Exact motion in noncentral electric fields

Exact motion in noncentral electric fields André Hautot

Citation: Journal of Mathematical Physics 14, 1320 (1973); doi: 10.1063/1.1666184 View online: http://dx.doi.org/10.1063/1.1666184 View Table of Contents: http://aip.scitation.org/toc/jmp/14/10 Published by the American Institute of Physics

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Exact motion in noncentral electric fields Andre Hautot* University of Liege, Institute Physics, Sort Tilman par 4000 Li~ge /, Belgium We study the problem of the motion of a charged particle in noncentral potentials of the type f«(J)/r2 + V(r). Newton's and Schrodinger's mechanics are considered. Exact solutions exist if V(r) = - H/r or Kr2 (i.e., Coulomb or harmonic oscillator potentials) whilef«(J) may have at least three different expressions as a function of (J if the problem is three-dimensional and seven expressions if it is two-dimensional. The classical trajectories are computed and the energy levels in the corresponding quantum problem are given. Analogies between the two treatments are discussed.

For a presumed complete bibliography about the problem of finding exact solutions to the equations of motion in the presence of unusual types of potentials see Refs. 1-10. Central magnetic fields were treated in a previous paper. 10 We now turn to noncentral electric potentials of the type J::::: (J.I/e) [j(6)/r 2

+

(1)

V(r)],

where J.I is the mass and e the charge of the particle, (r, 6, cp) its spherical coordinates. V represents its velocity,y its acceleration. i equals -J- 1. We shall perform the calculations both for Newton's and for Schrodinger's mechanics.

Let us define P written as

p2

V(r)]

+ 2/(6) r/r 4 - (l/r 4)/'(6) + y2tl/2,yz(x2 + y2t1/2,_(X2 + y2)1/2].

dP/dt

The conservation of energy implies v 2 + 2/(6)/r 2 + 2 V(r) = a

=-

Pz

(= const).

(3)

By scalar multiplication of (2) by r we obtain: rV'(r)

+

v2

Remembering that 2r o v cal integration rov

= (ar 2 -

=a -

= r2

J

rV'(r) - 2v(r).

= dr 2 /dt, we have after a

classi(4)

2r2 V - btl/2 dr

= F(r).

in the two classical cases:

1320

= Kr2

(7)

sine de

r

(5)

It is very remarkable that the radial motion is independent of the non central term in the potential (1). Equation (5) is exactly integrable by means of circular functions

V= V2

=d

= 0, and after integration (d = const).

: : : J.!(ar 2 -2r 2 V-bt 1 / 2 dr,

Finally,

V= V1 =-H/r

sin 2 ecp

+ y2)-1/2,x(x2 + y2)-1/2, 0].

-J[b - 2/ (e)] sin 2 e - d2

2r2V(r) _ b)1/2.

Jdt = Jr(ar2 -

(6)

2/(e).

Equations (6) and (7) allow us to find the two last integrations needed for the complete solution of this problem: 1=

+ 2/(e)/r 2 .

= roy +

=b-

[f'(e)/r 2)[- y (x2

From this equation we deduce d(rov)/dt

sin 2e 0 and r 3 V' - 2/ > O. The first equation determines the angular velocity while the second connects r and e, i.e., it gives the distance d = r cose between the plane of the trajectory and the plane of symmetry. As an example, let us investigate the case VCr) = Arn - 2 • Simple algebraic calculations show that d = ([2f(e) w= ±

+ f'(e)

U' (e)/sine

tane]/A(n - 2)}1/n cose,

where the parameter 5 may only take special values to be determined from (30). Case 2: V= V2 =Kr2. The radial equation (29) is the same as in the theory of the three-dimensional harmonic oscillator; the energy levels are given by

cose]1/2 {A(n - 2)/[2f(e) +f' tane]} 2/n.

e

(b) Another special motion is deduced from: = 0, cP = O. The trajectory is located on a straight line passing through the origin. Of course such a motion also requires special initial conditions. Furthermore, if we restrict ourselves to potentials of the type (1), we must ensure that r /\ y = 0 which leads to (- y ,x, O)f'= O. If f' = 0, f = const is fulfilled the problem may be solved exactly like a one-dimensional problem on account of the fact that the potential remains central. An exact straight line motion in a noncentral potential is also possible along the Oz axis because in this case x =y = O.

