- Email: [email protected]
Coherent states for the “isotonic oscillator”
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
COHERENT STATES FOR THE “ISOTONIC OSCILLATOR” Vincent P. GUTSCHICK Life Sciences Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, NM 87545, USA
and Michael Martin NIETO and L.M. SIMMONS Jr. Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, NM 87545, USA Received 3 December 1979
Gaussians are not, as a matter of principle, allowable wave functions for the “i.sotonic oscillator” system. Appropriate coherent states for nonharmonic potentials are minimum-uncertainty coherent states. Moreover, they provide a better approximation to the classical motion than do gaussians.
Recently Weissman and Jortner [1] discussed the classical and quantum mechanics for the one-dimensional potential ~ (l) V(x) = U0(z liz)2, z ax >0, U 2a2i2m. (2) 0=~ = ~0X(X + 1), ~ = 11 They called this system the “isotonic oscillator” *2 Among other things Weissman and Jortner considered a gaussian wave function, which they called a “coherent state” for this system, and compared the mean motion (z(t)> in this state to the trajectory of a classical partide with the same right-hand turning point. They found that (z(t)> deviates significantly from the classical trajectory near the left-hand turning point. In this note we first point out that a gaussian wavefunction is not appropriate for this system as a matter of principle. Further, with the aid of the coherent states for general potentials which we have defined [2] and
extensively studied elsewhere [3—7] we can physically understand the specific numerical results of ref. [1]. (We call our states minimum-uncertainty coherent states (MUCS). The name arises because they are a particular subset of those states that minimize the un~,
—
*1 In
the present
paper, the
quantitiesa, z, U 0, v, X, and
~t
are the same
as the quantities 1/a, t, V0, v/2, v—i, and r [1]. In ref. [4] we called this system the “harmonic oscillator with centripetal barrier” since it is related to the radial equa-
2 of ref.
~
certainty relations for certain “natural quantum operators” for the system at hand.) We observe that the gaussian wave function proposed in ref. [1] is not an acceptable wave function for this system. Even though a gaussian yields a finite (x(t)), since a gaussian does not vanish at the origin (H) is infinite because the (liz2) contribution to (V) is infinite. Moreover, even for the simple harmonic oscillator, not every gaussian is a coherent state. Only those gaussians that have the same value of (Ax) as the harmonic oscillator ground states are coherent states for a particular harmonic oscillator [2,3]. In particular, unless the gaussian has the correct value of (Ax) it will not have a constant shape. The motion of a classical particle in the eq. (1)
tion for the three-dimensional harmonic oscillator by a simple transformation.
Further numerical time evolution
results are exhibited in
a 13 minute, 16 mm, color, sound, computer-generated movie, coherent forBaker. generalIt potentials, byTime V.P. evolution Gutschick,ofM.M. Nietostates and F. is available form Cinesound Co., 915 Highland Ave., Hollywood, CA 90038. A review of this film is given by Nelson [8]. 15
Volume 76A, number 1
PHYSICS LETTERS
~
0
[-u— 0
=
(3)
0
\
2.6—
(4)
Taking the specific potential p = 1~ described*’ in ref. [1] the classical trajectory for a half period with an ,
~ There are a number of misprints in the classical motion discussion of ref. [1]. To make the equations above eq. (13) of ref. [1] consistent with eq. (13), the definition ofT should be (8V 2)’12t; in eq. (10) replace the factor 1/2 by 4; in the integral 0/ma of eq. (11) replace the factor 2’~ by 2; in eq. (12) delete the factor 2—3/2. *5 Here we use the ref. [1] phase convention for time, differing from the ref. [4] phase convention by ir/2, so that eq. (3 varies as cos ~
z
(8U~~a~/m)’I2 .
potential is [1,4]*4~5 zc2 1~l + E \ = rE I~! + E \1112 cos ~ct, —
3 March 1980
—
initial position at a right-hand turning point of z = 5 is shown by the solid curve in fig. 1. (The trajectory for ir ~ Wct~ 2ir is the mirror image of this curve about w t = iT.) For comparison with the classical motion, Weissman
and Jortner [1] took the gaussian state ~G
=
exp [— ~(z
a1/21r1/4
—
5)2]
(5)
,
/
6
(H)cs
n~2
1.8—
—
5
—
4
—
____i~L___
\\\\ I.0-
3-
\\,_
-
2 n~0
0.2— 0
0
—
I
0
I
2
3
I
I
I
I
4
5
6
7
8
Z
w~t 1
Fig. 1. For the potential labeled by v ~, starting from a righthand turning point of z = 5, the solid curve shows the classical trajectory zc(t), the short-dashed curve shows (z(t)) for the gaussian discussed in ref. [1] and given in our eq. (5), and the long-dashed curve shows (z (t)) for our minimum-uncertainty coherent state labeled by eq. (11).
