Animation of classical and quantum mechanical  motions of a charged particle in a Paul trap

The formulation of classical and quantum mechanical motions of an elementary particle or an ion in a Paul trap is derived by symbolic manipulation using Mathematica. Paul trap potential, the trajectory of classical motions, the quantum mechanical ground state and first excited state, and the squeezed states for a trapped particle are represented in Mathematica-generated animation.

by Min-Qi Bao

As a result of the improvement of experimental instruments and techniques,trapping and cooling has become a booming enterprise in atomic physics, quantum optics, and even nuclear physics, because it permits the observation of isolated elementary particles, ions, and atoms over a long period of time. Therefore, according to Heisenberg's uncertainty principle, it may enable us to measure physical quantities with extraordinary precision. The recent achievements of trapping and cooling include mass [1] and frequency [2] standards, highly charged ion (e . g .  (U^(92+))) traps [3] , and radiative atom traps [4] , which in turn provide the opportunity to detect very subtle physical effects which test our knowledge of special relativity, quantum electrodynamics, and nuclear  β decay respectively. Trapping laser-cooled  atoms with microwave radiation [5] may also lead to the achievement of  Bose-Einstein condensation [6].

In order to design an efficient trap, it is necessary to compute and visualize the ion motion in the trap. Some simulation of the behavior of charged particles in electromagnetic traps[7] has been carried out under Microsoft Windows. Mathematica[8] provides a convenient way to animate graphics. In this paper we use Mathematica's animation capability to demonstrate how a charged particle moves in one of the most important kinds of traps, the Paul trap[9],both classically and quantum mechanically. Such a system is another kind of Geonium atom[10], which is similar to the hydrogen atom, but with the atomic potential replaced by an external trapping field that can be adjusted. This simple system provides an opportunity to measure physical quantities of a single charged particle and allows the exclusion of many-particle effects in experiments.

1. POTENTIAL

The Paul trap[9] consists of a hyperbolically-shaped ring and two hyperbolic rotational caps,  as shown schematically in Figure 1, where the voltage (Subscript[ϕ, o])=U+V cos (Subscript[ω, o])t is applied between the caps and the ring electrode. The potential in the Paul trap can be written as[9]

ϕ (x, y, z; t)=(((Subscript[ϕ, o]) ((r^2) - 2  (z^2)))/((Subscript[r, o]^2) + 
                (Subscript[z, o]^2)))
                              =((m (Subscript[ω, o]^2))/(8  e))(a + 2  q cos (Subscript[ω, o]) t)((x^2)+(y^2)-(z^2))                                            (1)

where a and q are dimensionless parameters defined by

a=((4  
                    eU)/(m(Subscript[ω, o]^2) ((Subscript[r, o]^2) + 
                        2  (Subscript[z, o]^2))))'\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\ \ \ \ \ \ \ \ \)\)(2)
	
q=((2  
                eV)/(m(Subscript[ω, o]^2) ((Subscript[r, o]^2) + 2  (Subscript[z, o]^2))));                                                                                                                 (3)

and U is a dc voltage,V is the amplitude of an ac voltage with frequency (Subscript[ω, o]) in the radio frequency range, m and e are the mass and charge of the trapped particle, respectively, and (Subscript[r, o])and (Subscript[z, o]) are the geometric parameters of the apparatus shown in Figure 1. Atomic units (a.u.) are used throughout this paper, namely, the unit of m is the electron mass, the unit of e is the electron charge, and  ℏ = 1.

Figure 1. Schematic view of a Paul trap.

In each of the following animations, the package will load and automatically generate the animation with the indicated set of default values. For example, to show how the potential traps a charged particle when y = 0 and z = 0.1, in Equation (1), we pick a = 0.05, q = π × (10^(-9))(a . u . ), and set (Subscript[mω, o]^2)/8 e=1  for simplicity in these inputs. We fix the particle in the center of the Paul trap in these two cases for clarity. In later sections, we show how the particle is moving inside the Paul trap.

Figures 2 and 3 show typical scenes for the above two cases respectively. In either case, we find that the trap potential does not behave as indicated by the rotational model given in Figure 8 of Ref.[9] which represents a rotational frame and involves Coriolis force, fictitious force, etc.