E

= n 12K [2n

+ 1 + (1/4 - 5)1/2],

(32)

where again s is quantized. Of course when fee) = 0, s = - l(l + 1) and we obtain the classical formulas for the energy levels of the hydrogen atom or of the harmonic oscillator. It only remains to solve the f)-equation (which does not depend on the choice V = V 1 or V = V2 ) to discover the

allowed s-values.

The e-equation is exactly soluble by means of known transcendental functions only when f(f)) is given by (10), (11) or (12). (a) The quantum motion in the electric potential (17).

II. SCHROOINGER'S MECHANICS

We have already seen (see Remark 2) that the quantum problem involving potentials like (1) is soluble in three dimensions if fee) is given by (10), (11) or (12) and that it is soluble in two dimensions in all cases (10) to (16).

The e-equation (30) becomes 6" + cote 6' - (m2/sin 2 e)6

- (c:r cos 2f) + (3 cose + y) sin- 2f) 6 - sO A. The three-dimensional problem

We use spherical coordinates r, equation takes the form:

a2 1f; +

(2/r)

ar 2

alf; +

(1/r2)

alf; _

v = cos 2 (e/2),

6 = vP (1 - v)o T,

where

a2 1f;

(m2/r2 sin2e) If;

ae

x [E - j.Lf(f))/r 2 - j.LV(r)]l/I

+

2j.L

n2

p

= (1/2)(m 2 +

a

= (1/2)(m 2 + c:r

v(l - v)T"

= O.

+

[(2p

c:r - {3 + y)1/2,

+

2rR'

+

(2/J-/n2) r 2 (E - /J-V(r))R

+

sR

= 0,

(29)

+ 1;1 +

(30)

Case 1: V= V1

=-

H/r.

Equation (29) reduces to the radial equation of a hydro-

E

(11'12

/1/4 - s

+

+

+ {3/2 +

a

2a

+

2)v] T' 2p2

+ p-

c:r]T= O.

+ c:r -

=-

(3

{3

+ y)1/2 +

(m2

+ c:r +

{3

+ y)1/2

+ y)1/2;v),

c:r + (k + P + a + 1/2)2.

This relation must be introduced into (31) and (32) to obtain the energy levels when V = V1 or V::;: V2' respectively. In what follows we only consider V == V l ' (The other case is analogous.) The energy levels are by (31)

== - (j.L3H2/2n 2) {n + 1/2 + ../- c:r + [k + (1/2)(11'1 2 + c:r -

J. Math. Phys., Vol. 14, No. 10, October 1973

s

= F(- k,k + (m 2 + c:r -

= O.

As in the Newtonian formalism, the radial motion does not depend on the term f(e)/r 2 present in the potential. We next investigate the two cases mentioned in Sec. I.A.

+

(2p

We recognize the hypergeometric equation. The polynomial condition gives the allowed s-values. One findS (k = 0,1,2, ... ) T

6" + coW 6' - (m 2/sin 2f))6- (2j.L2/n 2)f(e)6- sO

+ {3 + y )1/2,

+ 1) -

- [2pa

The variables can be separated in the usual way: If; = exp(imcp) 6 (f))R(r) (m is the usual magnetic quantum number; we assume m > 0; calculations are analogous when m < 0). One has r2R"

(33)

We make the following substitutions: Schrodinger's

ae 2

ar

+ (cote /r 2 )

e, cp.

= O.

