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Two new adiabatically equivalent quantities for the linear harmonic oscillator
13 June 1994 PHYSICS
PhysicsLettersA
ELSEVIER
LETTERS
A
189 (1994) 171-175
Two new adiabatically equivalent quantities for the linear harmonic oscillator Jishan Hu, Jian-Min Mao Departmentof Mathematics,Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 15 February 1994; accepted for publication 6 April 1994 Communicated by A.R. Bishop
Abstract Two new quantities that are adiabatically equivalent to the adiabatic invariant for the linear harmonic oscillator are given in this paper. One measures the total phase shift of wave motions. The other measures the symmetry of solutions.
1. Introduction It has been known for more than three decades that the reflection of waves is transcendentally small if the energy is above the barrier. This problem can be formulated as follows. Consider the linear Schriidinger equation
(1.1) If we assume that the function w satisfies W(X)EP( 4x)
#(A
0(x)+0* dkMX) dXk
-CO, co), =(--OO, m), >o,
+. ’
X+kCCl,
(1.2)
X+fCO,
there is a solution of Eq. ( 1.1) that behaves asymptotically like y- Texp( -&-x/e), wexp( -iw+x/e)+R
x+-CO, exp(iw+x/e),
x+co
.
(1.3)
Under conditions ( 1.2 ), the reflection coefficient R is zero to all t orders. It has also been known for years that the reflection coefficient is related closely to the adiabatic invariant. The later quantity comes from Eq. ( 1.1) with the new variable t = E- lx, 0375-9601/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIO375-9601(94)00300-E
172
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-l 75
(1.4) The real solutions of Eq. ( 1.4) form a one-parameter family, with boundary behaviours y-ATexp(
-io_x/e)
-A( l+R*)
x-+--co,
+;l*T* exp(io_x/e),
exp( -io+x/t)+A*(
x--razz)
1+R) exp(io+x/e),
(1.5)
where Ais a parameter. The adiabatic invariant of Eq. ( 1.4) is the total change of the ratio of energy to frequency,
where y is any real solution of Eq. ( 1.4). In fact, the reflection coefficient is adiabatically equivalent to the adiabatic invariant under conditions ( 1.2) (see, e.g., Ref. [ 11). More precisely, for any real solution of Eq. (1.4),wehave E(e, m)lc&co)-E(e, -co)/cJJ(-co) E(e, m)/o(m)
~2lRl’, -2lRl
COSCX,
21R12+2lRI cosff = 1+ lRl2+2lRl coso
ifcosa=O, ifcos(Y#O,
(1.6)
where (Y= arg (R*/R ). We call them adiabatically equivalent since if one is a transcendentally small quantity, so is the other. In this paper, for the linear harmonic oscillator, we give two more quantities which are adiabatically equivalent to the reflection coefficient. They have obviously physical importance. One measures the total phase shift of wave motions. The other measures the symmetry of solutions.
2. Total relative phase shift Recall the linear Schriidinger equation, c2
2
(2.1)
+02(x)y=o.
Under conditions ( 1.2), we know the classical WKB approximation can be used to represent the solutions of (2.1) to all orders of e. Since the approximation indicates that the solutions consist of two oscillatory functions, for any real solution of Eq. (2.1) we write (2.2)
y=A(x; e) cos[$(x; t)] . The functions A and @can be taken as solutions of the system 2 t25@
+~~(x)A=c~
dx2
$f
0
A,
Ati+2!%=0 da? dxdx
*
(2.3)
If we take (2.4)
lim @‘(x; 6) =0-/e, X---m then, the second equation of system (2.3) gives lim,,_,
A (x; E) = const. We normalize it by requiring
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
lim A(x;e)=l. X--+--m
173
(2.5)
The phase of the solution y is #(x; e), comparing with the leading term of the phase J”w(s) h/e, which is given by the WKB approximation. Hence, we call the quantity
(2.6) the total relativephase shi#. Under the boundary conditions (2.4) and (2.5), the second equation of (2.3) gives d@ -=_ dx
m-/t A*’
(2.7)
Since the real function o approaches nonzero constants as x-+ f co, we can apply L’Hospital’s rule and obtain the total relative phase shift in a simpler manner, 6z+Xw-r(x)A-2(X;
E) I”, .
