Generalized coherent states

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Ronald F. Fox ... This is achieved by introducing genuine Gaussian Klauder coherent states that ... 7,8. These experiments have been refined 9–11 and decay and revival have ..... Using Eq. 45, we may convert the right-hand side of Eq. 63 into ..... n0. 3. 0. 2. 8 n0. 3 . 127. Now shift the integration variables to y x n0/2 and y x.
PHYSICAL REVIEW A

VOLUME 59, NUMBER 5

MAY 1999

Generalized coherent states Ronald F. Fox School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 ~Received 3 November 1998! Generalized coherent states are constructed for the Coulomb problem. Following a construction procedure proposed by Klauder @J. Phys. A 29, L293 ~1996!#, Rydberg atom coherent states are defined and analyzed. The relationship between decorrelation in time and delocalization in space is elucidated. Keplerian orbits are discussed. The connection with sharp Gaussian wave packets used to explain pump-probe experiments is made. This is achieved by introducing genuine Gaussian Klauder coherent states that are overcomplete, and permit a resolution of the identity operator. They decorrelate comparatively slowly, and remain spatially localized for many Keplerian periods. @S1050-2947~99!05605-X# PACS number~s!: 03.65.2w

I. INTRODUCTION

Ever since Schro¨dinger @1# introduced coherent states for the harmonic oscillator, attempts to generalize this idea have been made. The su~2! generalized coherent states @2# are an especially nice example of a successful extension of the coherent state idea. More challenging has been the objective of obtaining generalized coherent states for the Coulomb potential problem, as was originally proposed by Schro¨dinger @1#. Recently, significant progress has been made in this direction @3,4#. Nevertheless, criticism of this approach has been raised @5,6#. It was motivated by comparison with experiment using a pump-probe technique to detect the periodic return of a wave packet to a nucleus along an elliptical orbit @7,8#. These experiments have been refined @9–11# and decay and revival have been observed as well as fractional revivals. Gaussian wave packets @12–14# have successfully accounted for these fascinating observations. Gaussian wave packets, per se, are not generalized coherent states and lack the property of resolution of the identity operator that is so useful for genuine coherent states @2#. The Majumdar-Sharatchandra @4# states for the hydrogen atom do have a Gaussian approximation ~see Sec. IV D below! for a large principal quantum number, but its variance is predetermined by the structure of these states and is much larger than for the ad hoc Gaussian wave packets @12–14# that are consistent with experimental observations. The purpose of the present paper is to present genuine Gaussian generalized coherent states, and to critique the recent literature. These states allow a resolution of the identity operator and can have very small variances for selected operators. They should prove useful in contexts other than the present, such as for quantum-classical correspondence theory via Husimi-Wigner distributions @15–17# semiclassical theory @18#, and wavelets for signal processing @19,20#. This paper is organized as follows. In Sec. II, a review of coherent states for the harmonic oscillator and of generalized coherent states for angular momentum is presented. In Sec. III, Klauder’s construction of generalized coherent states for Hamiltonians with discrete spectra is given. Section IV, the longest section of the paper, is devoted to Rydberg atom coherent states, in accord with Klauder’s construction @3# but in parallel with the particular rendering given by Majumdar 1050-2947/99/59~5!/3241~15!/$15.00

PRA 59

and Sharatchandra @4#. Section IV is separated into seven subsections. Section IV D contains a quantum-mechanical derivation of Kepler’s third law for circular Rydberg coherent states. Section V deals with temporal decorrelation and the criticisms of Bellomo and Stroud @5,6#. Finally, Sec. VI contains our construction of genuine Gaussian generalized coherent states and natural generalizations of them. Section VI could be read directly after Secs. I, II, and III, since the intervening sections essentially provide motivation and context only. II. HARMONIC OSCILLATOR AND su„2… COHERENT STATES

The paradigms for generalized coherent states are the harmonic-oscillator coherent states ua& and the su~2! coherent states uu,f& @2#. The harmonic-oscillator coherent state ua& for complex parameter a is defined by

F G( A

u a & 5exp 2

uau2 2

`

n50

an n!

un&,

~1!

where un& denotes an eigenstate of the harmonic-oscillator Hamiltonian, and the sum is over integer n’s. These states are normalized

^ a u a & 51,

~2!

because `

( n50

u a u 2n 5exp@ u a u 2 # , n!

~3!

and they provide a resolution of the identity operator: 1 p

E

d 2 a u a &^ a u 5

2

E

`

(

n50

u n &^ n u 51

~4!

r 2n 51 n!

~5!

because `

0

3241

r dr exp@ 2r 2 #

©1999 The American Physical Society

3242

RONALD F. FOX

PRA 59

for all n where a 5r exp@if#. If H is the harmonic-oscillator Hamiltonian, then

F

exp 2

G

i Ht u a & 5exp@ 2i v t/2# u a e 2i v t & \ 5exp@ 2i v t/2# u re i ~ f 2 v t ! & ,

~6!

H u n & 5E n u n & 5\ v e n u n & ,

so that the e n ’s are dimensionless for some energy scale \v, and wherein for definiteness e 0 ,e 1 ,e 2 ,¯ . We define the generalized Klauder coherent state by `

which exhibits Klauder’s definition of ‘‘temporal stability’’ @3#. The su~2! generalized coherent states uu,f& are defined by @2,17,21#

F F

u j, u , f & 5exp i u

5exp 2 2j

5

(

p50

3

S

G

1 „sin~ f ! J x 2cos~ f ! J y … u j, j & \

G

21/2

u n 0 , f 0 & 5„N ~ n 0 ! …

r n5

ip f

e cos2 j2p ~ u ! sinp ~ u ! p!

D

1/2

u j, j2p & ,

~7!

@ J z ,J 6 # 56\J 6 ,

@ J 1 ,J 2 # 52\J z

~8!

~10!

and provide a resolution of the identity operator: 2 j11 4p

E

dV u u , f &^ u , f u 51,

E

`

0

dn 0

Ar n

e ie n f 0 u n & ,

K~ n0! n n , N~ n0! 0

`

N~ n0!5

~17!

(

n n0

n50

rn

~18!

.

^ n 0 , f 0 u n 0 , f 0 & 51.

~19!