{3

+ y)1/2 +

(1/2)(m 2

+ c:r + {3 +

y)1/2

+ 1/2]2}-2,

1324

Andre Hautot: Exact motion in noncentral electric fields

when n,k

1324

= 0,1,2,···.

w(1 - w}T" + [(2p + 1/2) - (2p + 2a + 3/2}w] T' -

(b) The quantum motion in the electric potential (20). The e-equation (30) becomes 6" + cote 6' - (m 2 /sin 2e)6 - (a cos 4e + (3 cos 2e + 1') x sin- 2e cos- 2e 6 - s6 == O. (34)

T

= F(- k,k

1/4 - s :::: - a + [2k + 1 + (1/2)(1 + 4y)1/2

where

+ (m 2 + a + {3 + y)1/2J2,

p :::: 1/4 + (1/4)(1 + 41')1/2,

= (1/2)(m 2 + a

a

+ 1 + (1/2}(1 + 41')1/2

+ (m 2 + a + {3 + 1')1/2; 1 + (1/2)(1 + 4y)1/2;w),

e = wP(I- w)o T,

cos 2 e,

= O.

The solution is again the hypergeometric function

We make the following substitutions: W ""

(1/4)(s + Spa + m 2 + (3 + 21' + 2a + 4p) T

with the energy levels:

+ {3 + 1')1/2,

E :::: - (j.L3H2/21l 2 ) {n + 1/2 + .../- a + [2k + 1 + (1/2)(1 + 41')1/2 + (m2 + a + j3+ y)1/2)2}-2,

r-

where n, k

= 0,1,2, ..•.

= O.

(c) The quantum motion in the electric potential (22)

6"- (2j.L2/n 2 )f(e)6

The e-equation (30) becomes

Once again the radial motion is independent of fee). V = V 1 or V2 and the radial equation is analogous to that encountered in the problem of the two-dimensional hydrogen atom 12 or that of the two-dimensional oscillator. Energy levels are given by

6" +

(m 2 /sin2e)

cote 6' -

- (a

6

cot 2 e + {3 cote +

y)6- s6:::: O.

We make the following substitutions: Case 1: z::::e 2te ,

6=zo(l-z)TT,

= (1/4)

T::::

(m 2

+ (1/2)(1/4 - y - s + i{3 + 01)1/2,

T=

(1/4) _ y + a _ (2k + 1 + 2.../m 2 + 01)4 - 4j32 , 4(2k + 1 + 2...jm 2 + 01)2

F[-

k,k + 1 + (1/4 - I' -

s + i{3 + 01)1/2 s + i{3 + a)l/2;z].

+ 2(m 2 + a)V2; 1 + (1/4 - y -

The energy levels are given by (31) (we only consider the case V:::: V1 ): E::::- (j.L3H2/2n2)t

+

j

Case 2:

Y - a

+

(36)

H/Y,

+ 1/2

V = V2

+

2s

+ 1)-2, (37)

= Ky2,

E:::: (p~/2j.L) + n..f1K(2n + s + I),

+ 01)1/2.

The T-equation is again hypergeometric. One finds (k :::: 0,1,2, ... ) S :::::

=-

s26

E :::: (p;/2/.L) - 2(j.L3H2 j1i2)(2n

where a

V= V1

+

(3S)

= 0,1,2"" while s may only take quantized values. These are found by solving (36). The e-equation is soluble in terms of known transcendental functions in seven cases: when fee) is given by (10) ••• or (16). Three of them have been treated in three dimensions. Therefore we shall omit the corresponding twodimensional treatments.

n

(d) The quantum motion in the electric potential (24) (two-dimensional). The 8-equation (36) becomes e"- (a sin2 e + (3 sine + 1') cos-2ee + s2e:::: O. We make the following substitutions:

====___'--

(2k + 1 + 2..Jm 2 + 01)4 - 4{32j-2 4(2k + 1 + 2..Jm 2 + 01)2 '

y

.0......_ _ _ _ _

where n,k

= (1- sine}/2,

where

= 0,1,2,···.

p:::: 1/4 + 1/4(1 + 401 + 4j3 + 41')1/2,

= 1/4 + 1/4(1 + 401 -

B. The two-dimensional problem

a

We use cylindrical coordinate Y, e,z. Schrodinger's equation is then written as

y(I-Y)T"