(2.8)
The governing equation of function A can be obtained from the first equation of ( 2.3 ) ,
(2.9) To see that the total relative phase shift is adiabatically equivalent to the adiabatic invariant, we calculate the ratio of energy to the frequency, 2E( t; e)
-
=o-*[
(A cos +A-'
sin $)2+02A2
cos2@]
Met)
cos@sin@+(_&!+oA2)
=~-‘[k~cos~@-_~A-
=W-‘(k2COS~+2AA-’
cos @sin @+a
sin2#]+oA2cos2@
sin2@)+ WA’ .
(2.10)
Here the overpoint means d/dt with x=et. The solution A of Eq. (2.9) with the boundary condition (2.5) satisfies dkA/dxk-tO, with ka 1, as x+ + 0~).Hence, we have the relation 2E(t; t)/a(et)
I”,=
-o+A:A!_&@,
(2.11)
with A +=lim,,+,A(x; E). Fig. 1 gives a numerical comparison of the total relative phase shift to the adiabatic invariant for a problem with o(x) = 1+ exp ( -x2). The same problem has been solved numerically in Ref. [ 2 ] in order to compare the leading term of the reflection coefficient, obtained by asymptotics beyond all orders. It has been shown that ]R] aexp( -2.17858/e)
cos(4.7088/~).
(2.12)
We numerically solve the nonlinear equation (2.9) with the initial conditions A( - 100; e) = 1 and A’( - 100; We then plot this quantity as a function of 1/ e) =O, and evaluate the change of amplitude o(x)A2(x, C) I !!!opoO. e, after multiplying with the factor exp(2.17858/~). The continuous curve in Fig. 1 is the numerical result of the quantity E(x/e; c)/o(x) I !!opoO exp(2.17858/e). In computing the later quantity, we use the initial conditions y( - 100) = 1, y’( - 100) = 0. The adiabatic invariant depends on which solution y is calculated. To compare the total relative phase shift with the adiabatic invariant, we have to make sure that both problems have equivalent initial conditions. The first condition y( - 100) = 1 implies cos [ @(- 100; e) ] = 1. Thus, we have to take
174
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
0
1
2
3
4
5
l/f
6
7
8
9
10
Fig. 1. A comparison of the total relative phase shift to the adiabatic invariant. The plotted points are the quantity exp(2.17858/ andthecurveisexp(2.17858/~))2E(x/t;~)/w(x)~!!!~~.
t)(0(x)A~(x;t)1!!$,,,1
3. Symmetry of solutions
For an uneven function w, Eq. ( 1.1) has no symmetric (or even) solution except zero. In this section, we assume that conditions ( 1.2) hold and, in addition, that the function o is even on the real axis. We show that the quantity y’(0) is asymptotically equivalent to the reflection coefficient for any solution behaving like a cosine function near -co. Since symmetric solutions must have this type of boundary behaviour, this asymptotic equivalence implies that for generic e, Eq. ( 1.1) does not have any non-trivial symmetric solution, provided that the reflection coefficient is nonzero. In particular, in many cases with even w, there exists no nontrivial symmetric solution. For the even function w, denote Q= w_ = w+ . Denote by w the modified Jost function of Eq. ( 1. 1 ), defined
by v-exp(
x+-a.
-iaxle),
(3.1)
Any solution behaving like a cosine function near -co is proportional to y=v/+v/*.
(3.2)
In virtue of ( 1.3), at any finite point x, to the leading term of E,we have iy-CTw-l/z(x)
exp( - i /
w(s)d.s
)
+&co-l/'(x)
0
exp(+ f jw(s) ds),
t-+0+.
