1. Introduction
The
method (we simply refer to the “
n-th order JWKB (or WKB)” by a simple abbreviation:
) is conventionally known to be a strong and effective semiclassical approximation method enabling accurate analytical solutions in quantum mechanical systems, i.e., [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10]. Quantum mechanical systems described by the Time Independent Schrodinger’s Equation (TISE), which is in the form of a linear second order homogenous (normal form) differential equation:
where these terms have their usual meanings (
m represents mass;
ℏ represents Planck’s constant divided by
;
E represents total energy; and
represents the potential function), have exact and approximate
solutions in the following forms:
where
and
are the arbitrary constants and,
and
are the exact and
complementary solutions, respectively. These constant coefficients in the exact and
general solutions can be found from given initial values. The
solution has a typical property that both complementary
solutions (and hence, the
general solution) diverge at a small region around the classical turning point where
, i.e., [
3,
4,
5]. Moreover, the
general solution in (
2b) can be accurate for the Classically Accessible Region (CAR) under some circumstances, but always needs asymptotic matching in the Classically Inaccessible Region (CIR) for accurate
solutions [
3,
4]. CAR is the classically accessible region where the particle can classically exist since its potential energy is smaller than its total energy:
; and CIR is the classically inaccessible region where it cannot classically exist since its potential energy is greater than its total energy:
. Conventional
asymptotic matching rules require either of the complementary solutions in (
2b) to be canceled in the CIR as follows [
3,
4]:
and the resultant asymptotically-matched solution should assure:
where
represents the asymptotically-matched
general solution. In other words, the asymptotically diverging term in the CIR should be canceled in the general solution so that (
4) can hold. The formal
approximation formula involving both complementary functions in (
2b) is actually in the form of an infinite series:
where
for the TISE and
represents the expansion terms given in [
3,
4], and the two-valuedness of these expansion terms gives two complementary
functions. However, only the first two terms (with indices
and 1) are used in the
approximation. Moreover,
gives accurate-enough solutions for slowly-changing potentials in the TISE, and a criterion for this is given as follows [
3,
4,
5]:
As the potential in the TISE in (
1b) gets sharper, (
6) fails, and some of the higher order terms can no longer be neglected; and the higher order JWKB approximation (=
) is required for accurate-enough solutions [
1,
2,
3,
4,
5,
6,
7]. Therefore, for a general potential
involving both smooth and sharp sub-domains in the corresponding normal form (the TISE), say
(for the smooth) and
(for the sharp), respectively, the
approximation always gives accurate general solution in
and inaccurate (but asymptotically matchable) solutions in
; however,
is required for
[
3,
4,
5]. Such a potential with obedient (smooth potential =
) and non-obedient (sharp potential =
) subdomains in the corresponding TISE is studied semiclassically by the first order Bessel Differential Equation,
, as a chosen model differential equation here. Our aim here is to show these physical results by a successful semiclassical analysis mathematically only (not physically).
Since the
method is generally applied to the quantum mechanical systems, the main principles of the existing asymptotic matching rules rely on the nature of the physical system under study, i.e., a physically-acceptable bound state wave function (solution of the TISE) in the CIR should not asymptotically diverge to infinity (which means Equation (
4)) so that it can be normalized in
[
3,
4,
5]. This result is the physical consequence of the asymptotic matching rule in the
for the
. Its semiclassical explanation for the Simple Linear Potential (SLP), as a model potential where the
applicability criterion is satisfied in the entire domain, was studied in terms of the
expansion terms in [
4]. In this work, similarly, a pure semiclassical analysis of the asymptotic matching rules is studied for the intentionally-chosen
where the
applicability criterion is now partially satisfied in some subdomains involving both CAR and CIR. In other words,
and
coexist in the whole domain of our
with a successive turning point so that
and
are guaranteed. In our analysis, appropriately-chosen associated initial values are used to compare the general asymptotically-unmatched and -matched
solutions with the related exact general solutions.
solutions of some quantum mechanical systems involving exponential potential-decorated TISE, which is associated with the
, were studied by the use of the common asymptotic matching rules given in (
4) in the literature [
10,
11,
12]. Our aim here is rather to search the asymptotic modifications of the
approximation for the
mathematically via the semiclassical theories where the physical nature of the system regarding the bound and unbound quantum mechanical system analysis is no longer interfered. We expect to find the same asymptotically-matched
wavefunction solution as in [
10,
11,
12]. It is also shown here for a specific case of the exponential potential decorated TISE in
Section 5.