2 2 2 y2 0 1/1 + Y 01/1 + 0 1/1 + y2 0 1/1 oy2 oY oe 2 oz2

+

4{3 + 41')1/2,

[(2p+ 1/2)- (2p

(1/2)(- 2S2 + P +

0"

= F(- k,k

20"+ l)y]T'

+ 1 + (1/2)(1 + 401 + 4{3 + 4y)1/2

+ (1/2)(1 + 401 - 4{3 + 41')1/2; 1 + (1/2)(1 + 401 + 4{3 + 4y)1/2;y), S2 :::: - a + [k + 1/2 + (1/4)(1 + 401 + 4j3 + 41')1/2 + (1/4)(1 +4a-4j3+ 41')1/2]2.


+

+ 4pa + I' - a) T :::: O.

The solution is hypergeometric: T

Variables can be separated in the usual way: 1/1 = exp(ip .. zj1i}R(Y)6(e). (PI! is the z component of the momentum; it is a constant of the motion.) One has

e =yP(I-y}oT,

1325


1325

The energy levels follow from E

=::

(P~/2p.) X

2(p.3 H2 /1l2)

7 {2 n + 1 + 2 .J:-_-a----:-+'[7k-:+--=-1/7.:2:-+"""'-;:(1:-C/77 J2 }-2 4 )"(:::-1--:-+-4:-Ca--:'"+-;4:-::::{3-:+:-4 y');-1/"'2-+-:-:(1-:/:7. 4 ):-7(1::--:-+---:4:-a-_--=47{3-:+-4y:-.7)1;-;/=2 wheren,k

= 0,1,2, .... The solution is hypergeometric:

(e) The quantum motion in the electric potential (25) (two-dimensional). The a-equation (36) becomes an - [a tan 2 (a/2)+{3 tan(a/2) + y] a + s2a

1 + (1 + 16a)1/2;2p + 1;z),

where

We make the following substitutions:

= _ i{3

p2 z=_e ie ,

= F(- k,k + 2p +

T

= o.

+ (1/4) {[k + (1/2) + (1/2)(1 + 16a )1/ 2]4 - 4{32}

a=zp(1-z)oT, X

where p

= (s2

+ a - i{3 - y)1/2,

One finds

a

= 1/2

+ (1/2)(1 + 16a)1/2,

S2

=_ a

z(1- z) Tn + [(2p + 1) - (2p + 2a + 1)z]T' -

(2pa

+

a

+ 4a -

2i(3) T

= O. +

[k + (1/2) + (1/2)(1 + 16a)1/2]4 - 4{32f2 ,

4[k + (1/2) + (1/2)(1 + 16a)1/2)2

(f) The quantum motion in the electric potential (27) (two-dimensional). The a-equation (36) becomes (a cot 2(a/2) + (3 cot(a/2) + y)a + s2a

It deduces from the

=::

p

- [2pa

=::

1

+

e 2ie ,

e

=::

T

O.


+ (1 +

4a)1/2;z)

= Y-

a +

[(1 + 4a)1/2 + 1 + 2k]4 - 4{32 4[(1 + 4a)1/2 + 1 + 2k)2

,

with energy levels

[(1 + 4a)I/2 + 1 + 2k]4 - 4{32}-2

C. Conditionally soluble quantum motions

cos-la

= O.

F(- k,k + 1 + (1 + 4a)1/2

s2

In this section we shall investigate the solubility of Schrodinger's equation when more complicated potentials are conSidered, precisely those which lead to elliptic functions in the classical theory. Since these potentials are numerous we shall restrict ourselves to a special case in view of illustrating what we have called in a previous paper lO the "conditional solubility".

= (1l 2/2p.2) a

=::

1

y-a+

I(a)

(i{3/2)] T

+ (s2 + a - i{3 - y)1/2;

zP(1- z)o T.

