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Semiclassical quantization conditions for the Coulomb and radial-harmonic-oscillator potentials
17 May 1982
PHYSICS LETTERS
Volume 89A, number 5
SEMICLASSICAL QUANTIZATION CONDITIONS FOR THE COULOMB AND RADIALHARMONICUSCILLATOR
POTENTIALS
Pablo A. VICHARELLI and C.B. COLLINS Center for Quantum Electronics, Universityof Texas at ?allas, Box 688, Richardson, TX 75080, USA Received 5 November 1981 Revised manuscript received 17 March 1982
It is shown that the Lange1 transformation is unique for the Coulomb and radial-harmonic+sciator bitrary values of angular momentum.
One of the most useful methods for determining approximate energy eigenvalues of the one-dimensional Schriidinger equation is the semiclassical JWKB approximation, which has received an increasing amount of attention in recent years [l-5]. The method is based on the Bohr-Sommerfeld quantization rules of the old quantum theory and is applicable to potentials defined on the entire real axis, -4 x
-42 [ .2m
2 +V(x)6x2
E’$(x)=O. 1
In the JWKB method, the Bohr-Sommerfeld zation condition then gives,
(1)
potentials for ar-
VW
xp
XI
I.
Fig. 1. A typical potential function for which bound-state eigenvalues exist.
j2
(n + ;)7r=
{2m[E
- V(X)]/A~}~/~
dx ,
(2)
Xl
if the potential is defined on the whole line (--I 00). However, for finite or semi-infinite intervals, it is necessary to incorporate a correction AP(x) into the potential. The quantization condition then becomes
quanti(n + :)n =
j2
Xl
(2m[E
-
V(x) - AP(x)]/A~)~/~ d.x .
(3) 215
Volume
89A, number
5
PHYSICS
A problem associated with this modified condition is the nonuniqueness of AV(x). The correction AV(x) depends on the transformation that maps the (a, b) or (a, -) interval into (--, -), and there are many such transformations [6]. A possible criterion for selecting the appropriate AV(x) was suggested by Adams and Miller [S] : One chooses the AV(x) that gives the correct quantum mechanical results when the potential V(x) is set to zero (i.e., particle in a box). According to this, the appropriate correction is AV(x) = A2/8mx2 ,
(4)
for (0, -), which is the usual Langer [7] correction, AV(x) =?! [a/@ - ,)12sec2 [&(x 8m
- ‘-:“)1
or (5)
for finite (a, b) intervals [5]. This is an intersting result because it eliminates the need for the integer Bohr-Sommerfeld quantization condition for the particle in a box
(n + 1)n = j2
(2m [E - V(x)] /G}1’2
dx
)
17 May 1982
LETTERS
same well-known, exactly solvable potentials of previous treatments. Our aim is to show that it is possible to derive exactly the modified Bohr-Sommerfeld quantization conditions in some cases. It is important to recognize that the RKR inversion as implemented by Adams and Miller was inteded to be complete only for symmetric potentials [S] . There are actually two RKR integrals which allow the calculation of the two turning points. Adams and Miller [5] evaluated the difference x2 - x1 from the first integral and used symmetry arguments to obtain the turning points. Since this approach is not applicable to the asymmetric potentials considered in this work, we now consider the full RKR inversion. The RKR formulas [ 121 for the turning points, with AV(x) built in, are
and
(6)
Xl
used by Froman and Froman [8]. Rosenzweig and Krieger [9] have employed a formalism proposed by Froman and Froman [lo] to demonstrate the exactness of certain modified quantization conditions. The method has the advantage that a particular AI’(x) can be proven to be exact without having to compare the results with analytic solutions of the Schrodinger equation. However, the method has the disadvantage that “it does not provide a means to determine whether a proposed quantization rule is not exact, i.e., either one can prove exactness by some clever choice of contour, or else, no definite conclusion can be drawn about the correctness of the proposed quantization condition” [ 111. It appears, then, that a better and much simpler approach is the straightforward application of the RKR [ 121 inversion technique, as proposed by Adams and Miller [5]. This method offers the advantage that the appropriate correction AV(x) is derived without any a priori assumptions about its functional form. It suffers, however, from the disadvantage that exact analytic expressions for the eigenvalues must be known in order to perform the inversion. For our purposes, this represents no problem since we are dealing with the 216
x
[(J(x) -
qn, z)]-“2 dn
(8)
)
where 4(x) = V(x) + AVx)
,
(9)
V(x) = V*(x) + Xh2/2mX2 )
(10)
h = Z(1t l), x1 and x2 are the classical turning points, n and I are the radial and angular momentum quantum numbers (no longer restricted to integral values), respectively, and n”is the quantum number which makes the integrand vanish. After integration, eq. (7) yields the function $(x) in terms of the turning points, one of which is subsequently eliminated by use of eq. (8). Here we must emphasize that the use of the full RKR method allows the derivation of quantization conditions for arbitrary values of the angular momentum quantum number 1. We now proceed to apply this method to two wellknown problems. We start with the Coulomb-like potential V0(x) = -A”2(2A2/m)“2/~, which gives rise to eigenvalues
O
(11)
PHYSICS LETTERS
Volume 89A, number 5
E(n,Z)=-A/(n+Z+l)$
n,z=0,1,2
)...)
