24. First of all observe that the differential equation we are
looking for is
(q
0
;q) = W(q). (2.26)
Then, recalling that q
0
= S
0
−1
0
◦ S
0
(q), we see that a necessary condition to satisfy
Equation 2.11 is that (q
0
;q) depends only on the first and higher derivatives of q
0
.
This in turn implies that for any constant B we have (q
a
+ B;q
b
) = (q
a
;q
b
) that,
together with Equation 2.18, gives
(q
a
+ B;q
b
) = (q
a
;q
b
) = (q
a
;q
b
+ B). (2.27)
The Equivalence Postulate of Quantum Mechanics 25
Let A be a non-vanishing constant and set h(A, q) = (Aq;q). By Equation 2.27 we
have h(A, q + B) = h(A, q), that is h(A, q) is independent of q. On the other hand,
by Equation 2.25 h(A,0)= 0 that, together with Equation 2.18, implies
(Aq;q) = 0 = (q;Aq). (2.28)
Equation 2.20 implies (q
a
;Aq
b
) = A
−2
((q
a
;q
b
) − (Aq
b
;q
b
)), so that by Equa-
tion 2.28
(q
a
;Aq
b
) = A
−2
(q
a
;q
b
). (2.29)
By Equations 2.18 and 2.29 we have
(Aq
a
;q
b
) =−A
−2
(∂
q
b
q
a
)
2
(q
b
;Aq
a
) =−(∂
q
b
q
a
)
2
(q
b
;q
a
) = (q
a
;q
b
),
that is
(Aq
a
;q
b
) = (q
a
;q
b
). (2.30)
Setting f (q) = q
−2
(q;q
−1
) and noticing that by Equations 2.18 and 2.30 f (Aq) =
−f (q
−1
), we obtain
(q;q
−1
) = 0 = (q
−1
;q). (2.31)
Furthermore, since by Equations 2.20 and 2.31 one has (q
a
;q
−1
b
) = q
4
b
(q
a
;q
b
), it
follows that
(q
−1
a
;q
b
) =−
∂
q
b
q
−1
a
2
(q
b
;q
−1
a
) =−
∂
q
b
q
a
2
(q
b
;q
a
) = (q
a
;q
b
),
so that
(q
−1
a
;q
b
) = (q
a
;q
b
) = q
−4
b
(q
a
;q
−1
b
). (2.32)
Since translations, dilatations, and inversion are the generators of the Möbius group,
it follows by Equations 2.27, 2.29, 2.30, and 2.32 that
Lemma 2.1
Up to a coboundary term, Equation 2.20 implies
(γ(q
a
);q
b
) = (q
a
;q
b
),
(q
a
;γ(q
b
)) =
∂
q
b
γ(q
b
)
−2
(q
a
;q
b
),
where γ(q) is an arbitrary PSL(2, C) transformation.
Now observe that since (q
a
;q
b
) should depend only on ∂
k
q
b
q
a
, k ≥ 1, we have
(q + f (q);q) = c
1
f
(k)
(q) +O(
2
), (2.33)
26 Quantum Trajectories
where q
a
= q + f (q), q ≡ q
b
and f
(k)
≡ ∂
k
q
f , k ≥ 1. Note that by Lemma 1 and
Equation 2.33
(Aq + Af (q);Aq) = (q + f (q);Aq)
= A
−2
(q + f (q);q) = A
−2
c
1
f
(k)
(q) +O(
2
); (2.34)
on the other hand, setting F (Aq) = Af (q), by Equation 2.33
(Aq + Af (q);Aq) = (Aq + F (Aq);Aq)
= c
1
∂
k
Aq
F (Aq) +O(
2
) = A
1−k
c
1
f
(k)
(q) +O(
2
),
which, compared with Equation 2.34 gives k = 3. The above scaling property gen-
eralizes to higher order contributions in . In particular, to order
n
the quantity
(Aq + Af (q);Aq) is a sum of terms of the form
c
i
1
...i
n
∂
i
1
Aq
F (Aq) ···∂
i
n
Aq
F (Aq) = c
i
1
...i
n
n
A
n−
i
k
f
(i
1
)
(q) ···f
(i
n
)
(q),
and by Equation 2.34
n
k=1
i
k
= n + 2. On the other hand, since (q
a
;q
b
) depends
only on ∂
k
q
b
q
a
, k ≥ 1, we have
i
k
≥ 1, k ∈[1, n],
so that either
i
k
= 3, i
j
= 1, j ∈[1, n], j = k,
or
i
k
= i
j
= 2, i
l
= 1, l ∈[1, n], l = k, l = j.