16
0
Fig. 2. This figure shows the potential labeled by v = ~, the first five eigenstate energies, and the classical energy Ec for a right-hand starting point of z = 5. The dashed curve shows the eq. (5) gaussian wave packet on the energy line defined by (H)G (with the infinite energy contribution not included). On the energy line labeled by (H>cs we show both the coherent state wave packet labeled by eq. (ii) and also its form when it has evolved to the left-hand turning point.
Volume 76A, number 1
PHYSICS LETTERS
and found that the resulting (Z(t))G deviates from the classical z~(t)as the short-dashed curve in fig. 1. Our minimum-uncertainty coherent states are [4] 2
~‘cs = [2aph/2]hI2e~u/2[1x÷i/
2(u1K~I)]_~1’ 12) ,
X eY/2yh141x+l/
(6)
2((2vCv)l y
vz2
=
(7)
,
C
u + iu(X)+ i(P)/(2v),
x
z2
—
(8)
A detailed look at the potential labeled by ~ = ~ and the states we have discussed helps one to understand these results. In fIg. 2 we show the potential, the eigenenergy levels, the classical energy Ec corresponding to a right-hand turning point z = 5, the guassian wave packet at the energy ~>G (ignoring the infinite contribution to ~>G from (l/z2)G), the MUCS wave packet at the energy (H)cs, and the form of the MUCS wave packet when it has evolved to the left-hand turning point.
(1 + H/2 U 0)
(9)
—i [z(d/dz) ÷ (d/dz)z].
P
3 March 1980
(10)
First observe that the MUCS energy (H>cs is closer to the classical energy Ec for the same right-hand turning point. This, too, makes the MUCS more nearly classical than is the gaussian * Next observe that, because the parameter v was chosen to be only the potential rises very rapidly at the left. Already for V at the ground state energy, E0, the potential is rising almost vertically. Our studies of the time evolution of coherent states [7] *3 have shown that in such a region, where the classical velocity is nearly discontinuous, it is very difficult for any wave packet to follow the classical motion. The problem becomes one of fmding a packet that can do as well as possible. The MUCS clearly follows zc(t) better than the “guessed” gaussian ~.
For the potential v of (Z(0))cs = S is
=
1/2, a right-hand starting point
12.4744. (11) With this C, (z(t))cs is given by the long-dashed curve in fig. 1, which follows the classical motion better than the short-dashed curve * 6 C
=
*6
We remark that, because the energy levels for this potential are equally spaced, the frequencies of the time-dependent eigenfunctions are commensurate (up to a common ground state contribution). Therefore [3,4], any wave packet will be restored to its original shape after one oscillation. Both of the wave packets considered here are constralned to have (2> = Zc at the right-hand turning point (of every oscillation), Therefore, the large z agreement of the curves in fig. 1 is guaranteed and is not a good measure of the “classicality” portant. In potentials with unequally spaced levels, the imof the states. It is the left-hand behavior which is most commensurability property is absent and all wave packets will eventually disperse. However, we have found none with better classical properties or longer coherence times than our MUCS wave packets [7].
~,
of ref. [1]. The fact that it is the low value of p which causes the quantum motion to deviate on the left from the classical motion can be seen by allowing v to rise. First it is useful to define an effective eigenstate label, neff’ 1-~>cs = E(fleff), where the eigenenergies are given by < *7
The narrower gaussian in table 1 of ref. [1] has an even higher (H>.
Table 1 For increasing values of v, C is chosen to give an neff = 2.53 and a starting position (z(0)> at the right-hand turning point. The values of (H>cs and the quantum left-hand turning points (z(t)>L are given. For comparison of these MUCS with the classical trajectories we list the classical energies Ec of particles starting at right-hand turning points zc = (z(0)> and their corresponding classical lefthand turning points zcL = 1/z~.Finally, E is the ratio of the size of the secondary maximum of the wave packet to the primary maximum of the wave packet at the left-hand turning point. v 0.5
1.5
3 10 30
C
(z(0)>
(H>cs
E~
(Z>L
ZcL
E
12.4844 4.6348 2.6607 1.1641 0.6134
5.00274
6.27 18.42 36.61 121.4 363.8
5.77 16.89 33.44 109.5 321.9
0.88 0.62 0.59 0.66 0.76
0.20 0.33 0.43 0.61 0.74
0.19 0.11 0.05 0.002 0.0000
3.06625
2.35266 1.65167 1.34275
17
Volume 76A, number 1
E(n) = c~ov [4n
+ 2X +
PHYSICS LETTERS
3
—
2v].
