WKB Approximation for a Deformed Schrodinger-like Equation and its Applications to Quasinormal Modes of Black Holes and Quantum Cosmology

0.42 MB·

by  Bochen Lv

CTP-SCU/2017002
WKB Approximation for a Deformed Schrodinger-like Equation
and its Applications to Quasinormal Modes of Black Holes and
Quantum Cosmology
Bochen Lv, Peng Wang, and Haitang Yang
∗ † ‡
Center for Theoretical Physics, College of Physical Science and Technology,
7 Sichuan University, Chengdu, 610064, China
1
0
Abstract
2

n
In this paper, we use the WKB approximation method to approximately solve a deformed
a
J
Schrodinger-like differential equation: ~2∂2g2( i~α∂ ) p2(ξ) ψ(ξ) =0, which are frequently
8 − ξ − ξ −
1
h i
dealt with in various effective models of quantum gravity, where the parameter α characterizes

]
c effects of quantum gravity. For an arbitrary function g(x) satisfying several properties proposed in
q
-
r the paper, we find the WKB solutions, the WKB connection formulas through a turning point, the
g
[
deformed Bohr–Sommerfeld quantization rule, and the deformed tunneling rate formula through a

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v potential barrier. Several examples of applyingthe WKB approximation to the deformed quantum
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9 mechanics are investigated. Inparticular, we calculate theboundstates of the Po¨schl-Teller poten-
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0 tial andestimate theeffects ofquantum gravity on thequasinormalmodes ofa Schwarzschild black
0
.
1 hole. Moreover, the area quantum of the black hole is considered via Bohr’s correspondence princi-
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7 ple. Finally, theWKB solutions of the deformed Wheeler–DeWitt equation for a closed Friedmann
1
: universe with a scalar field are obtained, and the effects of quantum gravity on the probability of
v
i
X sufficient inflation is discussed in the context of the tunneling proposal.
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
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Contents
I. Introduction 2
II. WKB Method 4
A. WKB Solutions 5
B. Connection Formulas 6
C. Bohr-Sommerfeld Quantization and Tunneling Rates 13
III. Examples 15
A. Harmonic Oscillator 17
B. Schwinger Effect 18
C. Po¨schl-Teller Potential and Quasinormal Modes of A Black Hole 21
D. Quantum Cosmology 25
IV. Conclusion 28
Acknowledgments 30
A. Contours in g(x) = tan(x) Case 30
x
References 31
I. INTRODUCTION
The WKB approximation, named after Wentzel, Kramers, and Brillouin, is a method for
obtaining an approximate solution to a one-dimensional Schrodinger-like differential equa-
tion:
~2∂2 p2(ξ) ψ(ξ) = 0, (1)
− ξ −
(cid:2) (cid:3)
where the real function p2(ξ) can be either positively or negatively valued. The WKB
approximation has a wide range of applications. Its principal applications are in calculating
bound-state energies and tunneling rates through potential barriers.
On the other hand, the construction of a quantum theory for gravity has posed one of the
most challenging problems of the theoretical physics. Although there are various proposals
for quantum gravity, a comprehensive theory is not available yet. Rather than considering a
2
full quantum theoryofgravity, we caninsteadstudy effective theoriesofquantum gravity. In
various effective models of quantum gravity, one always deals with a deformed Schrodinger-
like equation:
P2( i~∂ ) p2(ξ) ψ(ξ) = 0, (2)
ξ
− −
where P (x) = xg(αx). The (cid:2)properties of the fu(cid:3)nction g(x) will be discussed in section
II. Note that the parameter α characterizes effects of quantum gravity. For example, the
deformed Schrodinger-like equation (2) could appear in two effective models, namely the
Generalized Uncertainty Principle (GUP) and the modified dispersion relation (MDR). We
will briefly show that how it appears in these two models in section III.