The
is given in the standard form by:
whose exact general solution in (
2a) is the linear combination of two kinds of first order Bessel functions, namely:
Their expressions in terms of infinite series are given as follows [
13,
14]:
The
solution of the
in (
7) can similarly be written (when solved) as a linear combination of two complementary functions as given in (
2b). However, to follow this procedure, one has to face with the problem arising from the fact that one cannot find the
general solution given in (
2b) directly by the conventional methods since the
technique including the famous
connection formulas requires (rather than that in (
7)) a Linear Differential Equation (LDE) in the normal form given in (
1a). Therefore, we have to study it in the normal form with a suitable change of variable. Complementary
functions (solutions) in (
2b) can then be easily found by using the famous
connection formulas given in [
1,
2,
3,
4,
5]. Once the
solution of either region (CAR or CIR) is found, the other region can directly be determined via these connection formulas.
Therefore, our interest here can be summarized as follows: (i) to find the
general solution of the
whose structure is given in (
2b) by using some appropriate change of variable to transform into a normal form (which is not unique); (ii) to check its accuracy in the (sub)domains of the CAR and the CIR where the
applicability criterion in (
6) holds; and (iii) to find ways to do the correct asymptotic matching in the necessary (sub)domains by semiclassical analyses mathematically only. Our analyses should also show that asymptotic matching is required only for
as the physical requirement (
normalizability of quantum mechanical wave functions, which is in a bound state form here).
The
general solution of the
is obtained here after having been transformed into the normal form via change of the independent variable so that the conventional
connection formulas given in [
1,
2,
3,
4,
5] can be used along with the associated initial values, which have been intentionally chosen in the CAR. Comparisons with the exact solutions are being achieved by applying these carefully-chosen common initial values. Since our asymptotic matching rule gives a criterion for a semiclassically (not physically) acceptable
solution, we have the following semiclassical outcomes: (i) why
solutions are accurate in some domains in the CAR (
) and why they give inaccurate results in some other domains in the CIR (
); (ii) which complementary function in the general solution in (
2b) (and also in the corresponding transformed representation) should be canceled in order to give an accurate general solution as desired by a successful asymptotic matching. Therefore, it can be thought of as an alternative and more general semiclassical matching rule for the present conventional
theories.
In
Section 2, we give the statement and re-statement of our analyses based on the initial value-aided comparison in both standard and normal form representations.
solutions of the
and their asymptotic matching in the transformed representations (in the normal form) are studied in
Section 3, and the same calculations for the re-transformed representations (in the standard form) are studied in
Section 4. Our semiclassical asymptotic matching rule neither involving exact solutions nor reasoning about the physical (quantum mechanical) nature of the system should give the correct asymptotic matching in both representations. We use the exact solutions only to show the accuracy and reliability of our alternative pure semiclassical asymptotic matching rule given in
Section 3.2 (by Proposition 2). In
Section 5, a physical application of our semiclassical asymptotic matching rule for a specific case of the exponential potential decorated bound state problem is presented. Preliminary work regarding the part “Calculations in The Normal Form” of
Section 3 was discussed in the
19th International Conference on Applied Mathematics (AMATH’14)-Istanbul where some 2D analyses are available in [
15]. (Note that Equations: (
40) and (
41a)–(
41b) here are in the corrected form when compared with the misprints in [
15]).