(12)
partment of Energy. L.M.S. is grateful for the hospitality of the Aspen Center for Physics, where part of this work
In the p = case studied here, C = 12.4744 gives a value Of(H)cs corresponding to an fl6ff of 2.527 and a right.
was done.
hand starting point. In table 1 we show the results of numerical studies of the time evolution of MUCS for increasing values of v, with the MUCS parameter C chosen so that in each case neff = 2.53 and the state starts at the right-hand turning point. For comparison with ~>cs in each case we give the Ec defmed by a classical right-hand turning point zc = (z (0)) and compare the quantum left-hand turning points 2cL (z)L l/2c.with One the classical points left-hand turning sees that as pleft-hand increasesturning the quantum points (z(t)>L become closer to the classical left-hand turning points zcL = lIzc. Also, at the left-hand turning point the relative size of the secondary maximum of the packet to the primary maximum becomes smaller as v increases, Further results on the coherent states for this and other potentials are given in refs. [2—6].Detailed studies of the time evolution of minimum-uncertainty coherent states are in ref. [7]*3.
References
This work was supported by the United States De-
18
3 March 1980
[1] Y. Weissman and J. Jortner, Phys. Lett. 70A (1979) 177. [2] M.M. (1978)Nieto 207. and L.M. Simmons Jr., Phys. Rev. Lett. 41 [3] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D20 (1979) 1321; this paper contains a discussion ofthe general formalism of the MUCS. [4] M.M. and L.M. Simmons Jr., Phys.toRev. D20 (1979) 1332; Nieto this paper applies the formalism confming onedimensional potentials. Included is an extensive discussion of the “isotonic oscillator”.
[5] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D20 (1979) 1342; this paper discusses nonconfining one-dimensional potentials. [6] M.M. Nieto, Los Alamos preprint LA-UR-79-2 101; this paper discusses minimum-uncertainty coherent states for multidimensional systems. [7] V.P. Gutschick and M.M. Nieto, Los Alamos preprint LA-UR-79-2925; this paper contains the results of exten-
sive time evolution studies. [8] C.A. Nelson, Am. J. Phys. 47 (1979) 755.
PHYSICS LETTERS
3 March 1980
COHERENT STATES FOR THE “ISOTONIC OSCILLATOR” Vincent P. GUTSCHICK Life Sciences Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, NM 87545, USA
and Michael Martin NIETO and L.M. SIMMONS Jr. Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, NM 87545, USA Received 3 December 1979
Gaussians are not, as a matter of principle, allowable wave functions for the “i.sotonic oscillator” system. Appropriate coherent states for nonharmonic potentials are minimum-uncertainty coherent states. Moreover, they provide a better approximation to the classical motion than do gaussians.
Recently Weissman and Jortner [1] discussed the classical and quantum mechanics for the one-dimensional potential ~ (l) V(x) = U0(z liz)2, z ax >0, U 2a2i2m. (2) 0=~ = ~0X(X + 1), ~ = 11 They called this system the “isotonic oscillator” *2 Among other things Weissman and Jortner considered a gaussian wave function, which they called a “coherent state” for this system, and compared the mean motion (z(t)> in this state to the trajectory of a classical partide with the same right-hand turning point. They found that (z(t)> deviates significantly from the classical trajectory near the left-hand turning point. In this note we first point out that a gaussian wavefunction is not appropriate for this system as a matter of principle. Further, with the aid of the coherent states for general potentials which we have defined [2] and
extensively studied elsewhere [3—7] we can physically understand the specific numerical results of ref. [1]. (We call our states minimum-uncertainty coherent states (MUCS). The name arises because they are a particular subset of those states that minimize the un~,
—
*1 In
the present
paper, the
quantitiesa, z, U 0, v, X, and
~t
are the same
as the quantities 1/a, t, V0, v/2, v—i, and r [1]. In ref. [4] we called this system the “harmonic oscillator with centripetal barrier” since it is related to the radial equa-
2 of ref.
~
certainty relations for certain “natural quantum operators” for the system at hand.) We observe that the gaussian wave function proposed in ref. [1] is not an acceptable wave function for this system. Even though a gaussian yields a finite (x(t)), since a gaussian does not vanish at the origin (H) is infinite because the (liz2) contribution to (V) is infinite. Moreover, even for the simple harmonic oscillator, not every gaussian is a coherent state. Only those gaussians that have the same value of (Ax) as the harmonic oscillator ground states are coherent states for a particular harmonic oscillator [2,3]. In particular, unless the gaussian has the correct value of (Ax) it will not have a constant shape. The motion of a classical particle in the eq. (1)
tion for the three-dimensional harmonic oscillator by a simple transformation.