Hence
(q + f (q);q) =
∞
n=1
n
c
n
f
(3)
f
(1)
n−1
+ d
n
f
(2)
2
f
(1)
n−2
, d
1
= 0. (2.35)
Let us now consider the transformations
q
b
= v
ba
(q
a
), q
c
= v
cb
(q
b
) = v
cb
◦ v
ba
(q
a
), q
c
= v
ca
(q
a
).
Note that v
ab
= v
ba
−1
, and
v
ca
= v
cb
◦ v
ba
. (2.36)
We can express these transformations in the form
q
b
= q
a
+
ba
(q
a
),
q
c
= q
b
+
cb
(q
b
) = q
b
+
cb
(q
a
+
ba
(q
a
)), (2.37)
q
c
= q
a
+
ca
(q
a
).
The Equivalence Postulate of Quantum Mechanics 27
Since q
b
= q
a
−
ab
(q
b
), we have q
b
= q
a
−
ab
(q
a
+
ba
(q
a
)), which, compared
with q
b
= q
a
+
ba
(q
a
) yields
ba
+
ab
◦ (1 +
ba
) = 0,
where 1 denotes the identity map. More generally, Equation 2.37 gives
ca
(q
a
) =
cb
(q
b
) +
ba
(q
a
) =
cb
(q
b
) −
ab
(q
b
),
so that we obtain Equation 2.36 with v
yx
= 1 +
yx
ca
=
cb
◦ (1 +
ba
) +
ba
= (1 +
cb
) ◦ (1 +
ba
) − 1. (2.38)
Let us consider the case in which
yx
(q
x
) = f
yx
(q
x
), with infinitesimal. To first
order in (Equation 2.38) reads
ca
=
cb
+
ba
, (2.39)
in particular,
ab
=−
ba
. Since (q
a
;q
b
) = c
1
ab
(q
b
) + O
ab
(
2
), where
denotes
the derivative with respect to the argument, we can use the cocycle condition (Equa-
tion 2.20) to get
c
1
ac
(q
c
) + O
ac
(
2
) = (1 +
bc
(q
c
))
2
(c
1
ab
(q
b
) + O
ab
(
2
)
− c
1
cb
(q
b
) − O
cb
(
2
)), (2.40)
which, to firstorder in corresponds to Equation 2.39. We see that c
1
= 0. For, if
c
1
= 0, then by Equation 2.40, to second order in one would have
O
ac
(
2
) = O
ab
(
2
) − O
cb
(
2
), (2.41)
which contradicts Equation 2.39. In fact, by Equation 2.35 we have
O
ab
(
2
) = c
2
ab
(q
b
)
ab
(q
b
) + d
2
ab
2
(q
b
) + O
ab
(
3
),
that together with Equation 2.41 provides a relation which cannot be consistent with
ac
(q
c
) =
ab
(q
b
) −
cb
(q
b
). A possibility is that (q
a
;q
b
) = 0. However, this is ruled
out by the EP, so that
c
1
= 0.
Higher-order contributions due to a non-vanishing c
1
are obtained by using
q
c
= q
b
+
cb
(q
b
),
ac
(q
c
) =
ab
(q
b
) −
cb
(q
b
),
and
bc
(q
c
) =−
cb
(q
b
)inc
1
∂
3
q
c
ac
(q
c
) and in
c
1
(2∂
q
c
bc
(q
c
) + ∂
q
c
bc
(q
c
)
2
)∂
3
q
b
(
ab
(q
b
) −
cb
(q
b
)).