(12)
XIX2 = A(21+ 1)/2mw (13)
it immediately follows that the integral of eq. (7) becomes x2 - x1 = -(2fi+z)1’2[@(x) x (Z+ 1/2)/W)
t A/(Z + 1/2)2]“2
,
(14)
while the one of eq. (8) becomes x;’ - x2-’ = (8m/$)l/2 x [G(x) + A/(1 + 1/2)2 J 1’2/(z t l/2) .
x1x2
-(Fz2/2m)(Z + 1/2)%(x).
up>
Solving for x1 in terms of x2 E x it can easily be shown that AI’(x) = Zi2/8mx2, which is the wellknown Langer correction, derived here without recourse to any particular type of coordinate transformation or any assumptions other than those associated with the semiclassical JWKB method. Next, we consider the radial isotropic harmonic oscillator potential, given by V,(x) = (1/2)nrU%2 )
o
(17)
which has eigenvalues E=Ao(2ntZt3/2),
n,Z=0,1,2
,....
(18)
In this case aE(n, Z)/XI = Aw/(2Z + 1) ,
(20)
is obtained. Again, solving for x1 in terms of x2 E x after integration, we arrive at AV(x) = ti2/8m.x2. In summary, we have shown in a simple and straightforward manner, that Langer’s correction [7] is indeed unique for the two potentials considered here when arbitrary values of 1 are involved. In order to do so, the Adams-Miller [5] treatment for semiclassical eigenvalues corresponding to potentials defined on semi-infinite intervals has been extended to include nonzero angular momentum.
(19
Combining these last two expressions we arrive at the relationship =
in eq. (7), it can be verified by inspection that the re-
lationship
where A is a constant. Since in this case &F(n, z)/ax = A/(Z f l/2)@ + 1 t 1)3 )
17 May 1982
(19)
and since this term does not depend on the quantum number n, it can be taken outside the integral of eq. (8). Since the resulting integral is identical to the one
References
111C.C. Gerry and A. Inomata, Phys. Lett. 84A (1981) 172. 121MM. Nieto, L.M. Simmons Jr. and V.P. Gutschick, Phys. Rev. D23 (1981) 927. 131 C.K. Chan and P. Lu, Phys. Rev. A22 (1980) 1869. 141 M.W. Cole and R.H. Good Jr., Phys. Rev. A18 (1978) 1085. 151 J.E. Adams and WH. Miller, J. Chem. Phys. 67 (1977) 5775. 161 R. Engelke and C.L. Beckel, J. Math. Phys. 6 (1970) 1991. 171 R.E. Langer, Phys. Rev. 51 (1937) 669. [81 N. Framan and P.O. Fr8man, J. Math. Phys. 19 (1978) 1823. [91 C. Rosenzweig and J.B. Krieger, J. Math. Phys. 9 (1968) 849. [lOI N. Frbman and P.O. FrBman, JWKB approximation, contrbutions to the theory (North-Holland, Amsterdam, 1965). [Ill J.B. Krieger, J. Math. Phys. 10 (1969) 1455. [I21 See, for example, E.A. Mason and L. Monchick, Adv. Chem. Phys. 12 (1967) 351; for a more recent review of the RKR method see R.J. LeRoy, in: Semiclassical methods in molecular scattering and spectroscopy, ed. M.S. Child (Reidel, Dordrecht, 1980) pp. 109-126.