Further numerical time evolution
results are exhibited in
a 13 minute, 16 mm, color, sound, computer-generated movie, coherent forBaker. generalIt potentials, byTime V.P. evolution Gutschick,ofM.M. Nietostates and F. is available form Cinesound Co., 915 Highland Ave., Hollywood, CA 90038. A review of this film is given by Nelson [8]. 15
Volume 76A, number 1
PHYSICS LETTERS
~
0
[-u— 0
=
(3)
0
\
2.6—
(4)
Taking the specific potential p = 1~ described*’ in ref. [1] the classical trajectory for a half period with an ,
~ There are a number of misprints in the classical motion discussion of ref. [1]. To make the equations above eq. (13) of ref. [1] consistent with eq. (13), the definition ofT should be (8V 2)’12t; in eq. (10) replace the factor 1/2 by 4; in the integral 0/ma of eq. (11) replace the factor 2’~ by 2; in eq. (12) delete the factor 2—3/2. *5 Here we use the ref. [1] phase convention for time, differing from the ref. [4] phase convention by ir/2, so that eq. (3 varies as cos ~
z
(8U~~a~/m)’I2 .
potential is [1,4]*4~5 zc2 1~l + E \ = rE I~! + E \1112 cos ~ct, —
3 March 1980
—
initial position at a right-hand turning point of z = 5 is shown by the solid curve in fig. 1. (The trajectory for ir ~ Wct~ 2ir is the mirror image of this curve about w t = iT.) For comparison with the classical motion, Weissman
and Jortner [1] took the gaussian state ~G
=
exp [— ~(z
a1/21r1/4
—
5)2]
(5)
,
/
6
(H)cs
n~2
1.8—
—
5
—
4
—
____i~L___
\\\\ I.0-
3-
\\,_
-
2 n~0
0.2— 0
0
—
I
0
I
2
3
I
I
I
I
4
5
6
7
8
Z
w~t 1
Fig. 1. For the potential labeled by v ~, starting from a righthand turning point of z = 5, the solid curve shows the classical trajectory zc(t), the short-dashed curve shows (z(t)) for the gaussian discussed in ref. [1] and given in our eq. (5), and the long-dashed curve shows (z (t)) for our minimum-uncertainty coherent state labeled by eq. (11).
16
0
Fig. 2. This figure shows the potential labeled by v = ~, the first five eigenstate energies, and the classical energy Ec for a right-hand starting point of z = 5. The dashed curve shows the eq. (5) gaussian wave packet on the energy line defined by (H)G (with the infinite energy contribution not included). On the energy line labeled by (H>cs we show both the coherent state wave packet labeled by eq. (ii) and also its form when it has evolved to the left-hand turning point.
Volume 76A, number 1
PHYSICS LETTERS
and found that the resulting (Z(t))G deviates from the classical z~(t)as the short-dashed curve in fig. 1. Our minimum-uncertainty coherent states are [4] 2
~‘cs = [2aph/2]hI2e~u/2[1x÷i/
2(u1K~I)]_~1’ 12) ,
X eY/2yh141x+l/
(6)
2((2vCv)l y
vz2
=
(7)
,
C
u + iu(X)+ i(P)/(2v),
x
z2
—
(8)
A detailed look at the potential labeled by ~ = ~ and the states we have discussed helps one to understand these results. In fIg. 2 we show the potential, the eigenenergy levels, the classical energy Ec corresponding to a right-hand turning point z = 5, the guassian wave packet at the energy ~>G (ignoring the infinite contribution to ~>G from (l/z2)G), the MUCS wave packet at the energy (H)cs, and the form of the MUCS wave packet when it has evolved to the left-hand turning point.
(1 + H/2 U 0)
(9)
—i [z(d/dz) ÷ (d/dz)z].
P
3 March 1980
(10)
First observe that the MUCS energy (H>cs is closer to the classical energy Ec for the same right-hand turning point. This, too, makes the MUCS more nearly classical than is the gaussian * Next observe that, because the parameter v was chosen to be only the potential rises very rapidly at the left. Already for V at the ground state energy, E0, the potential is rising almost vertically. Our studies of the time evolution of coherent states [7] *3 have shown that in such a region, where the classical velocity is nearly discontinuous, it is very difficult for any wave packet to follow the classical motion. The problem becomes one of fmding a packet that can do as well as possible. The MUCS clearly follows zc(t) better than the “guessed” gaussian ~.