(3.3)
0
As E+ 0 + , the coefficients C-r and CR satisfy the relations
CR
c,
-4
C&3-‘/2-
1.
(3.4)
The conservation law of energy gives lC~l2=Q+lC~l*. Thus. we obtain
(3.5)
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
Y’(O) =
2uly(o)
(Im CT -1m C,) .
175
(3.6)
To see the quantity y’( 0) is asymptotically equivalent to the reflection coefficient, in virtue of (3.4)) we write Cr-&“2+6exp(i/I),
lRlQ ‘/‘exp(ia)
c,-
.
(3.7)
Then, (3.6) gives y’(O) u *a”2(0) e
(6sin/3-
(3.8)
lRl0sina).
The quantity 6 is adiabatically equivalent to IR 1, since ( 3.5) and ( 3.7) imply 62+26cos/3-
(3.9)
IR12.
Finally, we have Y’(O)%
20”2(0) ~
N 2o1r(o)
(-I)QIRI
sin&
(fl-Qsina!)lRl,
ifcosB#O, if cosj3=0 .
(3.10)
The adiabatic equivalence given in (3.10) indicates that a symmetric equation does not necessarily have a nontrivial symmetric solution. Recently, a non-linear example given by Byatt-Smith also confirmed this (see Ref. [ 3 ] ). All these require calculations for some quantities asymptotically beyond all orders.
Acknowledgement One of the authors (J.H. ) would like to acknowledge the financial support of the Research Grants Council of Hong Kong under grant no. DAG92/93. SC14.
References [ 1 ] P. Lochak and C. Meunier, Multiphase averaging for classical systems (Springer, Berlin, 1988 ). [2] J. Hu, Phys. Lett. A 167 (1992) 191. [ 31 J.C.B. Byatt-Smith, Stud. Appl. Math. 80 (1989) 109.
PhysicsLettersA
ELSEVIER
LETTERS
A
189 (1994) 171-175
Two new adiabatically equivalent quantities for the linear harmonic oscillator Jishan Hu, Jian-Min Mao Departmentof Mathematics,Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 15 February 1994; accepted for publication 6 April 1994 Communicated by A.R. Bishop
Abstract Two new quantities that are adiabatically equivalent to the adiabatic invariant for the linear harmonic oscillator are given in this paper. One measures the total phase shift of wave motions. The other measures the symmetry of solutions.
1. Introduction It has been known for more than three decades that the reflection of waves is transcendentally small if the energy is above the barrier. This problem can be formulated as follows. Consider the linear Schriidinger equation
(1.1) If we assume that the function w satisfies W(X)EP( 4x)
#(A
0(x)+0* dkMX) dXk
-CO, co), =(--OO, m), >o,
+. ’
X+kCCl,
(1.2)
X+fCO,
there is a solution of Eq. ( 1.1) that behaves asymptotically like y- Texp( -&-x/e), wexp( -iw+x/e)+R
x+-CO, exp(iw+x/e),
x+co
.