The resolution of the identity operator is given by

E

`

0

~11!

dn 0 K ~ n 0 ! lim

F→`

5

E

`

`

0

1 2F

dn 0

E

F

2F

d f 0 u n 0 , f 0 &^ n 0 , f 0 u `

n n0 K~ n0! u n &^ n u 5 u n &^ n u 51 N ~ n 0 ! n50 r n n50

(

(

where dV is differential solid angle. They are localized for large j in the sense that

^ u , f u J z u u , f & 5\ j cos~ u ! ,

~12!

^ u , f u J 6 u u , f & 5\ je 6i f sin~ u ! ,

~13!

1 1 2 2 Q 2 2 @^u,fuJ uu,f&2^u,fuJuu,f& #5 . \ j j

~16!

This guarantees that

~9!

for J 6 5J x 6iJ y , and where « i jk is completely antisymmetric and repeated indices are summed. These states are normalized

^ u , f u u , f & 51,

n50

N(n 0 ) is the normalization factor satisfying

where uj,m& denotes an eigenstate of J 2 and J z for the su~2! algebra of angular momentum operators. These operators satisfy the commutation identities @ J i ,J j # 5i\« i jk J k ,

(

n n/2 0

in which 2`, f 0 ,`. The parameters r n are moments of a positive weight function K(n 0 ) such that

u ~ J e 2i f 2J 2 e i f ! u j, j & 2\ 1

~ 2 j ! !p! ~ 2 j2p ! !

~15!

~14!

Thus uu,f& points in the direction of nˆ kˆ cos(u)1„iˆ cos(f) 1jˆ sin(f)… sin(u) with a ratio of its standard deviation to its average that vanishes with increasing j like 1/A j. III. KLAUDER COHERENT STATES

Klauder’s construction of generalized coherent states @3# for Hamiltonians with discrete spectra may be represented as follows. Let the Hamiltonian H have eigenstates and eigenenergies satisfying

~20! because

lim F→`

1 2F

E

F

2F

d f o e i ~ e n 2e n 8 ! f 0 5 d nn 8 .

~21!

One natural choice of weight function @3# K(n 0 ) is K(n 0 ) 51 for which r n 5n!. In this case, N(n 0 )5e n 0 , and we have precisely the Poisson coefficients used in the harmonic oscillator coherent states of Eq. ~1!. Notice that the extension of the f 0 domain from @2p,p# to ~2`,`! is essential for the resolution of the identity operator because it is required for the identity of Eq. ~21!. This is a key step in the Klauder construction. In order to obtain Gaussian generalized coherent states below ~see Sec. VI!, a similar extension will be required for the n 0 domain.

PRA 59

GENERALIZED COHERENT STATES

3243

IV. RYDBERG ATOM COHERENT STATES

In this section, a detailed account of Pauli’s su~2!3su~2! algebra @22# for the quantum Coulomb problem is given. It produces the Klauder states for Rydberg atoms in the form given by Mujumdar and Sharatchandra @4#. The special case of circular orbits is elucidated, and Kepler’s third law is derived quantum mechanically. This is followed by a study of dephasing in the azimuthal angle. These results enable us to critique the recent criticisms of Bellomo and Stroud @5,6#. The critique is presented in Sec. V.

~27!

n opu n,l,m & 5n u n,l,m & ,

~28!

and has the property

in which un,l,m& denotes a standard Rydberg atom state of the form u n,l,m & 5R nl ~ r ! Y m l ~ u,f !,

A Rydberg atom is described by the Hamiltonian p2 Ze 2 , 2 2m 0 r

~22!

angular momentum LW 5rW 3pW ,

~23!

and eccentricity vector ~also called the Runge-Lenz vector @23#! «W 5rˆ 2

which is rendered in spherical polar coordinates for later use. Instead of «W , we will use a renormalized variant defined by

S

D

~25!

in which the number operator appears and is defined by

X

K 6 5\n op sin~ u ! e 6i f 1

F S

~32!

~34!

L 6 56\ exp@ 6i f # „] u 6i cot an ~ u ! ] f …,

~35!

the well-known matrix element formulas follow: ~36!

^ n 8 ,l 8 ,m 8 u L 6 u n,l,m & 5 d n 8 n d l 8 l d m 8 m61 \ A~ l7m !~ l6m11 ! . ~37! These are paralleled by the following formulas:

S D GC D S D GC

F

FS

~33!

L z 52i\ ] f

a0 cos~ u ! L2 „cos~ u ! 1sin~ u ! ] u …] r 1 2 2 Z r \

a 0 6i f e Z

^ n 8 ,l 8 ,m 8 u K z u n,l,m & 52 d n 8 n d m 8 m \ d l 8 l21

@ L i ,K j # 5i\« i jk K k ,

and

~26!

X

~31!

^ n 8 ,l 8 ,m 8 u L z u n,l,m & 5 d n 8 n d l 8 l d m 8 m \m,

~ Ze 2 ! 2 m 0 , 2\ 2 ~ n op! 2

K z 5\n op cos~ u ! 1

@ K i ,K j # 5i\« i jk L k ,

Using the well-known formulas

where E is the Rydberg atom energy given by E52

~30!

@ L 2 1K 2 ,K i # 50⇒ @ n op ,L i # 5 @ n op ,K i # 50.

1/2

«W 5\n op«W ,

@ L i ,L j # 5i\« i jk L k ,

@ L 2 ,L i # 5 @ K 2 ,L i # 50,

1 a0 W 1r ] r ¹ W 2rW ¹ 2 ! , ~ pW 3LW 2LW 3pW ! 5rˆ 2 ~ ¹ 2 2Ze m 0 Z ~24!

~ Ze 2 ! 2 m 0 W5 K 2uEu

~29!

in which the spherical harmonics have the standard form @24# and the radial functions are the standard hydrogenlike functions @25# for ZÞ1. W satisfy the commutation relations The operators LW and K

A. Pauli’s algebra

H5

1 AL 2 1K 2 1\ 2 \

n op5

sin~ u ! 2cos~ u ! ] u 7

~ n 2 2l 2 !~ l2m !~ l1m ! ~ 2l11 !~ 2l21 !

D

i sin~ u ! L2 ] f ] r1 2 2 sin~ u ! r \

1/2

1 d l 8 l11

S

~38!

,

,

„n 2 2 ~ l11 ! 2 …~ l2m11 !~ l1m11 ! ~ 2l11 !~ 2l13 !

~39!

DG 1/2

,

~40!