28 Quantum Trajectories
Note that one can also consider the case in which both the first- and second-order con-
tributions to (q
a
;q
b
) are vanishing. However, this possibility is ruled out by a similar
analysis. In general, one has that if the first non-vanishing contribution to (q
a
;q
b
)is
of order
n
, n ≥ 2, then, unless (q
a
;q
b
) = 0, the cocycle condition (Equation 2.20)
cannot be consistent with the linearity of Equation 2.39. Observe that we proved that
c
1
= 0 is a necessary condition for the existence of solutions (q
a
;q
b
) of the cocycle
condition Equation 2.20, depending only on the first and higher derivatives of q
a
.
Existence of solutions follows from the fact that the Schwarzian derivative {q
a
, q
b
}
solves Equation 2.20 and depends only on the first and higher derivatives of q
a
.
The fact that c
1
= 0 implies (q
a
;q
b
) = 0 can also be seen by explicitly evaluating
the coefficients c
n
and d
n
. These can be obtained using the same procedure consid-
ered above to prove that c
1
= 0. Namely, inserting the expansion Equation 2.35
in Equation 2.20 and using q
c
= q
b
+
cb
(q
b
),
ac
(q
c
) =
ab
(q
b
) −
cb
(q
b
) and
bc
(q
c
) =−
cb
(q
b
), we obtain
c
n
= (−1)
n−1
c
1
, d
n
=
3
2
(−1)
n−1
(n − 1)c
1
, (2.42)
which in fact are the coefficients one obtains expanding c
1
{q+f (q), q}. However, we
now use only the fact that c
1
= 0, as the relation (q+f (q);q) = c
1
{q +f (q), q}can
be proved without making the calculations leading to Equation 2.42. Summarizing,
we have
Lemma 2.2
If
q
a
= q
b
+
ab
(q
b
),
the unique solution of Equation 2.20, depending only on the first and higher deriva-
tives of q
a
,is
(q
a
;q
b
) = c
1
ab
(q
b
) + O
ab
(
2
), c
1
= 0.
It is now easy to prove that, up to a multiplicative constant and a coboundary
term, the Schwarzian derivative is the unique solution of the cocycle condition Equa-
tion 2.20. Let us first note that
[q
a
;q
b
]=(q
a
;q
b
) − c
1
{q
a
;q
b
},
satisfies the cocycle condition
[q
a
;q
c
]=
∂
q
c
q
b
2
(
[q
a
;q
b
]−[q
c
;q
b
]
)
.
In particular, since both (q
a
;q
b
) and {q
a
;q
b
} depend only on the first and higher
derivatives of q
a
, we have, as in the case of (q + f (q);q), that
[q +f (q);q]=˜c
1
f
(3)
(q) +O(
2
),
The Equivalence Postulate of Quantum Mechanics 29
where either ˜c
1
= 0or[q + f (q);q]=0. However, since {q + f (q);q}=
f
(3)
(q) + O(
2
) and (q + f (q);q) = f
(3)
(q) + O(
2
), we have ˜c
1
= 0 and the
lemma yields [q +f (q);q]=0. Therefore, we have that the EP univocally implies
that
(q
a
;q
b
) =−
β
2
4m
{q
a
, q
b
},
where for convenience we replaced c
1
by −β
2
/4m. This concludes the proof of The-
orem 1.
We observe that despite some claims [15], we have not been able to find in the
literature a complete and closed proof of the above theorem (see also Ref. [16]). We
thank D.B. Fuchs for a bibliographic comment concerning the above theorem.
In deriving the equivalence of states we considered the case of one-particle states
with identical masses. The generalization to the case with different masses is straight-
forward. In particular, the right hand side of Equation 2.20 gets multiplied by m
b
/m
a
,
so that the cocycle condition becomes
m
a
(q
a
;q
c
) = m
a
∂
q
c
q
b
2
(q
a
;q
b
) + m
b
(q
b
;q
c
),
explicitly showing that the mass appears in the denominator and that it refers to the
label in the first entry of (·;·), that is
(q
a
;q
b
) =−
2
4m
a
{q
a
;q
b
}. (2.43)
The QSHJE (Equation 2.21) follows almost immediately by Equation 2.43 [1–6].