(1.3)
Under conditions ( 1.2 ), the reflection coefficient R is zero to all t orders. It has also been known for years that the reflection coefficient is related closely to the adiabatic invariant. The later quantity comes from Eq. ( 1.1) with the new variable t = E- lx, 0375-9601/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIO375-9601(94)00300-E
172
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-l 75
(1.4) The real solutions of Eq. ( 1.4) form a one-parameter family, with boundary behaviours y-ATexp(
-io_x/e)
-A( l+R*)
x-+--co,
+;l*T* exp(io_x/e),
exp( -io+x/t)+A*(
x--razz)
1+R) exp(io+x/e),
(1.5)
where Ais a parameter. The adiabatic invariant of Eq. ( 1.4) is the total change of the ratio of energy to frequency,
where y is any real solution of Eq. ( 1.4). In fact, the reflection coefficient is adiabatically equivalent to the adiabatic invariant under conditions ( 1.2) (see, e.g., Ref. [ 11). More precisely, for any real solution of Eq. (1.4),wehave E(e, m)lc&co)-E(e, -co)/cJJ(-co) E(e, m)/o(m)
~2lRl’, -2lRl
COSCX,
21R12+2lRI cosff = 1+ lRl2+2lRl coso
ifcosa=O, ifcos(Y#O,
(1.6)
where (Y= arg (R*/R ). We call them adiabatically equivalent since if one is a transcendentally small quantity, so is the other. In this paper, for the linear harmonic oscillator, we give two more quantities which are adiabatically equivalent to the reflection coefficient. They have obviously physical importance. One measures the total phase shift of wave motions. The other measures the symmetry of solutions.
2. Total relative phase shift Recall the linear Schriidinger equation, c2
2
(2.1)
+02(x)y=o.
Under conditions ( 1.2), we know the classical WKB approximation can be used to represent the solutions of (2.1) to all orders of e. Since the approximation indicates that the solutions consist of two oscillatory functions, for any real solution of Eq. (2.1) we write (2.2)
y=A(x; e) cos[$(x; t)] . The functions A and @can be taken as solutions of the system 2 t25@
+~~(x)A=c~
dx2
$f
0
A,
Ati+2!%=0 da? dxdx
*
(2.3)
If we take (2.4)
lim @‘(x; 6) =0-/e, X---m then, the second equation of system (2.3) gives lim,,_,
A (x; E) = const. We normalize it by requiring
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
lim A(x;e)=l. X--+--m
173
(2.5)
The phase of the solution y is #(x; e), comparing with the leading term of the phase J”w(s) h/e, which is given by the WKB approximation. Hence, we call the quantity
(2.6) the total relativephase shi#. Under the boundary conditions (2.4) and (2.5), the second equation of (2.3) gives d@ -=_ dx
m-/t A*’
(2.7)
Since the real function o approaches nonzero constants as x-+ f co, we can apply L’Hospital’s rule and obtain the total relative phase shift in a simpler manner, 6z+Xw-r(x)A-2(X;
E) I”, .
(2.8)
The governing equation of function A can be obtained from the first equation of ( 2.3 ) ,
(2.9) To see that the total relative phase shift is adiabatically equivalent to the adiabatic invariant, we calculate the ratio of energy to the frequency, 2E( t; e)
-
=o-*[
(A cos +A-'
sin $)2+02A2
cos2@]
Met)
cos@sin@+(_&!+oA2)
=~-‘[k~cos~@-_~A-
=W-‘(k2COS~+2AA-’
cos @sin @+a
sin2#]+oA2cos2@
sin2@)+ WA’ .
(2.10)
Here the overpoint means d/dt with x=et. The solution A of Eq. (2.9) with the boundary condition (2.5) satisfies dkA/dxk-tO, with ka 1, as x+ + 0~).Hence, we have the relation 2E(t; t)/a(et)
I”,=
-o+A:A!_&@,
(2.11)
with A +=lim,,+,A(x; E). Fig. 1 gives a numerical comparison of the total relative phase shift to the adiabatic invariant for a problem with o(x) = 1+ exp ( -x2). The same problem has been solved numerically in Ref. [ 2 ] in order to compare the leading term of the reflection coefficient, obtained by asymptotics beyond all orders. It has been shown that ]R] aexp( -2.17858/e)
cos(4.7088/~).