F

^ n 8 ,l 8 ,m 8 u K 6 u n,l,m & 5 d n 8 n d m 8 m61 \ 6 d l 8 l11 7 d l 8 l21

S

S

„n 2 2 ~ l11 ! 2 …~ l6m12 !~ l6m11 ! ~ 2l11 !~ 2l13 !

~ n 2 2l 2 !~ l7m !~ l7m21 ! ~ 2l11 !~ 2l21 !

DG

D

1/2

1/2

.

~41!

3244

RONALD F. FOX

Equations ~36! and ~37! imply

Since

^ n 8 ,l 8 ,m 8 u L 2 u n,l,m & 5 d n 8 n d l 8 l d m 8 m \ 2 l ~ l11 ! ,

~42!

W 1N W !•~ M W 2N W ! 5LW •K W 50, M 2 2N 2 5 ~ M

and Eqs. ~40! and ~41! imply

^ n 8 ,l 8 ,m 8 u K 2 u n,l,m & 5 d n 8 n d l 8 l d m 8 m \ 2 „n 2 2 ~ l 2 1l11 ! ….

~43!

Together, these identities imply

^ n 8 ,l 8 ,m 8 u ~ L 2 1K 2 1\ 2 ! u n,l,m & 5 d n 8 n d l 8 l d m 8 m \ 2 n 2 ,

~44!

which justifies Eqs. ~27! and ~28!. W and N W defined by @23# Introduce operators M

W 5 21 ~ LW 1K W! M

PRA 59

W 5 21 ~ LW 2K W !. N

and

@ N i ,N j # 5i\«

i jk

Nk ,

@ M i ,N j # 50.

n 2opu j,m M & u j,m N & 5 5

W and N W are labeled by u j M ,m M & and the eigenstates of M u j N ,m N & , respectively, with j M 5 j N 5 j. The last equality in Eq. ~49! follows directly from the differential operator repW . While the eigenstates of L 2 and L s resentations of LW and K depend only on the angles u and f, the eigenstates of K 2 and K z depend on r as well. Thus we may express the states for the Rydberg atom as product states @4#, 2j

u j,m M & u j,m N & 5

m 1m C lj m j m u 2 j11,l,m M 1m N & , ( l50 M

M

N

N

~46! ~47! ~48!

in which the right-hand side gives the Clebsch-Gordon expansion in terms of the Rydberg states of Eq. ~29!. The fact that these Rydberg states all have principal quantum number 2 j11 follows from the operator n op . According to Eq. ~28!, n 2opu 2 j11,l,m M 1m N & 5 ~ 2 j11 ! 2 u 2 j11,l,m M 1m N & , ~51! whereas, according to Eqs. ~27!, ~45!, and ~48!

1 1 2 2 2 2 W W 2 W W 2 2 ~ L 1K 1\ ! u j,m M & u j,m N & 5 2 „~ M 1N ! 1 ~ M 2N ! 1\ …u j,m M & u j,m N & \ \ 1 ~ 2M 2 12N 2 1\ 2 ! u j,m M & u j,m N & 5„2 j ~ j11 ! 12 j ~ j11 ! 11…u j,m M & u j,m N & \2

5 ~ 2 j11 ! 2 u j,m M & u j,m N & .

In order to construct coherent states, we follow the procedure used to generate generalized, su~2! coherent states @17#. This requires obtaining the ‘‘highest weight’’ state, which we now prove is given by u j, j & u j, j & 5 u 2 j11,2j,2j & .

N 1 u j, j & u j, j & 50

L 1 u j, j & u j, j & 50

and

K 1 u j, j & u j, j & 50

L z u j, j & u j, j & 52\ j u j, j & u j, j & ~57!

and ~54!

~56!

and

~53!

In su~2!3su~2!, the highest weight state satisfies and

~52!

From Eq. ~45!, it follows that

B. Highest weight and Helgason’s identity

M 1 u j, j & u j, j & 50

~50!

~45!

These operators satisfy the commutation relations of the algebra su~2!3su~2!: @ M i ,M j # 5i\« i jk M k ,

~49!

K z u j, j & u j, j & 50. The four conditions of Eqs. ~56! and ~57! imply that

and M z u j, j & u j, j & 5\ j u j, j & u j, j &

u j, j & u j, j & 5 u 2 j11,2j,2j & .

~55!

and N z u j, j & u j, j & 5\ j u j, j & u j, j & .

~58!

The proof of this assertion involves explicit calculation using the differential forms in Eqs. ~34!, ~35!, ~38!, and ~39! and the functional form @23–25#

PRA 59

GENERALIZED COHERENT STATES

u 2 j11,2j,2j & 5

S D

Z Ap a 0 1

2 j13/2

1

~ 2 j11 !

2 j12

~ 2 j !!

F

r 2 j exp 2

3245

G

Zr sin2 j ~ u ! e i2 j f . ~ 2 j11 ! a 0

~59!

For fixed j, we may construct a coherent state factor using Helgason’s identity @17,26# to expand the generator:

F F

u j, u M , f M , u N , f N & 5exp i u M

5exp 2 2j

5

S

G

G

uM uN ~ M 1 e 2i f M 2M 2 e i f M ! exp 2 ~ N e 2i f N 2N 2 e i f N ! u j, j & u j, j & 2\ 2\ 1

2j

( ( p50 q50 3

G F G F

1 1 ~ sin f M M x 2cos f M M y ! exp i u N ~ sin f N N x 2cos f N N y ! u j, j & u j, j & \ \

e ip f M 1iq f N cos2 j2p ~ u M ! cos2 j2q ~ u N ! sinp ~ u M ! sinq ~ u N ! p!q!

~ 2 j ! !p! ~ 2 j ! !q! ~ 2 j2p ! ! ~ 2 j2q ! !

D

1/2

u j, j2 p & u j, j2q & .

~60!

Equation ~50! can be used to convert the ket outer product in the last line of Eq. ~60!, but this requires application of the Racah formula @27# for the construction of the Clebsch-Gordon coefficients which are not otherwise given in closed form. An alternative construction utilizes the properties of the operators, M 2 , N 2 , L 2 , and K 2 . From M 2 u j,m M & 5\ A~ j1m M !~ j2m M 11 ! u j,m M 21 & ,

~61!

it follows that M k2 u j, j & 5\ k

S

~ 2 j ! !k! ~ 2 j2k ! !

D

1/2

u j, j2k & ,

~62!

and similarly for N 2 . Therefore @recall Eq. ~48!#, u j, j2p & u j, j2q & 5

1 \ p1q

S

~ 2 j2 p ! ! ~ 2 j2q ! ! ~ 2 j ! !p! ~ 2 j ! !q!