The WKB approximation in deformed space and its applications have been considered
in effective models of quantum gravity. For example, in the framework of GUP, the WKB
wave functions were obtained in [1]. Moreover, the deformed Bohr–Sommerfeld quantization
rule and tunneling rate formula were used to calculate bound states of Harmonic oscillators
and Hydrogen atoms [1], α-decay [2, 3], quantum cosmogenesis [2], the volume of a phase
cell [4], and electron emissions [3], for some specific function g(x). In the context of both
GUP and MDR, the deformed Bohr–Sommerfeld quantization rule was used to compute the
number of quantum states to find the entanglement entropy of black holes in the brick wall
model [5–7]. In [3, 4], we found the WKB connection formulas and proved the deformed
Bohr–Sommerfeld quantization rule and tunneling rate formula for the g(x) = √1+x2
case. In this paper, we will consider the case with an arbitrary function g(x), for which the
WKB connection formulas, Bohr–Sommerfeld quantization rule and tunneling rate formula
are obtained.
The organization of this paper is as follows. In section II, the deformed Schrodinger-
like differential equation (2) are first approximately solved by the WKB method. After
the asymptotic behavior of exact solutions of eqn. (2) around turning points are found,
we obtain the WKB connection formulas through a turning point by matching these two
solutions in the overlap regions. Accordingly, the Bohr–Sommerfeld quantization rule and
tunneling rate formula are also given. In section III, the formulas obtained in section II are
used to investigate several examples, namely harmonic oscillators, the Schwinger effect, the
Po¨schl-Teller potential, and quantum cosmology. Section IV is devoted to our discussion
and conclusion. In the appendix, we plot the contours used to compute the asymptotic
behavior in the g(x) = tan(x) case.
x
3
II. WKB METHOD
We now apply the WKB method to approximately solve the deformed Schrodinger-like
differential equation (2). In what follows, we choose that argp(ξ) = 0 for p2(ξ) > 0 and
argp(ξ) = π for p2(ξ) < 0. Moreover, we could rewrite P (x) in terms of a new function
2
g(x) as
P (x) = xg(αx). (3)
To study the WKB solutions of eqn. (2) and the connection formulas through a turning
point, we shall impose the following conditions on the function g(x):
In the complex plane, g(z) is assumed to be analytic except for possible poles. We
•
assume that g(0) = 1.
For a positive real number a > 0, each of the equations
•
sg( ias) = eiπk/2, with k = 0, 1, 2, 3, (4)
−
possess only one regular solution λ (a), which is regular as a 0 and becomes
k
→
λ (0) = eiπk/2, (5)
k
(i)
and the possible runaway solutions η (a), which becomes
k
(i)
η
η(i)(a) k , (6)
k ∼ a
when a 1. We also assume that for small enough value of a, there exists a c > 0
1
≪
such that for all possible i and k,
c
1 (i)
< η (a) . (7)
a k
(cid:12) (cid:12)
(cid:12) (cid:12)
If there is no runaway solution, we simply(cid:12) set c1(cid:12)= .
∞
Finally, we assume that there exists a c > 0 such that
2
•
1
g2( is) 1 for s c . (8)
2
− − ≤ 2 | | ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
For example, g(x) = 1 x2 satisfies the above conditions with 0 < c < 1 and 0 < c
1 2
± ∃ ≤
1/√2. The function g(x) = tanx also satisfies the above conditions with 0 < c < π and
x ∃ 1
0 < c arctan(3/2).
2
≤
4
A. WKB Solutions
To find an approximate solution via the WKB method, we could make the change of
variable
iS(ξ)
ψ(ξ) = e ~ (9)
for some function S(ξ), which can be expanded in power series over ~
~
S(ξ) = S (ξ)+ S (ξ)+ . (10)
0 1
i ···
Plugging eqn. (10) into eqn. (2) gives
P2(S (ξ)) = p2(ξ),
0′
~ i~
P2(S0′ (ξ)) ′ iS1(ξ) = 2 P2(S0′ (ξ)) ′′S0′′(ξ), (11)
(cid:2) (cid:3) (cid:2) (cid:3)
where the prime denotes derivative with respect to the argument of the corresponding func-
tion. The first equation in eqn. (11) can be solved for S (ξ). In particular, when p2(ξ) > 0,
0′
S (ξ) = i p(ξ) λ (α p(ξ) ), with k = 1 and 3, (12)
0′ − | | k | |
and when p2(ξ) < 0,
S (ξ) = i p(ξ) λ (α p(ξ) ), with k = 0 and 2, (13)
0′ − | | k | |
where λ (a) are regular solutions of eqn. (4). It is noteworthy that there are other possible
k
solutions, namely
S (ξ) = i p(ξ) η (α p(ξ) ). (14)
0′ − | | k | |
These solutions are called ”runaways” solutions since they do not exist in the limit of α 0.