We choose the special case

+p +a -

The solution is hypergeometric:

We make the following substitutions: z

(1/2) + (1/2)(1 + 4a)1/2,

z(1 - z) Tn + [- (2p + 2a + 1)z]T'

(g) The quantum motion in the electric potential (28) (two-dimensional). The a-equation (36) becomes =::

=::

a = (1/2)(s2 + a - i{3 - y)1/2,

O.

a- equation of Sec. II. B e by the a. Therefore, the wavefunction is

(a tan2a + (3 tana + y) a + s2e

where n,k = 0,1,2,···.

where

substitution a -7 1f obtained through the same procedure while the energy levels are given by the same formula.

en -

+ y + [k + (1/2) + (1/2)(1 + 16a)1/2]4 - 4{32 . 4[k + (1/2) + (1/2)(1 + 16a)1/2]2

The energy levels follow from

y-a

an -

[k + (1/2) + (1/2)(1 + 16a)]-2.

4[(1 + 4a)1/2 + 1 + 2k)2

,

wheren,k

= 0,1,2,· ...

so that we investigate the quantum motion in the electric potential J

= (p./€)

[a(Il/2p.2)/(r2 cosa)

+

V(r)].

ConSidering the three-dimensional problem so that one has in spherical coordinates, the a-equation (30) can be written as follows in spherical coordinates:

en + cote 6' - m 2 sin-2 a a - a cos-Ia e - sa

=

o.

1326


1326

We make the following substitutions:

u

= cose,

e = (1- u 2 )-m/2 T;

one finds u(l- u)(- 1- u) T" + (2 - 2m)u 2 T'

+

[(rn2- m + s)u + a]T= O.

This equation is of the general type u(l - u)(a - u)f" + (au 2 + bu + elf' + (d + eu)f = O.

We have studied it previously.6 Polynomials solutions exist which ensure the integrability of 1e 12 provided the four following conditions are fulfilled: 10 a

+

aa 2

+ e = - j' (1 - a) + ba + e = - j" a(a

b

- 1)

(where j' and j" are integers., 0)

e

=-

n(n + a - 1)

+ a "continuant" condition (see Ref. 10). Then one has T = z i '+1 (1 - z) i"+1 p(v), where denotes a polynominal of degree II. The first two conditions are satisfied if j' The third implies m 2 - m + s

=-

(II

p(v)

= j" = m -

+ 2m)(m +

II

1.

+ 1)

which gives the allowed s-values. The continuant condition is expressed by the vanishing of a continuant of order 11+ 1. We have previously seen10 that each value of II must be analyzed separately leading to a quantization of the parameter a entering into the definition of the potential. For example, when II = 2 the determinant is of order three: d

e

o

e

d+b

2(e - 1)

o

a+e

d

+

= O.

2b

In this case, the problem is soluble only if a = ± 2..J4m + 2. Performing the same operation for each value of II we arrive at the list of the allowed values for a. Inversely a being fixed (among the allowed values of course) II, m and s are also fixed so that arbitrary angular momentum states are automatically forbidden. We retrieve the conclusions of our preceding paper. 10