D

1/2 p q M2 N 2 u j, j & u j, j & .

~63!

Equations ~30!, ~31!, and ~32! imply @ L 2 ,K 2 # 50.

~64!

Using Eq. ~45!, we may convert the right-hand side of Eq. ~63! into

S

1 ~ 2 j2p ! ! ~ 2 j2q ! ! u j, j2p & u j, j2q & 5 p1q ~ 2\ ! ~ 2 j ! !p! ~ 2 j ! !q!

D

1/2 p

(

a50

q

p! q! p1q2a2b a1b K 2 u 2 j11,2j,2j & . ~ 21 ! q L 2 a! ~ p2a ! ! b50 b! ~ q2b ! ! ~65!

(

The actions of L 2 and K 2 are given by Eqs. ~37! and ~41!, respectively. By inspection of these formulas, it is clear that the equality of the m components, i.e., j2p1 j2q52 j2 p2q, is guaranteed, and is consistent with Eq. ~50!. Following Klauder’s lead @3,4# for the j sum, we obtain the Rydberg coherent state ~we have scaled the phase f 0 slightly differently than in Sec. III in anticipation of Kepler’s third law below! `

u Ryd,n 0 , f 0 ,t & 5

( j50

G F F GA F S D S D S D S DS

exp 2

3cos2 j2p

n0 2

n 0j

~ 2 j !!

exp i

n 30 f 0

2 ~ 2 j11 !

2

exp i

uM uN uM uN cos2 j2q sinp sinq 2 2 2 2

Z 2R y t \ ~ 2 j11 ! 2

G( ( 2j

2j

exp@ ip f M 1iq f N # p!q!

p50 q50

~ 2 j ! !p! ~ 2 j ! !q! ~ 2 j2 p ! ! ~ 2 j2q ! !

D

1/2

u j, j2p & u j, j2q & ,

~66!

3246

RONALD F. FOX

in which the j sum is over half-integer values of j, and the terminal ket product may be replaced by standard Rydberg atom un,l,m& states in accord with the Clebsch-Gordon expansion in Eq. ~50!, or by the method of Eq. ~65!. The j coefficients must satisfy two roles simultaneously @3#. They must guarantee normalization of u Ryd,n 0 , f 0 ,t & , and provide a resolution of the identity operator for the Hilbert space of bound states @3,4#. The time evolution operator for the Hamiltonian of Eq. ~22! evolves the state u Ryd,n 0 , f 0 & 5 u Ryd,n 0 , f 0 ,0& into u Ryd,n 0 , f 0 12Vt/n 30 & 2 5 u Ryd,n 0 , f 0 ,t & , where V5Z R y /\, and Ryd is the Rydberg constant.

PRA 59

^ Ryd,n 0 , f 0 ,t u KW u Ryd,n 0 , f 0 ,t & 5\„ 21 n 0 ~ nˆ M 2nˆ N ! …, ^ Ryd,n 0 , f 0 ,t u « 2 u Ryd,n 0 , f 0 ,t & 5

1 „11 ~ g 21 ! e 2n 0 2Ei~ n 0 ! e 2n 0 1ln~ n 0 ! e 2n 0 n0 1 ~ 12nW M •nW N !@ n 0 221 ~ 22 g ! e 2n 0 1Ei~ n 0 ! e 2n 0 2ln~ n 0 ! e 2n 0 # …,

5

The following expectation values are exact consequences of Eq. ~66!, albeit after considerable computation:

5\ 2 „ 21 ~ n 0 1n 20 ! 1n 0 1 21 ~ n 0 1n 20 ! nˆ M •nˆ N …, ~67!

^ Ryd,n 0 , f 0 ,t u K 2 u Ryd,n 0 , f 0 ,t &

XS

~72!

in which nˆ M and nˆ N are radial unit vectors given in terms of u M and f M and u N and f N , respectively. Equations ~70! and ~72! differ by more than a factor of \n 0 , because Eqs. ~25!, ~26!, and ~28! imply that the j sum in Eq. ~66! is affected. In Eq. ~71!, g is the Euler constant, and Ei is the exponential integral function given by `

Ei~ z ! 5 g 1ln~ z ! 1

~68!

~69!

C

D

1 1 2 ~ 12exp@ 2n 0 # ! ~ nˆ M 2nˆ N ! , 2 2n 0

5\ 2 „ 21 ~ n 0 1n 20 ! 1n 0 2 21 ~ n 0 1n 20 ! nˆ M •nˆ N …,

^ Ryd,n 0 , f 0 ,t u LW u Ryd,n 0 , f 0 ,t & 5\„ 21 n 0 ~ nˆ M 1nˆ N ! …,

~71!

^ Ryd,n 0 , f 0 ,t u «W u Ryd,n 0 , f 0 ,t &

C. Properties of Rydberg atom coherent states

^ Ryd,n 0 , f 0 ,t u L 2 u Ryd,n 0 , f 0 ,t &

~70!

zn

( n51 n!n

~73!

for positive z. To obtain these results, we have repeatedly used the fundamental identity

W 6N W ! u j, j2p & zj, j2q‹ Šj 8 , j 8 2q 8 z^ j 8 , j 8 2p 8 u ~ M W 6N W ! u j, j2p & zj, j2q‹ 5 d j 8 j Šj, j2q 8 z^ j, j2p 8 u ~ M

S

F

\ \ 5 d j 8 j kˆ d p 8 p d q 8 q \„j2p6 ~ j2q ! …1 ˆi d p 8 p21 d q 8 q Ap ~ 2 j2 p11 ! 6 d p 8 p d q 8 q21 Aq ~ 2 j2q11 ! 2 2 1 d p 8 p11 d q 8 q

F

\ \ A~ 2 j2p !~ p11 ! 6 d p 8 p d q 8 q11 A~ 2 j2q !~ q11 ! 2 2

G

\ \ 1 ˆj d p 8 p21 d q 8 q Ap ~ 2 j2p11 ! 6 d p 8 p d q 8 q21 Aq ~ 2 j2q11 ! 2i 2i 2 d p 8 p11 d q 8 q

\ \ A~ 2 j2p !~ p11 ! 7 d p 8 p d q 8 q11 A~ 2 j2q !~ q11 ! 2i 2i

In performing the p and q sums, care must be taken with the limits of the summations since, for example, d p 8 p21 requires that p>1, so that p 8 is not less than zero. After carefully adjusting the limits and shifting the indices appropriately, we then use two identities @26# to finish the computations: 2 j21

( p50

~ 2 j2 p !