→
In [8], it was argued that these ”runaways” solutions were not physical and hence should be
discarded. A similar argument was also given in the framework of the GUP [9]. Therefore,
we will discard the ”runaways” solutions and keep only the solutions (12) and (13) in this
paper. Solving the second equation in eqn. (11) gives
1
S1(x) = ln P2(x) ′ x=S′(ξ) . (15)
−2 | 0
(cid:12) (cid:12)
(cid:2) (cid:3)
(cid:12) (cid:12)
The expression for the WKB solutions are (cid:12) (cid:12)
ψ (ξ) = C ψ1 (ξ)+C ψ3 (ξ) for p2(ξ) > 0 (16)
WKB 1 WKB 3 WKB
5
and
ψ (ξ) = C ψ0 (ξ)+C ψ2 (ξ) for p2(ξ) < 0, (17)
WKB 0 WKB 2 WKB
where C are constants, and we define
i
1 1
ψk (ξ) = exp p(ξ) λ (α p(ξ) )dξ . (18)
WKB ~ | | k | |
[x2g2(αx)]′|x=−i|p(ξ)|λk(α|p(ξ)|) (cid:18) Z (cid:19)
q
(cid:12) (cid:12)
These WKB solutions are valid if the RHS of the second equation in eqn. (11) is much less
(cid:12) (cid:12)
than that of the first one. Specifically, they are valid when
~
p2(ξ) ≫ 2 P2(S0′ (ξ)) ′′S0′′(ξ) . (19)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:2) (cid:3)
(cid:12) (cid:12)
However, the condition (19)(cid:12)fails n(cid:12)ear a turning point where P (x) = 0. In the following of
(cid:12) (cid:12)
this section, we will derived WKB connection formulas through the turning points.
B. Connection Formulas
We first investigate the asymptotic behavior of solutions of the differential equation
∂2g2( iα˜∂ )ψ ρψ = 0, (20)
ρ − ρ −
where α˜ > 0. To solve this equation, it is useful to Laplace transform it via
ψ(ρ) = eρtψ˜(t)dt, (21)
ZC
˜
where the contour C in the complex plane will be discussed below. The equation for ψ(t)
in terms of the complex variable t reads
˜
dψ(t)
+t2g2( iα˜t)ψ˜(t) = 0, (22)
dt −
where we use the integration by parts to obtain the second term. Note the integration by
parts used in eqn. (22) requires that eρtψ˜(t) vanishes at endpoints of C. Up to an irrelevant
pre-factor, its solution is
t
ψ˜(t) = exp t2g2( iα˜t)dt . (23)
′ ′ ′
− −
(cid:18) Z0 (cid:19)
1
To apply the saddle point method, we make the change of variables t = ρ 2 s and rewrite
| |
the Laplace transformation in eqn. (21) as
1 3
ψ(ρ) = ρ 2 exp ρ 2 f (s) ds, (24)
| | | | ±
ZC h i
6
where we define a = α˜ ρ 1/2 and
| |
s
f (s) = s s2g2( ias)ds, (25)
′ ′ ′
± ± − −
Z0
with + for ρ > 0 and for ρ < 0. The contour C in eqn. (24) is chosen so that the integrand
−
vanishes at endpoints of C.
Nowconsider a largecircle C ofradius R = c, where c = min c , c . The saddle points
R a { 1 2}
(i)
of f (s) (f (s)) are λ (a) and η (a) with k = 0 and 2 (1 and 3). Thus, all the saddle
+ k k
−
points except λ (a) are outside the circle C . To discuss the properties of the steepest
k R
descent contours passing through λ (a), we first prove two propositions. In the following,
k
let C denote the steepest descent contours passing through λ (a).