For the potential v of (Z(0))cs = S is
=
1/2, a right-hand starting point
12.4744. (11) With this C, (z(t))cs is given by the long-dashed curve in fig. 1, which follows the classical motion better than the short-dashed curve * 6 C
=
*6
We remark that, because the energy levels for this potential are equally spaced, the frequencies of the time-dependent eigenfunctions are commensurate (up to a common ground state contribution). Therefore [3,4], any wave packet will be restored to its original shape after one oscillation. Both of the wave packets considered here are constralned to have (2> = Zc at the right-hand turning point (of every oscillation), Therefore, the large z agreement of the curves in fig. 1 is guaranteed and is not a good measure of the “classicality” portant. In potentials with unequally spaced levels, the imof the states. It is the left-hand behavior which is most commensurability property is absent and all wave packets will eventually disperse. However, we have found none with better classical properties or longer coherence times than our MUCS wave packets [7].
~,
of ref. [1]. The fact that it is the low value of p which causes the quantum motion to deviate on the left from the classical motion can be seen by allowing v to rise. First it is useful to define an effective eigenstate label, neff’ 1-~>cs = E(fleff), where the eigenenergies are given by < *7
The narrower gaussian in table 1 of ref. [1] has an even higher (H>.
Table 1 For increasing values of v, C is chosen to give an neff = 2.53 and a starting position (z(0)> at the right-hand turning point. The values of (H>cs and the quantum left-hand turning points (z(t)>L are given. For comparison of these MUCS with the classical trajectories we list the classical energies Ec of particles starting at right-hand turning points zc = (z(0)> and their corresponding classical lefthand turning points zcL = 1/z~.Finally, E is the ratio of the size of the secondary maximum of the wave packet to the primary maximum of the wave packet at the left-hand turning point. v 0.5
1.5
3 10 30
C
(z(0)>
(H>cs
E~
(Z>L
ZcL
E
12.4844 4.6348 2.6607 1.1641 0.6134
5.00274
6.27 18.42 36.61 121.4 363.8
5.77 16.89 33.44 109.5 321.9
0.88 0.62 0.59 0.66 0.76
0.20 0.33 0.43 0.61 0.74
0.19 0.11 0.05 0.002 0.0000
3.06625
2.35266 1.65167 1.34275
17
Volume 76A, number 1
E(n) = c~ov [4n
+ 2X +
PHYSICS LETTERS
3
—
2v].
(12)
partment of Energy. L.M.S. is grateful for the hospitality of the Aspen Center for Physics, where part of this work
In the p = case studied here, C = 12.4744 gives a value Of(H)cs corresponding to an fl6ff of 2.527 and a right.
was done.
hand starting point. In table 1 we show the results of numerical studies of the time evolution of MUCS for increasing values of v, with the MUCS parameter C chosen so that in each case neff = 2.53 and the state starts at the right-hand turning point. For comparison with ~>cs in each case we give the Ec defmed by a classical right-hand turning point zc = (z (0)) and compare the quantum left-hand turning points 2cL (z)L l/2c.with One the classical points left-hand turning sees that as pleft-hand increasesturning the quantum points (z(t)>L become closer to the classical left-hand turning points zcL = lIzc. Also, at the left-hand turning point the relative size of the secondary maximum of the packet to the primary maximum becomes smaller as v increases, Further results on the coherent states for this and other potentials are given in refs. [2—6].Detailed studies of the time evolution of minimum-uncertainty coherent states are in ref. [7]*3.
References
This work was supported by the United States De-
18
3 March 1980
[1] Y. Weissman and J. Jortner, Phys. Lett. 70A (1979) 177. [2] M.M. (1978)Nieto 207. and L.M. Simmons Jr., Phys. Rev. Lett. 41 [3] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D20 (1979) 1321; this paper contains a discussion ofthe general formalism of the MUCS. [4] M.M. and L.M. Simmons Jr., Phys.toRev. D20 (1979) 1332; Nieto this paper applies the formalism confming onedimensional potentials. Included is an extensive discussion of the “isotonic oscillator”.
[5] M.M. Nieto and L.M. Simmons Jr., Phys. Rev. D20 (1979) 1342; this paper discusses nonconfining one-dimensional potentials. [6] M.M. Nieto, Los Alamos preprint LA-UR-79-2 101; this paper discusses minimum-uncertainty coherent states for multidimensional systems. [7] V.P. Gutschick and M.M. Nieto, Los Alamos preprint LA-UR-79-2925; this paper contains the results of exten-
sive time evolution studies. [8] C.A. Nelson, Am. J. Phys. 47 (1979) 755.