~ 2 j !! x 2p 52 j ~ 11x 2 ! 2 j21 , ~ 2 j2p ! !p!

~75!

2j

GD

.

~74!

~ 2 j !!

( ~ j2 p ! ~ 2 j2 p ! !p! x 2p 5 j ~ 12x 2 !~ 11x 2 ! 2 j21 . p50

~76!

We may choose to have the conserved angular momentum along the z axis and the conserved eccentricity vector along the x axis. It is straightforward to show that this can be achieved by setting

PRA 59

GENERALIZED COHERENT STATES

u M 5 u N 5¯u

and

f M 50

and

f N5 p ,

~77!

3247

^ « 2& 5

1 ~ 12e 2n 0 ! 2cos2 ~ ¯u ! n0 3

using ^¯& to denote the Rydberg coherent state expectation value, from Eqs. ~67!–~72! we obtain

1 „Ei~ n 0 ! 2 g 2ln~ n 0 ! …e 2n 0 n0

S

2 ~ 12e 2n 0 ! , n0

D

~83!

^ L 2 & 2 ^ LW & • ^ LW & 5\ 2 n 0 „11cos2 ~ ¯u ! …,

~84!

^ K 2 & 2 ^ KW & • ^ KW & 5\ 2 n 0 „11sin2 ~ ¯u ! …,

~85!

1sin2 ~ ¯u ! 12

^ LW & 5\n 0 cos~ ¯u ! kˆ ,

~78!

^ KW & 5\n 0 sin~ ¯u ! ˆi ,

~79!

^ « 2 & 2 ^ «W & • ^ «W &

S

D

1 ^ «W & 5 12 ~ 12e 2n 0 ! sin~ ¯u ! ˆi , n0

5

~80!

1 1 ~ 12e 2n 0 ! 2cos2 ~ ¯u ! „Ei~ n 0 ! 2 g 2ln~ n 0 ! … n0 n0 3e 2n 0 2sin2 ~ ¯u !

^ L 2 & 5\ 2 n 0 „11 ~ 11n 0 ! cos2 ~ ¯u ! …,

~81!

^ K 2 & 5\ 2 n 0 „11 ~ 11n 0 ! sin2 ~ ¯u ! …,

`

u circ,n 0 , f 0 ,t & 5

( j50

F GA

n 0j

~ 2 j !!

~ 12e 2n 0 ! 2 .

~86!

D. Circular Rydberg atom coherent states

A circle is produced when ¯u 50 is chosen. The general Rydberg coherent state in Eq. ~66! simplifies considerably ~only the p50 and q50 terms need to be kept!, becoming

~82!

n0 exp 2 2

1 ~ n0!2

F

exp i

n 30 f 0 2 ~ 2 j11 !

2

G F

exp i

G

Z 2R y t u 2 j11,2j,2j & , \ ~ 2 j11 ! 2

~87!

in which the j sum is again over half-integers. The position vector expectation value is now `

^ circ,n 0 , f 0 ,t u rnˆ u circ,n 0 , f 0 ,t & 5

`

( ( j 50 j50 8

e 2n 0

~ n 0 ! j1 j 8

A~ 2 j ! ! ~ 2 j 8 ! !

FS

exp i Vt1

n 30 f 0 2

DS

1 1 22 ~ 2 j11 ! ~ 2 j 8 11 ! 2

3 ^ 2 j 8 11,2j 8 ,2j 8 u rnˆ u 2 j11,2j,2j & ,

DG ~88!

in which V5Z 2 R y /\. The matrix elements, by lengthy but straightforward computation, yield

^ 2 j 8 11,2j 8 ,2j 8 u rnˆ u 2 j11,2j,2j & 5

FS D S D

a0 Z 1

ˆj ˆi ~ 2 j11 ! 2 j14 ~ 2 j12 ! 2 j13 1 d 2 j 8 2 j11 2 2i ~ 2 j13/2! 4 j15

G

ˆj ˆi ~ 2 j11 ! 2 j12 ~ 2 j ! 2 j13 2 d 2 j 8 2 j21 . 2 2i ~ 2 j11/2! 4 j13

~89!

Thus Eq. ~88! becomes `

^ circ,n 0 , f 0 ,t u rnˆ u circ,n 0 , f 0 ,t & 5

F XS DF

n 30 f 0 a 0 2n e 0 P ~ j ! ˆi cos Vt1 Z 2 j50

(

XS

1 ˆj sin Vt1

n 30 f 0 2

DF

1 1 22 ~ 2 j11 ! ~ 2 j12 ! 2

1 1 22 ~ 2 j11 ! ~ 2 j12 ! 2

G CG

,

GC ~90!

3248

RONALD F. FOX

in which

t 52 p

~ n 0 ! 2 j11/2

~ 2 j11 ! 2 j14 ~ 2 j12 ! 2 j13 P~ j !5 ~ 2 j13/2! 4 j15 ~ 2 j ! ! A2 j11

S D

pn0 ' P~ n0! 2

1/2

F

S DG

1 exp 2 2 pn0 2

n0 j2 2 ~ n 0 /4!

S D

1/2

e n0

A2 p n 0

n 0 @1

2

`

F S

D S

DG

a 0 2 \ 2 n 20 n 5 , Z 0 Zm 0 e 2

u j, j21 & u j, j & 5

.

~94!

S D

u j, j & u j, j21 & 5

1/2

r

~95!

3/2

u ellip,n 0 , f 0 ,t & 5

n 0j

F

( e 2n /2A~ 2 j ! ! exp j50 0

n 30 f 0

2 ~ 2 j11 ! 2

G

FS S FS S

a0 Z 1

^ 2 j 8 11,2j 8 ,2j 8 u rnˆ u 2 j11,2j21,2j21 & 5

1/2

~98!

.

a0 Z 1

1 &

~ u 2 j11,2j,2j21 &

1 &

~99!

~ u 2 j11,2j,2j21 &

@ u 2 j11,2j,2j & 1¯u A j u 2 j11,2j21,2j21 & ].

We now need variations of the matrix element given in Eq. ~89!:

^ 2 j 8 11,2j 8 21,2j 8 21 u rnˆ u 2 j11,2j21,2j21 & 5

m0 Ze 2

~100!