5. A Physical Application: Exponential Potential Decorated Bound State Problem
It might be interesting to test the success of our alternative asymptotic matching rules in a real quantum mechanical (physical) problem corresponding to the normal form representation via the TISE. Therefore, let us apply these calculations to a specific case of the exponential potential decorated bound state problem studied in [
10], from which we have the followings (note that we use different symbols here so that we can refer to the calculations in this work, namely:
in [
10] are replaced by
here, respectively):
For the TISE in (
1a) and (
1b), but in variable
τ, the potential under study is given by:
Necessarily,
for the bound states and by a simple change of variable:
we have the TISE in (
1a) and (
1b) in
ρ:
Now, the definitions of dimensionless
a and
b:
give the TISE in
ρ with:
Then, the change of the independent variable:
gives the
b-th order Bessel differential equation,
, namely:
whose exact general solution is the linear combination of two kinds of the
b-th order Bessel functions (similar to (
19a) and (
19b)), namely:
Note that, second complementary solutions for both regions in (
64) are given in [
10] (instead of
) as
, since
and
are two linearly independent solutions for non-integer Bessel functions (just like
and
). Indeed, from Bessel theories, we have [
13,
14]:
and for also integer
b values, as in our study here with
, we can use Equation (
64) as a general expression involving both integer and non-integer
b values [
13,
14]. Now, (
22) takes the form:
and its comparison with (
64) gives the following relation:
Now, let us take
as a special case and apply the asymptotic matching rules we have already found in the previous sections. Our restriction then becomes:
so that we have:
which is just as the form we have been using here as in (
17a) and (
17b). Additionally, its solution from (
64) gives:
which is just the same as our result in (
22). Our intention here is not to find accurate
eigenenergies (
) and eigenfunctions (
) where some correction terms (like Friedrich-Trost (F-T) corrections) are involved [
8,
11,
12], but to apply our asymptotic matching rules under the assumption in (
68) where the problem simplifies to the
being studied here.
Conventional asymptotic modification rules in (
4) were applied in [
10] to give:
which means:
in our calculations. We have already shown half of it for
by our alternative semiclassical asymptotic matching rule in (
41a), (
41b) and (
42) via
Figure 3, but let us see it for (
69) in the whole domain explicitly:
Now, expansion terms in (
38a)–(
38c) for (
69) give just the same as (
44a)–(
44c) for
, but these results require
for
, namely:
from which (42) gives:
Graphs of functions
f and
g in both representations are given in
Figure 8, from which we can see that
solutions are consistent with the exact solutions, except for the non-obedient narrow regions (from (
29a)):
and the remaining wide regions (from (
29b)):
need asymptotic matching (
). Graphs of
and
for some
c values are given in
Figure 9. We can obviously see that both (
41a) and (
41b) hold in the CAR simultaneously, but they do not hold in the CIR simultaneously and, hence, are in need of a cancellation in the CIR. Here, we have non-obedient
term (and hence, a cancellation of
for
(see the left column in
Figure 9) and
term (and hence, a cancellation of
) for
(see the right column in
Figure 9) in these CIRs. This is because, as stated above, (
39) with (
38a)–(
38c) has an implication that
(or
) contributes to
and
(or
) contributes to
. Therefore, the non-obedient
terms requiring a cancellation of
for
and
terms requiring a cancellation of
for
give the same right asymptotic matching as given in (
71b) of [
10].
6. Conclusions
In this work, the
has been chosen as a mathematical model for a pure semiclassical analysis (without interference of the physical nature of the system) since there exist subdomains where the
applicability criterion both holds (in
) and fails (in
). The hereby presented generalized asymptotic matching rules regarding the
matrix elements obtained from the
expansion terms show that the
general solution of the
for carefully chosen initial values needs asymptotic matching in the transformed (normal form) representation in the CIR where the
applicability criterion holds (
). Moreover, there is no need for asymptotic matching in the CAR where the
applicability criterion holds (again
). These results, obtained by our pure semiclassical analysis, are consistent with the present conventional asymptotic matching rules given in (
3) and (
4) via [
3,
4], which is a natural consequence of the physical nature of the related quantum mechanical system (the normalizability requirement of quantum mechanical wave functions). The generalized asymptotic matching rules suggested in (
41a) and (
41b) with the generalized definition of
in (
42) are the results of our pure semiclassical analyses where the physical nature of the system is not interfered.