type (1) which allow a complete integration of the equations of motion in both classical and quantum nonrelativistic mechanics. Our first conclusion is that the threedimensional problem is completely soluble if the particle experiences the potentials (17), (20), or (22) while the two-dimensional problem is soluble when potentials (17), (20), (22), (24), (25), (27), or (28) are considered. To our knowledge these potentials were not treated before in the literature. It is interesting to compare the classical and the quantum treatments. As in our previous paper 10 it is possible to exhibit analogies from two different points of view: (a) Firstly, there are some purely formal analogies: the resolution of Newton's equation and that of Schrodinger's equation offer many common points in spite of their well distinct origins. Variables separate in both equations for the same potentials. The changes of variables needed for the complete calculation are often identical. The analogy is sometimes very suggestive, e.g., the classical handling of potentials (22) or (25) leads to a trajectory whose equation contains complex quantities but so mixed that the overall result is real. The quantum equation leads to the same result: it is impossible to avoid the use of complex numbers in the calculation of the energy levels though the final expression is of course real. Beside the potentials mentioned above, there is a large class of potentials which allow a complete integration of the classical equation of motion in terms of elliptic functions. The corresponding problem in quantum mechanics leads to a conditional solubility analog to that previously encountered in a paper dealing with the motion in various magnetic fields.1 o (b) Physical analogies also exist. Firstly we note that in classical mechanics potentials (17), (20), and (22) allow an exact solution whatever the choice of the axis of reference while potentials (24), (25), (27), (28) are soluble only for a special choice of these axes. In quantum mechanics the potentials of the first category are soluble in three dimensions (also in two) while those of the second category are soluble in two dimensions only. A Schrodinger's equation soluble by means of elementary transcendental functions (with energy levels) corresponds in the claSSical theory to a bounded trajectory expressible by means of circular functions: the parameters entering into the definition of the potential are submitted to analogous conditions in both mechanics in order to warrant a stable motion. Before terminating let us illustrate this last point with an example. Consider the motion in the potential (17); the quantum solution is given in Sec. II. A(a). The formula which determines the energy levels is only valid under the conditions

II = 2, one finds s = - (m + 2)(m + 3) and the energy levels are

If

a -

{3

+ Y ., 0,

a + (3 + Y ., O.

Remark 3: This formula is analogous to the one

giving the hydrogen spectrum except for the fact that the ground state and the first excited state are missing. Such a truncated hydrogen-like spectrum is found for every value of II.

(39)

It is not difficult to show tqat these conditions are suffi-

cient to ensure the stability of the classical trajectory. Let us return to Sec. I. B(a). It is easily seen that the 8integration leads to circular functions only if b + a1'i 2 / 1J.2 > O. Otherwise, the quadrature leads ~o ~o?arith.ms and the trajectory spirals to r = 0 or to mfmlty. Smce b - 2f(8) = p2 > 0 it is sufficient that

III. DISCUSSION AND CONCLUSION

In this paper we examined the various potentials of the J. Math. Phys., Vol. 14, No. 10, October 1973

b

+

a1'i2/1J. 2 ., b - 2f(8),

where f(8) is given by (10).


1327

After some reductions one obtains the condition O!

+

{3

cose + y "" 0

1327

imposed. In this case no equivalence seems to exist in quantum mechanics.

to be compared with (39). In conclusion, the connection between the classical and the quantum treatments of the problem here investigated can be stated as follows: when the classical trajectory is stable (0 < r min < r < r max) and when it is expressible by means of circular functions, the corresponding quantum problem is exactly soluble in terms of known functions and the energy spectrum contains a discrete part. If the trajectory is stable but can only be expressed by means of elliptic functions, the connection with the classical motion disappears. However, in that case the example treated in Sec. II. C indicates that the quantum energy spectrum may be discrete if the parameters entering in the definition of the potential are limited to discrete values. Finally, it must be pointed out that exact motions in classical mechanics sometimes occur when very special initial conditions are


·Presently Professor at the National University of Zaire, Kinshasa. IE. T. Whittaker, A Treatise on the Analytical Dynamics of Particles (Cambridge U.P., Cambridge, 1960). 2Andre Hautot, Physica (Utr.) 58, 37 (1972). 3N. D. Sengupta, Satyendranath Bose. 70th birthday Commemoration Volume. p. 55, Calcutta (1965). 4L. Lam, Phys. Lett. A 31, 406 (1970). 5Andre Hautot, Phys. Lett. A 35, 129 (1971). 6Andre Hautot, I. Math. Phys. 13, 710 (1972). 7Harish-Chandra, Phys. Rev. 74, 883 (1948). 8p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953). 9N. D. Sengupta, Ind. 1. Pure App!. Phys. 8, 765 (1970). IOAndre Hautot, I. Math. Phys. 14 (1973) (to be published). IIA. Armenti and P. Havas, Relativity and Gravitation (Gordan and Breach, New York, 1971). 12B. Zaslow and M. E. Zandler, Am. I. Phys. 35, 1118 (1967).