217
PHYSICS LETTERS
Volume 89A, number 5
SEMICLASSICAL QUANTIZATION CONDITIONS FOR THE COULOMB AND RADIALHARMONICUSCILLATOR
POTENTIALS
Pablo A. VICHARELLI and C.B. COLLINS Center for Quantum Electronics, Universityof Texas at ?allas, Box 688, Richardson, TX 75080, USA Received 5 November 1981 Revised manuscript received 17 March 1982
It is shown that the Lange1 transformation is unique for the Coulomb and radial-harmonic+sciator bitrary values of angular momentum.
One of the most useful methods for determining approximate energy eigenvalues of the one-dimensional Schriidinger equation is the semiclassical JWKB approximation, which has received an increasing amount of attention in recent years [l-5]. The method is based on the Bohr-Sommerfeld quantization rules of the old quantum theory and is applicable to potentials defined on the entire real axis, -4 x
-42 [ .2m
2 +V(x)6x2
E’$(x)=O. 1
In the JWKB method, the Bohr-Sommerfeld zation condition then gives,
(1)
potentials for ar-
VW
xp
XI
I.
Fig. 1. A typical potential function for which bound-state eigenvalues exist.
j2
(n + ;)7r=
{2m[E
- V(X)]/A~}~/~
dx ,
(2)
Xl
if the potential is defined on the whole line (--I 00). However, for finite or semi-infinite intervals, it is necessary to incorporate a correction AP(x) into the potential. The quantization condition then becomes
quanti(n + :)n =
j2
Xl
(2m[E
-
V(x) - AP(x)]/A~)~/~ d.x .
(3) 215
Volume
89A, number
5
PHYSICS
A problem associated with this modified condition is the nonuniqueness of AV(x). The correction AV(x) depends on the transformation that maps the (a, b) or (a, -) interval into (--, -), and there are many such transformations [6]. A possible criterion for selecting the appropriate AV(x) was suggested by Adams and Miller [S] : One chooses the AV(x) that gives the correct quantum mechanical results when the potential V(x) is set to zero (i.e., particle in a box). According to this, the appropriate correction is AV(x) = A2/8mx2 ,
(4)
for (0, -), which is the usual Langer [7] correction, AV(x) =?! [a/@ - ,)12sec2 [&(x 8m
- ‘-:“)1
or (5)
for finite (a, b) intervals [5]. This is an intersting result because it eliminates the need for the integer Bohr-Sommerfeld quantization condition for the particle in a box
(n + 1)n = j2
(2m [E - V(x)] /G}1’2
dx
)
17 May 1982
LETTERS
same well-known, exactly solvable potentials of previous treatments. Our aim is to show that it is possible to derive exactly the modified Bohr-Sommerfeld quantization conditions in some cases. It is important to recognize that the RKR inversion as implemented by Adams and Miller was inteded to be complete only for symmetric potentials [S] . There are actually two RKR integrals which allow the calculation of the two turning points. Adams and Miller [5] evaluated the difference x2 - x1 from the first integral and used symmetry arguments to obtain the turning points. Since this approach is not applicable to the asymmetric potentials considered in this work, we now consider the full RKR inversion. The RKR formulas [ 121 for the turning points, with AV(x) built in, are
and
(6)
Xl
used by Froman and Froman [8]. Rosenzweig and Krieger [9] have employed a formalism proposed by Froman and Froman [lo] to demonstrate the exactness of certain modified quantization conditions. The method has the advantage that a particular AI’(x) can be proven to be exact without having to compare the results with analytic solutions of the Schrodinger equation. However, the method has the disadvantage that “it does not provide a means to determine whether a proposed quantization rule is not exact, i.e., either one can prove exactness by some clever choice of contour, or else, no definite conclusion can be drawn about the correctness of the proposed quantization condition” [ 111. It appears, then, that a better and much simpler approach is the straightforward application of the RKR [ 121 inversion technique, as proposed by Adams and Miller [5]. This method offers the advantage that the appropriate correction AV(x) is derived without any a priori assumptions about its functional form. It suffers, however, from the disadvantage that exact analytic expressions for the eigenvalues must be known in order to perform the inversion. For our purposes, this represents no problem since we are dealing with the 216
x
[(J(x) -
qn, z)]-“2 dn
(8)
)
where 4(x) = V(x) + AVx)
,
(9)
V(x) = V*(x) + Xh2/2mX2 )
(10)
h = Z(1t l), x1 and x2 are the classical turning points, n and I are the radial and angular momentum quantum numbers (no longer restricted to integral values), respectively, and n”is the quantum number which makes the integrand vanish. After integration, eq. (7) yields the function $(x) in terms of the turning points, one of which is subsequently eliminated by use of eq. (8). Here we must emphasize that the use of the full RKR method allows the derivation of quantization conditions for arbitrary values of the angular momentum quantum number 1. We now proceed to apply this method to two wellknown problems. We start with the Coulomb-like potential V0(x) = -A”2(2A2/m)“2/~, which gives rise to eigenvalues
O
(11)
PHYSICS LETTERS
Volume 89A, number 5
E(n,Z)=-A/(n+Z+l)$
n,z=0,1,2
)...)