Therefore, a slightly eccentric coherent state is given by

Vt1 i

5

2 u 2 j11,2j21,2j21 & ).

where k is the strength of the 1/r potential. In the present case,

`

1/2

~97!

1 u 2 j11,2j21,2j21 & ),

Kepler’s third law relates the period t to the radius r: m0 t 52 p k

~96!

,

To obtain a slightly eccentric elliptical orbit, we choose ¯u slightly larger than 0, and keep the p51 and q50 and q 51 and p50 terms in Eq. ~66! as well as the p50 and q 50 term used for the circle case. The equivalent of the Clebsch-Gordon coefficients can be obtained by using propW, M W , and N W operators expressed in Eq. ~65!. erties of the LW , K In particular,

~93!

2Z 2 R y 2Z 2 R y a0 2 ˆ n 0 ˆi cos t1 f 0 3 t1 f 0 1 j sin Z \n 0 \n 30

Z 2m 0e 4

E. Slightly eccentric Rydberg atom coherent states

For the circle case, we obtain ~for n 0 @1)

5

\ 3 n 30

Even the coefficient agrees exactly. The transition from reciprocal squares of the principal quantum number in the exponentials of Eq. ~90! to reciprocal cubes in Eq. ~94! results from the interference of adjacent energy levels in the expansion of Eq. ~87! caused by the couplings of 2 j to 2 j61 created by the matrix elements on the right-hand side of Eq. ~88!. This is a manifestation of the traditional Bohr correspondence principle @28#.

~92!

.

( f ~ j ! →2 E0 dy f ~ y ! . j50 ^ circ,n 0 , f 0 ,t u rnˆ u circ,n 0 , f 0 ,t &

2Z R y

52 p

S D S D

~91!

,

2

m0 k

These limiting approximations permit us to replace the sum in Eq. ~90! by an integral, provided that we observe that the half-integer values for j in the sum imply a ‘‘density-ofstates’’ factor of 2, i.e., `

\n 30

r5

wherein P ~ n 0 ! ——→

PRA 59

D

S D S DG

ˆj ˆi j ~ 2 j11 ! 2 j14 ~ 2 j12 ! 2 j12 1 d 2 j 8 2 j11 4 j14 2 2i j11/2 ~ 2 j13/2!

D

ˆj ˆi ~ 2 j11 ! 2 j11 ~ 2 j ! 2 j13 j21/2 2 d 2 j 8 2 j21 2 2i j ~ 2 j11/2! 4 j12

D

ˆj ˆi 1 d „23 ~ 2 j11 ! A j… 2 2i 2 j 8 2 j

D

G

ˆj ˆi ~ 2 j21 ! 2 j12 ~ 2 j11 ! 2 j11 Aj , 2 d 2 j 8 2 j22 2 2i ~ 2 j ! 4 j12

~101!

1/2

1/2

,

~102!

~103!

PRA 59

GENERALIZED COHERENT STATES

^ 2 j11,2j,2j u rnˆ u 2 j 8 11,2j 8 21,2j 8 21 & 5

FS D S D

ˆj ˆi 1 d „23 ~ 2 j11 ! A j… 2 2i 2 j 8 2 j

a0 Z

1

3249

G

ˆj ˆi ~ 2 j11 ! 2 j14 ~ 2 j13 ! 2 j13 A j11 . 2 d 2 j 8 2 j12 2 2i ~ 2 j12 ! 4 j16

~104!

In the sums for d 2 j 8 2 j21 and d 2 j 8 2 j22 , lower limit restrictions on j are required so that j 8 >0. When these are imposed, j can be shifted so that the new j runs from 0 to ` as before. After lengthy computation, the result is

FX S XS

D

S

D C

2Z 2 R y 4Z 2 R y a0 2 « 3« ˆ ^ ellip,n 0 , f 0 ,t u rnˆ u ellip,n 0 , f 0 ,t & 5 n 0 i cos 3 t1 f 0 1 cos 3 t12 f 0 2 Z 2 2 \n 0 \n 0 1 ˆj sin

2Z 2 R y

Because eccentricity only introduces simple harmonics of the fundamental frequency, 2V/n 30 , Kepler’s third law remains exact. We can show that Eq. ~105! represents an « perturbation of the circular orbit described by Eq. ~94!. By changing variables from r to u51/r, one may show that the classical equation of motion is (k5Ze 2 , and L is the angular momentum! @30# d2 « 2 21 3 k 4 L2 2 u 23 2 u 5 . 2 u54k 2 u 15 dt 2L m0 m0

~106!

Writing u5u 0 1«u 1 1« 2 u 2 , we find

u 05

m 0k 1 5 , L2 rc

u 15

1 cos~ v c t ! , rc

\n 30

D

S

4Z 2 R y « t1 f 0 1 sin t12 f 0 2 \n 30

D CG

.

~105!

radius and 2V/n 30 for the classical frequency, Eq. ~105! may be used to show that the Rydberg atom electron radius magnitude is r5r c A11 25 « 2 22« cos~ v c t ! 2 23 « 2 cos~ 2 v c t ! >r c „12« cos~ v c t ! 2« 2 cos~ 2 v c t ! 1« 2 …,

~109!

wherein we have used A11x>11 21 x2 81 x 2 , in which x stands for all of the « terms. This is precisely the first-order inversion of the results in Eq. ~107!, i.e., 1/(11x)>12x. So far, we have been unable to obtain comparable closed-form results for arbitrary eccentricity. However, the results here strongly suggest that higher powers of the eccentricity and higher harmonics of the fundamental frequency will make up such general results.

~107! u 25

F. Dephasing of the azimuthal angle

1 „cos~ 2 v c t ! 21…, rc

in which r c is the classical radius and v c is the classical frequency. The boundary conditions used for the solution just given are that this solution agrees with the orbital equation at t50, i.e., with

r5

a ~ 12« 2 ! , 11« cos~ u !

~108!

where a is the semimajor axis and the numerator is equal to the classical radius @29#. Using Eq. ~97! for the classical

While the results above show that the expected value of the position executes circular or slightly eccentric orbital motion, it is also important to determine the rate at which uncertainty in the coordinates grows. In this section, we investigate this issue for the circular Rydberg coherent states. We show that these states remain tightly compact in both r and u, but exhibit dephasing in f. To do this, we need the explicit coordinate dependence given by Eqs. ~59! and ~87!. Define c circ(r, u , f ,t) by

c circ~ r, u , f ,t ! 5 ^ r, u , f u circ,n 0 , f 0 ,t & .