λk k
Proposition 1 For small a, if C intersects C at Reiθ∗, then there exists an n 0,1,2
λk R ∈ { }
such that
2nπ π
θ + (a). (26)
∗
− 3 ≤ 18 O
(cid:12) (cid:12)
(cid:12) (cid:12)
Moreover, one finds (cid:12) (cid:12)
(cid:12) (cid:12) c3
Ref Reiθ∗ . + a 2 . (27)
−
± −6a3 O
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. For f (s), we have
±
s 3ρ(s)ei[3θ+σ(s)]
f (s) = s | | , (28)
± ± − 3
where
s = s eiθ,
| |
3 as
f (s)eiα(s) = x2 g2( ix) 1 dx, (29)
a3s3 − −
Z0
(cid:2) (cid:3)
ρ(s)eiσ(s) = 1+f (s)eiα(s).
Since g2( ix) 1 1 for x aR, one finds for s R that
| − − | ≤ 2 | | ≤ | | ≤
3 as 1
f (s) | | x2 g2( ix) 1 d x , (30)
≤ as 3 − − | | ≤ 2
| | Z0
(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
and hence
1 3
1 f (s) ρ(s) 1+f (s) ,
2 ≤ − ≤ ≤ ≤ 2
7
1 π
sinσ(s) f (s) σ(s) arcsinf (s) arcsin = . (31)
| | ≤ ⇒ | | ≤ ≤ 2 6
Suppose that the contour C intersects C at s = Reiθ∗. Since C is also a constant-
λk R ∗ λk
phase contour, C is determined by
λk
Imf (s) = Imf (λ (a)). (32)
k
± ±
At s = s , this equation becomes
∗
ρ(s)sin[3θ +σ(s )]
ca2sinθ c3 ∗ ∗ = a3Imf (λ (a)), (33)
∗ k
± − 3 ±
where we use R = c. For small a, one has Imf (λ (a)) (a) and hence
a ± k ∼ O
σ(s ) 2nπ (2n+1)π
∗
θ + = + (a) or + (a), (34)
∗
3 3 O 3 O
where n 0,1,2 . However for θ + σ(s∗) = (2n+1)π + (a), we find at s = s that
∈ { } ∗ 3 3 O ∗
ρ(s )c3
Ref±(s∗) ∼ 3a∗3 ≫ Ref±(λk(a)) ∼ O(a), (35)
which contradicts C being the steepest descent contour. Thus, for the steepest descent
λk
contour C , eqn. (34) gives for some n 0,1,2 that
λk ∈ { }
2nπ σ(s ) π
∗
θ = | | + (a) + (a).
∗
− 3 3 O ≤ 18 O
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
It can be easily shown that
(cid:12) (cid:12)
ρ(s )c3 c3
Ref (s ) = ∗ + a−2 + a−2 , (36)
± ∗ 3a3 O ≤ −6a3 O
(cid:0) (cid:1) (cid:0) (cid:1)
where we use ρ(s ) 1.
∗ ≥ 2
Proposition 2 On the circle C , Ref Reiθ c3 + (a 2) for θ 2nπ π + (a),
R ± ≤ −12a3 O − − 3 ≤ 18 O
where n 0,1,2 . (cid:0) (cid:1) (cid:12) (cid:12)
∈ { } (cid:12) (cid:12)
Proof. Since σ(s) π on the C , if θ 2nπ π + (a), we have
| | ≤ 6 R − 3 ≤ 18 O
(cid:12) (cid:12)
σ(s)(cid:12) 2nπ(cid:12) π
θ+ + (a), (37)
3 − 3 ≤ 9 O
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
which leads to
(cid:12) (cid:12)
π 1
cos[3θ+σ(s)] cos = + (a). (38)
≥ 3 2 O
(cid:16) (cid:17)
8
4
2
Λ2H0LΘ=2Π(cid:144)3 Λ1H0L Λ0H0L
-3 -2 -1 1 2 3
Θ=4Π(cid:144)3
Λ3H0L
-2
-4
FIG. 1: The saddle points (red dots) and steepest descent contours (blue thick lines) of f (s)
±
when a = 0.
Thus, it shows that on θ 2nπ π + (a),
− 3 ≤ 18 O
R3(cid:12)(cid:12)ρ Reiθ (cid:12)(cid:12)cos 3θ+σ Reiθ c3
Ref Reiθ = + a 1 + a 2 , (39)
− −
± − 3 O ≤ −12a3 O
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where we use ρ Reiθ 1.