The above investigation may be applied to Conformal Field Theory (CFT). Let
us consider a local conformal transformation of the stress tensor in a 2D CFT. The
infinitesimal variation of T is given by
δ
T (w) =−
1
12
c∂
3
w
(w) − 2T (w)∂
w
(w) − (w)∂
w
T (w), (2.44)
where c is the central charge. The finite version of such a transformation is
˜
T (w) = (∂
w
z)
2
T (z) +
c
12
{w, z}. (2.45)
While it is immediate to see that Equation 2.45 implies Equation 2.44, the converse
is not evident. A possible way to prove Equation 2.45 is just to set
˜
T (w) = (∂
w
z)
2
T (z) +k(w;z), (2.46)
and then to impose the cocycle condition which will show that (w;z) is proportional
to {w, z}. Comparison with the infinitesimal transformation Equation 2.44 fixes the
constant k.
In Ref. [7] it has been shown that the cocycle condition fixes the higher dimensional
version of the Schwarzian derivative. In this respect we observe that its definition
30 Quantum Trajectories
seems an open question in the mathematical literature. While in the one-dimensional
case the QSHJE reduces to a unique differential equation, this is not immediate in the
higher dimensional case. However, it turns out that such a reduction exists upon
introducing an antisymmetric tensor [7] (in this respect it is worth noticing that
some authors introduce a connection to define the higher dimensional Schwarzian
derivative).
A basic feature of the cocycle condition is that it implies, as it should, the higher
dimensional Möbious invariance with respect to q
a
in (q
a
;q
b
) (with similar properties
with respect to q
b
). In particular, in Ref. [7] it has been shown that
(q
a
;q
b
) =−
2
2m
(p
b
|p
a
)
∆
a
R
a
R
a
−
∆
b
R
b
R
b
. (2.47)
It would be interesting to consider such a definition in the context of the transformation
properties of the stress tensor in higher dimensional CFTs.
2.4 PROOF OF THEOREM 2
The QSHJE is equivalent to
{w, q}=−
4m
2
W(q), (2.48)
where w = ψ
D
/ψ with ψ
D
and ψ two real linearly independent solutions of the
Schrödinger equation. The existence of this equation requires some conditions on the
continuity properties of w and its derivatives. Since the QSHJE is a consequence of
the EP, we can say that the EP imposes some constraints on w = ψ
D
/ψ. These con-
straints are nothing but the existence of the QSHJE (Equation 2.21) or, equivalently,
of Equation 2.48. That is, implementation of the EP imposes that {w, q}exists, so that
w = cnst, w ∈ C
2
(R) and ∂
2
q
w differentiable on R. (2.49)
These conditions are not complete. The reason is that, as we have seen, the imple-
mentation of the EP requires that the properties of the Schwarzian derivative be
satisfied. Actually, its very properties, derived from the EP, led to the identification
(q
a
;q
b
) =−
2
{q
a
, q
b
}/4m. Therefore, in order to implement the EP, the transforma-
tion properties of the Schwarzian derivative and its symmetries must be satisfied. In
deriving the transformation properties of (q
a
;q
b
) we noticed how, besides dilatations
and translations, there is a highly non-trivial symmetry under inversion. Therefore,
we have that Equation 2.48 must be equivalent to
{w
−1
, q}=−
4m
2
W(q).