~110!

The probability density associated with c circ(r, u , f ,t) is given by

3250

RONALD F. FOX `

P ~ r, u , f ,t ! 5

( ( j 50 j50 8

3

n 0j1 j 8

`

e 2n 0

S D

1 Z p a0

A~ 2 j ! ! ~ 2 j 8 ! !

2 j12 j 8 13

FS

exp i Vt1

1 1 Zr 1 a 0 2 j11 2 j 8 11

0

E

p

0

E

`

0

d f e i ~ 2 j22 j 8 ! f 52 p d j j 8 ,

d u sin2 j12 j 8 11 ~ u ! 52

F S S D D

dr r 2 j12 j 8 12 exp 2

5 ~ 2 j12 j 8 12 ! !

S

1 1 Zr 1 a 0 2 j11 2 j 8 11

a0 Z

2p

DG

0

3

~111!

In parallel with Eqs. ~91! and ~92!, we find

e 2n 0

n 20 j ~ 2 j !!

'

1 2

F

exp 2

1 ~ j2n 0 /2! 2 2 ~ n 0 /4!

A2 p ~ n 0 /4!

G

.

~116!

This implies that

sin4 j ~ u ! 'sin2n 0 ~ u ! 5exp@ 2n 0 ln sin~ u !#

F

'exp 2

2 j12 j 8 13

~114!

.

`

d f P ~ r, u , f ,t ! 52

e i ~ 2 j22 j 8 ! f sin2 j12 j 8 ~ u !

~113!

In Eq. ~113! j1 j 8 is even; for j1 j 8 odd multiply by p/2. It is now clear that the f integration produces reduced distributions that are independent of time. The reduced distribution for r and u is given by

E

DG

.

2 j12 j 8 13

~ 2 j11 !~ 2 j 8 11 ! 3 2 j12 j 8 12

Q ~ r, u ! 5

DG

~112!

~ 2 j12 j 8 ! !! , ~ 2 j12 j 8 11 ! !!

1 1 2 2 ~ 2 j11 ! ~ 2 j 8 11 ! 2

~ 2 j ! ! ~ 2 j 8 11 ! 2 j 8 12 ~ 2 j 8 ! !

We can reduce this to distributions in one coordinate at a time by integrating the other two coordinates. The required integrals are 2p

2

DS 1

2 j12

F S

E

n 30 f 0

1 ~ 2 j11 !

3r 2 j12 j 8 exp 2

PRA 59

( j50

e 2n 0

n 20 j

S D

Z ~ 2 j !! a0

G

1 ~ u 2 p /2! 2 . 2 ~ 1/2n 0 !

~117!

This means that the root-mean-square deviation compared to the mean is

4 j13

A^ ~ D u ! 2 & p /2

5

&

p An 0

~118!

.

1 ~ 2 j11 ! 4 j14 @~ 2 j ! ! # 2

F S DG

2 Zr 3r 4 j exp 2 a 0 2 j11

sin4 j ~ u ! .

~115!

S D F S DG S D Zr a0

4j

exp 2

2 Zr a 0 2 j11

'

Zr z0

2n 0

Thus, for sufficiently large n 0 , u is confined to be very close to p/2, i.e., in the azimuthal plane. Equation ~116! also implies that

F S DG F F

exp 2

Zr 2 a0 n0

5exp 2

'exp@ 22n 0 12n 0 ln n 20 # exp 2

2Zr 12n 0 ln~ Zr/a 0 ! n 0a 0

G

G

1 ~ r2r 0 ! 2 , 2 ~ n 30 a 20 /2Z 2 !

~119!

wherein r 05

a0 2 n , z 0

and the variance is clearly n 30 a 20 /2Z 2 . This means that the root-mean-square deviation compared to the mean is

~120!

PRA 59

GENERALIZED COHERENT STATES

A^ ~ Dr ! 2 &

5

r0

1

A2n 0

3251

~121!

.

Thus, for sufficiently large n 0 , r is confined to relatively very close to the circle radius of Eq. ~97!. In contrast to these time-independent results for r and u, the reduced distribution for the angle f is time dependent. Using Eqs. ~111!, ~113!, and ~114!, we obtain F ~ f ,t ! 5

E

`

dr r 2

0

E

p

0

`

d u sin~ u ! P ~ r, u , f ,t ! 5

3 ~ 2 j12 j 8 12 !@~ 2 j12 j 8 ! !! # 2

n 0j1 j 8

`

2 2n e 0 p j 8 50

( j50 ( A~ 2 j ! ! ~ 2 j 8 ! !

~ 2 j11 ! 2 j 8 11 ~ 2 j 8 11 ! 2 j11

FS

exp i Vt1

1

n 30 f 0 2

DS

1 1 22 ~ 2 j11 ! ~ 2 j 8 11 ! 2

DG

e i ~ 2 j22 j 8 ! f ,

~122!

~ 2 j12 j 8 ! !! 5 ~ 2 j12 j 8 12 !@~ 2 j12 j 8 ! !! # 2 . ~ 2 j12 j 8 11 ! !!

~123!

~ 2 j12 j 8 12 !

~ 2 j !!~ 2 j 8!!

2 j12 j 8 13

wherein we have used the identity ~ 2 j12 j 8 12 ! !

We now use the following approximations:

e 2n 0

n 0j1 j 8

'

A~ 2 j ! ! ~ 2 j 8 ! !

pn0

F

A2 p n 0

G F

1 ~ j2n 0 /2! 2 1 ~ j 8 2n 0 /2! 2 exp 2 2 ~ n 0 /2! 2 ~ n 0 /2!

exp 2

Ap n 0

Ap n 0

G

~124!

,

~ 2 j11 ! 2 j 8 11 ~ 2 j 8 11 ! 2 j11

@~ 2n 0 ! !! # 2 ~ n 0 11 ! n 0 11 ~ n 0 11 ! n 0 11 ' ~ 2 j12 j 8 12 !@~ 2 j12 j 8 ! !! # ~ n 0! !2 ~ 2n 0 12 ! 2n 0 12 ~ 2 j12 j 8 12 ! 2 j12 j 8 13 ~ 2 j ! ! ~ 2 j 8 ! ! 2

'2 2n 0 2 2 ~ 2n 0 12 ! 52 22 ,

FS

exp i Vt1

n 30 f 0 2

DS

1 1 22 ~ 2 j11 ! ~ 2 j 8 11 ! 2

DG F S

'exp i Vt1

n 30 f 0 2

F S

'exp 2i Vt1

DS

~125!