(2.12)
We numerically solve the nonlinear equation (2.9) with the initial conditions A( - 100; e) = 1 and A’( - 100; We then plot this quantity as a function of 1/ e) =O, and evaluate the change of amplitude o(x)A2(x, C) I !!!opoO. e, after multiplying with the factor exp(2.17858/~). The continuous curve in Fig. 1 is the numerical result of the quantity E(x/e; c)/o(x) I !!opoO exp(2.17858/e). In computing the later quantity, we use the initial conditions y( - 100) = 1, y’( - 100) = 0. The adiabatic invariant depends on which solution y is calculated. To compare the total relative phase shift with the adiabatic invariant, we have to make sure that both problems have equivalent initial conditions. The first condition y( - 100) = 1 implies cos [ @(- 100; e) ] = 1. Thus, we have to take
174
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
0
1
2
3
4
5
l/f
6
7
8
9
10
Fig. 1. A comparison of the total relative phase shift to the adiabatic invariant. The plotted points are the quantity exp(2.17858/ andthecurveisexp(2.17858/~))2E(x/t;~)/w(x)~!!!~~.
t)(0(x)A~(x;t)1!!$,,,1
3. Symmetry of solutions
For an uneven function w, Eq. ( 1.1) has no symmetric (or even) solution except zero. In this section, we assume that conditions ( 1.2) hold and, in addition, that the function o is even on the real axis. We show that the quantity y’(0) is asymptotically equivalent to the reflection coefficient for any solution behaving like a cosine function near -co. Since symmetric solutions must have this type of boundary behaviour, this asymptotic equivalence implies that for generic e, Eq. ( 1.1) does not have any non-trivial symmetric solution, provided that the reflection coefficient is nonzero. In particular, in many cases with even w, there exists no nontrivial symmetric solution. For the even function w, denote Q= w_ = w+ . Denote by w the modified Jost function of Eq. ( 1. 1 ), defined
by v-exp(
x+-a.
-iaxle),
(3.1)
Any solution behaving like a cosine function near -co is proportional to y=v/+v/*.
(3.2)
In virtue of ( 1.3), at any finite point x, to the leading term of E,we have iy-CTw-l/z(x)
exp( - i /
w(s)d.s
)
+&co-l/'(x)
0
exp(+ f jw(s) ds),
t-+0+.
(3.3)
0
As E+ 0 + , the coefficients C-r and CR satisfy the relations
CR
c,
-4
C&3-‘/2-
1.
(3.4)
The conservation law of energy gives lC~l2=Q+lC~l*. Thus. we obtain
(3.5)
J. Hu, J.-M. Mao /Physics Letters A 189 (1994) 171-I 75
Y’(O) =
2uly(o)
(Im CT -1m C,) .
175
(3.6)
To see the quantity y’( 0) is asymptotically equivalent to the reflection coefficient, in virtue of (3.4)) we write Cr-&“2+6exp(i/I),
lRlQ ‘/‘exp(ia)
c,-
.
(3.7)
Then, (3.6) gives y’(O) u *a”2(0) e
(6sin/3-
(3.8)
lRl0sina).
The quantity 6 is adiabatically equivalent to IR 1, since ( 3.5) and ( 3.7) imply 62+26cos/3-
(3.9)
IR12.
Finally, we have Y’(O)%
20”2(0) ~
N 2o1r(o)
(-I)QIRI
sin&
(fl-Qsina!)lRl,
ifcosB#O, if cosj3=0 .
(3.10)
The adiabatic equivalence given in (3.10) indicates that a symmetric equation does not necessarily have a nontrivial symmetric solution. Recently, a non-linear example given by Byatt-Smith also confirmed this (see Ref. [ 3 ] ). All these require calculations for some quantities asymptotically beyond all orders.
Acknowledgement One of the authors (J.H. ) would like to acknowledge the financial support of the Research Grants Council of Hong Kong under grant no. DAG92/93. SC14.
References [ 1 ] P. Lochak and C. Meunier, Multiphase averaging for classical systems (Springer, Berlin, 1988 ). [2] J. Hu, Phys. Lett. A 167 (1992) 191. [ 31 J.C.B. Byatt-Smith, Stud. Appl. Math. 80 (1989) 109.