A property of the Schwarzian derivative is duality between its entries
{w, q}=−
∂w
∂q
2
{q, w}. (2.50)
The Equivalence Postulate of Quantum Mechanics 31
This shows that the invariance under inversion of w results in the invariance, up to a
Jacobian factor, under inversion of q. That is {w, q
−1
}=q
4
{w, q}, so that the QSHJE
Equation 2.48 can be written in the equivalent form
{w, q
−1
}=−
4m
2
q
4
W(q). (2.51)
In other words, starting from the EP one can arrive at either Equation 2.48 or Equa-
tion 2.51. The consequence of this fact is that since under
q →
1
q
,
0
±
maps to ±∞, we have to extend Equation 2.49 to the point at infinity. In other
words, Equation 2.49 should hold on the extended real line
ˆ
R = R∪{∞}. This aspect
is related to the fact that the Möbius transformations, under which the Schwarzian
derivative transforms as a quadratic differential, map circles to circles. We stress that
we are considering the systems defined on R and not
ˆ
R. What happens is that the
existence of the QSHJE forces us to impose smoothly joining conditions even at
±∞, that is Equation 2.49 must be extended to
w = cnst, w ∈ C
2
(
ˆ
R) and ∂
2
q
w differentiable on
ˆ
R. (2.52)
One may easily check that w is a Möbius transformation of the trivializing map [1–6].
Therefore, Equation 2.50, which is defined if and only if w(q) can be inverted, that
is if ∂
q
w = 0, ∀q ∈ R, is a consequence of the cocycle condition Equation 2.19. By
Equation 2.51 we see that also local univalence should be extended to
ˆ
R. This implies
the following joining condition at spatial infinity
w(−∞) =
w(+∞), for w(−∞) =±∞,
−w(+∞), for w(−∞) =±∞.
(2.53)
As illustrated by the non-univalent function w = q
2
, the apparently natural choice
w(−∞) = w(+∞), one would consider also in the w(−∞) =±∞case, does not
satisfy local univalence.
We saw that the EP implied the QSHJE Equation 2.21. However, although this
equation implies the SE, we saw that there are aspects concerning the canonical vari-
ables which arise in considering the QSHJE rather than the SE. In this respect a
natural question is whether the basic facts of Quantum Mechanics (QM) also arise
in our formulation. A basic point concerns a property of many physical systems such
as energy quantization. This is a matter of fact beyond any interpretational aspect
of QM. Then, as we used the EP to get the QSHJE, it is important to understand how
energy quantization arises in our approach. According to the EP, the QSHJE con-
tains all the possible information on a given system. Then, the QSHJE itself should
be sufficient to recover the energy quantization including its structure. In the usual
approach the quantization of the spectrum arises from the basic condition that in the
case in which lim
q→±∞
W > 0, the wavefunction should vanish at infinity. Once
32 Quantum Trajectories
the possible solutions are selected, one also imposes the continuity conditions whose
role in determining the possible spectrum is particularly transparent in the case of
discontinuous potentials. For example, in the case of the potential well, besides the
restriction on the spectrum due to the L
2
(R) condition for the wavefunction (a conse-
quence of the probabilistic interpretation of the wavefunction), the spectrum is further
restricted by the smoothly joining conditions. Since the SE contains the term ∂
2
q
ψ,
the continuity conditions correspond to an existence condition for this equation. On
the other hand, also in this case, the physical reason underlying this request is the
interpretation of the wavefunction in terms of probability amplitude. Actually, strictly
speaking, the continuity conditions come from the continuity of the probability den-
sity ρ =|ψ|
2
. This density should also satisfy the continuity equation ∂
t
ρ +∂
q
j = 0,
where j = i(ψ∂
q
¯
ψ −
¯
ψ∂
q
ψ)/2m. Since for stationary states ∂
t
ρ = 0, it follows that
in this case j = cnst. Therefore, in the usual formulation, it is just the interpretation
of the wavefunction in terms of probability amplitude, with the consequent meaning
of ρ and j, which provides the physical motivation for imposing the continuity of the
wavefunction and of its first derivative.