<< "pxz`"

[Editor's Note: selected frames from the animations are shown. Complete animations are generated in the electronic version of this article.]

Figure 2.  Paul trap potential in x and z directions. (Four frames selected.)

<< "pxy`"

Figure 3.  Paul trap potential in x and y directions.

2. CLASSICAL MOTIONS

The Hamiltonian for the Paul trap with a potential described by Equation (1) is

H=(1/2)m(((OverDot[x])^2) +((OverDot[y])^2)+((OverDot[z])^2))+((m (Subscript[ω, o]^2))/(8  e))(a+2q cos (Subscript[ω, o])t)((x^2) + (y^2) - (z^2)).                             (4)

The equations of classical motion for a charged particle with unit charge (e=1 a.u.) are therefore

\(\(x\& .. \) + \(ω\_o\%2\/4\) \((a + 
                  2  q\ cos\ \(ω\_o\) t)\) x = 0\),                                                                         (5)
\(\(y\& .. \) + \(ω\_o\%2\/4\) \((a + 
                  2  q\ cos\ \(ω\_o\) t)\) y = 0\),                                                                         (6)
(Overscript[z, ..])-((Subscript[ω, o]^2)/4) (a + 2  q cos (Subscript[ω, o]) t) 
            z=0.                                                                           (7)

Because Equations. (5) and (6) are exactly the same and differ from Equation (7) by a negative sign, which is just a π difference in phase, we only need to solve Equations. (5) and (7). The motions in three dimensions are separable and the motion in each dimension is similar.

Letting (Subscript[ω, o])t/2  = τ,  x(t) = f(τ) and z(t) = g(τ), Equations(5) and (7) become

(((d^2) f (τ))/d)(τ^2) + (a + 
                    2  q cos 2  τ) f (τ) = 0,                                                          (8)

(((d^2) g (τ))/d)(τ^2) - (a + 
                2  q cos 2  τ) g (τ)=0.                                                            (9)

These are Mathieu equations[11],  which give rise to a stable motion only for some specific values of a and q. 
 In this paper,  we pick the case in which a = 0 and q≪1, so that  the analytical solution of Equations. (8) and
(9) can be obtained approximately  [11, 12] as:

x (t)=(([1 + (q/2) cos(Subscript[ω, o]) t])/(1 + 
                    q/2)) cosωt,                                                                    (10)
			
z(t)=(([1 - (q/2) cos(Subscript[ω, o]) t])/(1 - q/2)) 
            cos (-ωt),                                                          (11)

where

ω=(((Subscript[ω, o]) q)/(2^(3/2)))≪(Subscript[ω, o]).                                                                                                              (12)

Figure 4 , produced by the following input package, shows x(t) versus t in which one finds the micro-motion with a radio frequency (Subscript[ω, o])=π × (10^(-9)) (a . u . ) superimposed on the secular macro-motion with much lower frequency ω.

<< "CLASSIC'"

Figure 4  Time evolution of classical motion in a Paul trap.

This one-dimensional classical motion can be animated for a charged particle with a unit charge in the trap potential:

<< "CLASSIC1D`"

Figure 5.  One-dimensional classical motion in a Paul trap.

One of the typical graphics for the animation is shown in Figure 5, in which both micro-motion and macro-motion can be seen clearly. To see how this particle moves in the trap, we animate its three-dimensional classical motion:

<< "CLASSIC3D`"

Figure 6.  Three dimensional classical motion in a Paul trap.

III. QUANTUM MECHANICAL STATES

Although there have been continuous efforts to implement and improve the  trapping of charged particles since the 1920s, most of these experiments can be explained at a classical mechanics level. Only now that experiments have trapped a single ion, which is cooled into the quantum ground state in a Paul trap, is there a need complete quantum mechanical treatment[13].

The quantum mechanical treatments for a charged particle in a Paul trap have been carried out by Combescure[14], Brown[15], and Glauber[16]. Quasi-stationary quasi-energy[17,18] is one of the most effective approaches in solving a Schrödinger equation for quantum mechanical system subjected a classical force which is a periodic function in time. Quasi-stationary state is a set of solutions of this Schrödinger equation with certain energy level shift and broadening,comparing with the solutions for the same system without the periodic force.