1

2 n 20 „11 ~ 2 j2n 0 11 ! /n 0 …2

D

S

1 n 20 „11 ~ 2 j 8 2n 0 11 ! /n 0 …2

G

D

DG

n 30 f 0 8 n 30 f 0 3 „~ 2 j ! 2 2 ~ 2 j 8 ! 2 … , 3 ~ 2 j22 j 8 ! 1i Vt1 2 n0 2 n 40 ~126!

wherein we have used 1/(11x) 2 ;122x13x 2 1¯ . Replacing the two sums by integrals in accord with Eq. ~93!, we find

S DE

2n 0 F ~ f ,t ! ' p

1/2

`

0

F

exp 2 dx 8

F X S F X S

S

D

n 30 f 0 3 1 ~ x 8 2n 0 /2! 2 2i Vt1 ~ 2x 8 ! 2 2 2 n 40 ~ n 0 /2!

Ap n 0

D CG E D CG

n 30 f 0 8 3exp 2i2x 8 f 2 Vt1 2 n 30 3exp i2x f 2 Vt1

n 30 f 0 8 2 n 30

F

exp 2

`

dx

S

F ~ f ,t ! '

S DE E S F 2n 0 p

1/2

`

2`

dy 8

`

2`

dy

F

3exp i ~ 2y22y 8 ! f 2Vt

2 n 30

D

n 30 f 0 3 1 ~ x2n 0 /2! 2 1i Vt1 ~ 2x ! 2 2 ~ n 0 /2! 2 n 40

Ap n 0

0

G ~127!

.

Now shift the integration variables to y5x2n 0 /2 and y 8 5x 8 2n 0 /2, and obtain exp 2

G

S

D

G F

S

D

n 30 f 0 3 n 30 f 0 3 1 ~ y 8!2 1 ~ y !2 2 2i Vt1 2y exp 2 1i Vt1 ! ~ ~ 2y ! 2 8 2 ~ n 0 /2! 2 n 40 2 ~ n 0 /2! 2 n 40

2f0

DG

Ap n 0 .

Ap n 0

G

~128!

3252

RONALD F. FOX

PRA 59

By performing the Gaussian integrals and rearranging the results, a normalized Gaussian reduced distribution is produced:

F

1 exp 2 ~ f 2Vt2/n 30 2 f 0 ! 2 2 F ~ f ,t ! '

F S 2p

The issue of the correspondence principle can be approached by treating celestial dynamics by the Schro¨dinger equation, and comparing the resulting description with that of Newtonian classical mechanics. In this section, we do this for the Earth, Mars, and Saturn. The strength of attraction, Ze 2 , for Rydberg atoms need only be replaced by GMm for celestial bodies where G56.6731028 dyn cm2/gm2, Newton’s gravitational constant; M 51.8931033 gm, the mass of the Sun; and m5m e 55.9831027 gm, the mass of the Earth. The masses of Mars and Saturn are 0.108m e and 95.2m e , respectively. This change in attractive strength is enormous: Ze 2 ;Z323.04310220 erg cm and GM m e ;7.538 31053 erg cm, about 72 orders of magnitude larger. The Bohr radius \ 2 /m 0 e 2 is 5.2931029 cm, whereas the celestial analog \ 2 /GM mm is 2.443102136 cm for m5m e , about 127 orders of magnitude smaller. Similarly, the Bohr orbital period 2 p \ 3 /e 4 m 0 is 1.5310216 s, whereas the ce-

5

U(

>

j50

e

F S DG 11

n 30

2

F

U

C ~ t ! 5 ^ c u exp 2

G U

2 i Ht u c & , \

~130!

where uc& denotes either a generalized coherent state or a wave packet. For the circular Rydberg coherent states, this yields

F

G

i Ht u circ,n 0 , f 0 & \

GU S D

Z 2R y exp i t \ ~ 2 j11 ! 2 ~ 2 j !!

1 6Vt

~129!

.

Bellomo and Stroud @5,6# used the time autocorrelation function proposed by Nauenberg @13,14#,

C ~ t ! 5 ^ circ,n 0 , f 0 u exp 2 n 20 j

1/2

V. TEMPORAL DECORRELATION

F

U

DG

DG

lestial analog 2 p \ 3 /G 2 M 2 m 2 m, is 2.143102216 s for m 5m e , about 200 orders of magnitude smaller. Since we know the orbital radius and period for the Earth ~for the present purpose, we can ignore the eccentricity of the Earth’s orbit!, it is a simple matter to determine the principal quantum number in accord with the celestial analogs of Eqs. ~96! and ~97!. For the Sun-Earth system we know that t 53.16 3107 s, and that r51.5031013 cm. Equation ~96! implies that n SE52.5331074, and Eq. ~97! implies that n SE52.53 31074. This is an enormous principal quantum number. Corresponding results for Mars and Saturn yield n SM53.37 31073 and n SS57.4331076, respectively. Looking back at Eq. ~129!, we see that for the Earth the variance grows by a factor of about 1422 3(square of the number of periods). Since each period is a year, the variance will not reach order unity for n SE52.53 31074, until about 1036 years have elapsed. This is so much longer than the age of the universe that we can conclude that a Rydberg coherent state treatment of the Sun-Earth system yields a compact, localized state in all three spherical polar coordinates for the entire lifetime of the system. In this limit of extremely large principal quantum numbers, the quantummechanical treatment of celestial dynamics reproduces the classical mechanical description with very great precision.

G. Celestial bodies as Rydberg coherent states

2n 0

1 9 1 ~ Vt2/n 30 1 f 0 ! 2 4n 0 n 0

9 1 1 ~ Vt2/n 30 1 f 0 ! 2 4n 0 n 0

This distribution clearly shows that the averaged value of f changes linearly with time in accord with the result in Eq. ~94!. At first glance, it would appear that the variance grows quadratically in time with a 1/n 70 dependence. This would seem to be negligible for sufficiently large n 0 . However, we have expressed this growing term in a form that shows that after exactly one period of the orbital revolution, the variance increases from 1/4n 0 to (1/4n 0 )1(9/n 0 )(2 p ) 2 , or by a factor of 1421.5. Thus, for typically obtained experimental Rydberg atom states with 50