(12)
XIX2 = A(21+ 1)/2mw (13)
it immediately follows that the integral of eq. (7) becomes x2 - x1 = -(2fi+z)1’2[@(x) x (Z+ 1/2)/W)
t A/(Z + 1/2)2]“2
,
(14)
while the one of eq. (8) becomes x;’ - x2-’ = (8m/$)l/2 x [G(x) + A/(1 + 1/2)2 J 1’2/(z t l/2) .
x1x2
-(Fz2/2m)(Z + 1/2)%(x).
up>
Solving for x1 in terms of x2 E x it can easily be shown that AI’(x) = Zi2/8mx2, which is the wellknown Langer correction, derived here without recourse to any particular type of coordinate transformation or any assumptions other than those associated with the semiclassical JWKB method. Next, we consider the radial isotropic harmonic oscillator potential, given by V,(x) = (1/2)nrU%2 )
o
(17)
which has eigenvalues E=Ao(2ntZt3/2),
n,Z=0,1,2
,....
(18)
In this case aE(n, Z)/XI = Aw/(2Z + 1) ,
(20)
is obtained. Again, solving for x1 in terms of x2 E x after integration, we arrive at AV(x) = ti2/8m.x2. In summary, we have shown in a simple and straightforward manner, that Langer’s correction [7] is indeed unique for the two potentials considered here when arbitrary values of 1 are involved. In order to do so, the Adams-Miller [5] treatment for semiclassical eigenvalues corresponding to potentials defined on semi-infinite intervals has been extended to include nonzero angular momentum.
(19
Combining these last two expressions we arrive at the relationship =
in eq. (7), it can be verified by inspection that the re-
lationship
where A is a constant. Since in this case &F(n, z)/ax = A/(Z f l/2)@ + 1 t 1)3 )
17 May 1982
(19)
and since this term does not depend on the quantum number n, it can be taken outside the integral of eq. (8). Since the resulting integral is identical to the one
References
111C.C. Gerry and A. Inomata, Phys. Lett. 84A (1981) 172. 121MM. Nieto, L.M. Simmons Jr. and V.P. Gutschick, Phys. Rev. D23 (1981) 927. 131 C.K. Chan and P. Lu, Phys. Rev. A22 (1980) 1869. 141 M.W. Cole and R.H. Good Jr., Phys. Rev. A18 (1978) 1085. 151 J.E. Adams and WH. Miller, J. Chem. Phys. 67 (1977) 5775. 161 R. Engelke and C.L. Beckel, J. Math. Phys. 6 (1970) 1991. 171 R.E. Langer, Phys. Rev. 51 (1937) 669. [81 N. Framan and P.O. Fr8man, J. Math. Phys. 19 (1978) 1823. [91 C. Rosenzweig and J.B. Krieger, J. Math. Phys. 9 (1968) 849. [lOI N. Frbman and P.O. FrBman, JWKB approximation, contrbutions to the theory (North-Holland, Amsterdam, 1965). [Ill J.B. Krieger, J. Math. Phys. 10 (1969) 1455. [I21 See, for example, E.A. Mason and L. Monchick, Adv. Chem. Phys. 12 (1967) 351; for a more recent review of the RKR method see R.J. LeRoy, in: Semiclassical methods in molecular scattering and spectroscopy, ed. M.S. Child (Reidel, Dordrecht, 1980) pp. 109-126.
217