≥ 2
For a = 0, w(cid:0)e plo(cid:1)t saddle points (red dots) of f (s) and the steepest descent contours
±
(blue thick lines) passing through them in FIG. 1. When a > 0, more possible saddle
pointsandpoles couldappear andthese steepest descent contours couldchange dramatically
around them, e.g. a contour that goes to infinity in the case a = 0 could change to the one
that ends at a new saddle point or a pole. However within C , there are no new saddle
R
points or poles, and hence C would change continuously as a is varied away from 0. FIG.
λk
1 shows that for a = 0, C approaches θ = 2π and 4π for a large value of s . So when a > 0,
λ2 3 3 | |
C will intersect C twice, and the intersections Reiθ∗ are within θ 2π π + (a) and
λ2 R ∗ − 3 ≤ 18 O
θ∗ − 43π ≤ 1π8+O(a), respectively. Similarly, Cλ1 intersects CR w(cid:12)(cid:12)ithin |θ∗(cid:12)(cid:12)| ≤ 1π8+O(a) and
(cid:12)(cid:12)θ∗ − 23π(cid:12)(cid:12) ≤ 1π8+O(a), Cλ3 intersects CR within |θ∗| ≤ 1π8+O(a) and θ∗ − 43π ≤ 1π8+O(a),
a(cid:12)(cid:12)nd Cλ0(cid:12)(cid:12)ends at λ2(a) and intersects CR within |θ∗| ≤ 1π8 +O(a). (cid:12)(cid:12) (cid:12)(cid:12)
Since the saddle points and hence the solutions in eqn. (24) depend on the sign of ρ,
it is convenient to choose different contours in the complex plane for ρ > 0 and ρ < 0,
making sure that they are deformable to each other. In FIG. 2(a), the contour considered
in the ρ > 0 case is the steepest descent contour C through λ (a). For simplicity, we
λ2 2
assume argλ (a) = kπ/2 to illustrate the contours in FIG. 2. In FIG. 2, Ci denotes the
k λk
9
Θ=2Π(cid:144)3 Θ=2Π(cid:144)3
CΛ12
C1
R
CR CR
Λ1HaL
CΛi1
Λ2HaL CR3 Λ2HaL CΛi0 Λ0HaL CΛ10
C4
R
CΛi2 CΛi3 CΛi12 CΛi3
Λ3HaL Λ3HaL
C2 C2
R R
CΛ22 CΛ22
Θ=4Π(cid:144)3 Θ=4Π(cid:144)3
(a) Contours of ψ1(ρ). The contour used in the (b) Contours of ψ2(ρ). The contour used in the
ρ>0 (ρ<0) case is the one passing through ρ>0 (ρ<0) case is the one passing through
the saddle point(s) λ2(a) (λ1(a) and λ3(a)). the saddle point(s) λ0(a) and λ2(a) (λ3(a)).
FIG. 2: Contours (blue thick lines) and saddle points (red dots) of ψ (ρ) and ψ (ρ) in the ρ > 0
1 2
and ρ < 0 cases.
part of C inside the circle C while C1,2 denotes the parts of C outside C . Note that
λk R λk λk R
f (s) when one moves away from the saddle point, and hence the corresponding
+
→ −∞
integrand in eqn. (24) vanishes at endpoints of C . Since C is a steepest descent contour,
λ2 λ2
when 1 ρ α 2, the dominant contribution to the integral over C in eqn. (24) comes
≪ ≪ − λ2
from the neighborhood of the saddle point λ (a). Thus by the method of steepest descent,
2
one has for 1 ρ α˜ 2 that
−
≪ ≪
1 3
ψ1(ρ) = |ρ|2 exp |ρ|2 f+(s) ds ∼ Iλ2(a), (40)
ZC> h i
where C = C , and I is the contribution from the saddle point λ (a). Using Watson’s
> λ2 λk(a) k
lemma, we find
3
√πexp ρ 2 f (λk(a)) 2
I | | ± , (41)
λk(a) ∼ h |ρ|14 is[s2g2(−ias)]′|s=λk(a)
where k = 0 and 2 is for +, and k = 1 and 3 for . To study the asymptotic behavior of
−
10