Now observe that in our formulation the continuity conditions arise from the
QSHJE. In fact, Equation 2.52 implies continuity of ψ
D
, ψ, with ∂
q
ψ
D
and ∂
q
ψ
differentiable, that is
EP → (ψ
D
, ψ) continuous and (ψ
D
, ψ
) differentiable. (2.54)
In the following we will see that if V (q) >E, ∀q ∈ R, then there are no solutions
such that the ratio of two real linearly independent solutions of the SE corresponds to
a local self-homeomorphism of
ˆ
R. The fact that this is an unphysical situation can also
be seen from the fact that the case V>E, ∀q ∈ R, has no classical limit. Therefore,
if V>Eboth at −∞and +∞, a physical situation requires that there are at least two
points where V −E = 0. More generally, if the potential is not continuous, V (q) −E
should have at least two turning points. Let us denote by q
−
(q
+
) the lowest (highest)
turning point. Note that by Equation 2.23 we have
−∞
q
−
dxκ(x) =−∞,
+∞
q
+
dxκ(x) =+∞,
where κ =
2m(V −E)/. Before going further, let us stress that what we actually
need to prove is that, in the case Equation 2.23, the joining condition (Equation 2.53)
requires that the corresponding SE has an L
2
(R) solution. Observe that while Equa-
tion 2.52, which however follows from the EP, can be recognized as the standard
condition Equation 2.54, the other condition Equation 2.53, which still follows from
the existence of the QSHJE, and therefore from the EP, is not directly recognized in
the standard formulation. Since this leads to energy quantization, while in the usual
approach one needs one more assumption, we see that there is quite a fundamental
difference between the QSHJE and the SE. We stress that Equations 2.52 and 2.53
guarantee that w is a local self-homeomorphism of
ˆ
R.
Let us first show that the request that the corresponding SE has an L
2
(R) solution
is a sufficient condition for w to satisfy Equation 2.53. Let ψ ∈ L
2
(R) and denote
The Equivalence Postulate of Quantum Mechanics 33
by ψ
D
a linearly independent solution. As we will see, the fact that ψ
D
∝ ψ implies
that if ψ ∈ L
2
(R), then ψ
D
/∈ L
2
(R). In particular, ψ
D
is divergent both at q =−∞
and q =+∞. Let us consider the real ratio
w =
Aψ
D
+ Bψ
Cψ
D
+ Dψ
,
where AD − BC = 0. Since ψ ∈ L
2
(R), we have
lim
q→±∞
w = lim
q→±∞
Aψ
D
+ Bψ
Cψ
D
+ Dψ
=
A
C
, (2.55)
that is w(−∞) = w(+∞). In the case in which C = 0 we have
lim
q→±∞
w = lim
q→±∞
Aψ
D
Dψ
=± ·∞,
where =±1. The fact that Aψ
D
/Dψ diverges for q →±∞follows from the
mentioned properties of ψ
D
and ψ. It remains to check that if lim
q→−∞
Aψ
D
/Dψ =
−∞, then lim
q→+∞
Aψ
D
/Dψ =+∞, and vice versa. This can be seen by observing
that
ψ
D
(q) = cψ(q)
q
q
0
dxψ
−2
(x) + dψ(q),
c ∈ R\{0}, d ∈ R. Since ψ ∈ L
2
(R) we have ψ
−1
∈ L
2
(R) and
+∞
q
0
dxψ
−2
=+∞,
−∞
q
0
dxψ
−2
=−∞, implying that ψ
D
(−∞)/ψ(−∞) =− ·∞=−ψ
D
(+∞)/
ψ(+∞), where = sgn c.
We now show that the existence of an L
2
(R) solution of the SE is a necessary
condition to satisfy the joining condition Equation 2.53. We give two different proofs
of this, one is based on the WKB approximation while the other one uses Wronskian
arguments. In the WKB approximation, we have
ψ =
A
−
√
κ
exp
−
q
q
−
dxκ
+
B
−
√
κ
exp
q
q
−
dxκ
, q q
−
, (2.56)
and
ψ =
A
+
√
κ
exp
−
q
q
+
dxκ
+
B
+
√
κ
exp
q
q
+
dxκ
, q q
+
. (2.57)
In the same approximation, a linearly independent solution has the form
ψ
D
=
A
D
−
√
κ
exp
−
q
q
−
dxκ
+
B
D
−
κ
exp
q
q
−
dxκ
, q q
−
.
Similarly, in the q q
+
region we have
ψ
D
=
A
D
+
√
κ
exp
−
q
q
+
dxκ
+
B
D
+
√
κ
exp
q
q
+
dxκ
, q q
+
.