As explained in the work in detail, physics (nature) requires asymptotic matching to the full mathematical solution. Physical systems of second order are generally of two kinds:
(here, both are given in the normal form). We have two initial values or boundary values for both as common. However, physics require a third initial value or a third boundary value (or more correctly, an asymptotic constraint) given in (
3) and (
4), which is called asymptotic matching since
in (
4). In this work, it is studied and proven mathematically via the JWKB theories stating that JWKB expansion terms should be descending and bounded as given and studied in [
3,
4]. Therefore, the main idea here is the semiclassical requirement that
expansion terms should be asymptotically decreasing as the term index increases and bounded by the
-th indexed term as stated in [
3,
4]. The two valuedness of the
expansion terms as given in (
38a)–(
38c) enables definitions of two different sets (corresponding to two complementary functions as in (
39)) according to (
42) for them so that the semiclassical requirements for the asymptotic matching can be tested for these two sets accordingly. Equation (
39) shows that
(hence,
in our comparison function) contributes to
, and similarly,
(hence,
in our comparison function) contributes to
. Therefore, any violation of
in (
41a) requires a cancellation of
, and any violation of
in (
41b) requires a cancellation of
in the related non-obedient (sub)domain provided that the
applicability criterion holds (=
). The same definition in (
42) gives (
43) in the transformed normal form representation and (
52) in the re-transformed standard form representation, respectively. When they are used in our alternative asymptotic matching rules given in (
41a) and (
41b), this gives consistent results with the present asymptotic matching rules given in (
3) and (
4). Moreover, our alternative analyses enable a correct determination of which complementary function (solution) and where in the semiclassically solvable (sub)domain to cancel. As a result, asymptotic matching rules suggested here gives enhanced
solutions for the specifically chosen
in both normal form and standard form representations. Since both standard and normal forms correspond to different sub-domains obeying the JWKB applicability criterion in (
6) and (
29b) being for the normal form and (
50b) being for the standard form), both should require different asymptotic matching results for these different domains. In deed, our suggested asymptotic matching rule has explained both correctly and has given the correct results as the existing asymptotic matching rules given in (
3) and (
4). Applications of our pure semiclassical asymptotic matching rules to the exponential potential decorated bound state problem presented in
Section 5 also give the same asymptotically-matched solutions as in [
10], where the conventional asymptotic matching rules in (
3) and (
4) are applied. It can also be reminded here that the addition of an asymptotic constraint as a third initial value or boundary value does not change the correctness of the mathematical solution for sure, but matches it to the physical requirement of nature. We here have not applied the physical nature of the system, but only semiclassical theories, and the result is the correct asymptotic matching as nature requires. It may provide us deeper understanding of the relationship between the requirements of nature and mathematics. Therefore, the asymptotic matching rules in (
41a), (
41b) and (
42) obtained by our pure semiclassical analyses here without applying (or consulting) any of the physical (quantum mechanical) nature of the system seem to be an alternative (and also more general) equivalent asymptotic matching rules besides the present conventional rules given in (
3) and (
4) via [
3,
4] in the real domain. It should be noted here that we here studied the asymptotic matching near the turning point locally via the essential principle that the JWKB expansion terms should be descending and bounded. The Stokes phenomenon where the complementary solutions flip according to the Stokes and anti-Stokes lines as a result of the argument of the phase, i.e., ([
3], pp. 112–117, [
18,
19,
20,
21]), which is beyond the scope of our work here, would be involved in the far asymptotic analyses globally in the complex plane. However, JWKB connection formulas, which are known to be as a result of the Stokes phenomenon (with the restricted complex path in the calculation of the integrals in (
32a)) [
18], are used here. Indeed, the phase integral analyses for the higher order JWKB, where the conventional connection formulas are not valid (and the integrands in (
32a) are different), would obviously require a careful consideration of the Stokes phenomenon. Note that phase integral approximation with the integrands in (
32a), which are called the phase integrands, and hence, with the integrals in (
32a), which are called the phase integrals, give the conventional JWKB approximation studied here [
18].