This energy is known as quasi-energy because it has a complex value with its real part as the new energy level and its imaginary part as its width which is related to the decay rate of this state[17,18]. For the parameters a = 0 and q ≪ 1, the quasi-stationary quasi-energy solution[15] for the Paul trap is

(Subscript[ψ, n]) (x, t)=\(\((m\ ω/π)\)\^\(1/4\)\/\(\((\(2\^n\) \(n!\))\)\^\(1/2\) | f \((t)\)\( | \^\(1/2\)\)\)\)(Subscript[H, n])\((\(\(\@\(m\ ω\)\) x\)\/\(\(|\)\(f \((t)\)\)\(|\)\))\)\(exp[\(i  χ\) \((t)\) x\^2 - \(1\/2\) 
                m\ ω\ x\^\(\(2\)\(\ \)\)\/\(\(|\)\(f \((t)\)\)\( | \^2\)\) - inωt]\)        (13)

where

\(χ \((t)\) = \(m\/4\) \((f\& . \/f + \(f\^*\)\& . \/\(f\^*\))\)\),                                                                                  (14)

and

f (t)=(([1 + (q/2) cos (Subscript[ω, o]) t])/(1 + q/2))exp(iωt).                                          (15)

This is the complex solution of the classical motion for a charged particle in a Paul trap under the initial condition f(0)=1 and (OverDot[f])(0)=i ω, which was suggested in Ref.[16]. This selection of f(t) greatly simplifies the formulation of trapped squeezed states, as we show in the next section. In Equation(13) the (Subscript[H, n])(x) is one Hermite polynomial, where n denotes the quantum number of the state,with a physical meaning similar to the quantum number for the simple harmonic oscillator wave function.

Consider a proton or antiproton with a mass of 1836.152 a.u. moving in a Paul trap with a = 0 and q = 0.4.We show in animated form the probability density of the ground-state (n = 0) wave function and the  probability density of the first-excited-state (n = 1) wave function by the following:

<< "wf0`"

\(
\)

Figure 7  The probability density of ground-state wave function in a Paul trap.

<< "wf1`"

Figure 8  The probability density of first-excited-state wave function in a Paul trap.

Typical graphics generated by the above two inputs are shown in Figures 7 and 8.For the wave function of higher excited states (n ≥ 2), one can produce their animations by simply changing the value of n in the corresponding Mathematica animation notebook.

4. SQUEEZED STATES IN A PAUL TRAP

The squeezed states [19,20] for simple harmonic oscillation have been widely applied to various fields such as quantum optics, atomic physics, etc.[21]   Because there is a one-to-one correspondence[16] between the simple harmonic wave function and the Paul trap wave function, it is possible to constructs squeezed states in the Paul trap.

Suppose a charged particle such as a proton or an antiproton is dropped into Paul trap as a Gaussian wavepacket of width (Subscript[ω, p])  and a momentum boost  (Subscript[k, p]) initially:

(Subscript[ϕ, i])(x) =(((((m (Subscript[ω, p]))/π)))^(1/4)) exp [ i (Subscript[k, p])x-(1/2)(Subscript[mω, p]) (((x-(Subscript[x, o])))^2) ].                             (16)

We expand this in terms of the initial quasi-stationary quasi-energy states, as

(Subscript[ψ, n]) (x, 0) = ((((m (Subscript[ω, p])/π))^(1/4))/(((((2^n) n!))^(1/
                  2))))(Subscript[H, n])((Sqrt[m ω]) x)  exp[-(1/2)(Subscript[mω, p]) (((x-(Subscript[x, o])))^2) ].                        (17)

so that

(Subscript[ϕ, i]) (x) = (Overscript[(∑_(n = 0)), ∞]) (Subscript[C, n]) (Subscript[ψ, n])(x, 0).                                                                                                               (18)

Using the integral on page 837 of Ref.[22],

(∫_(-∞)^(+∞)) (e^(-b (((x - y))^2))) (Subscript[H, n]) (a x) dx =(((1-((a^2)/b)))^((n/2)))(Sqrt[(π/b)])(Subscript[H, n])((ay/(((1 - ((a^2)/b)))^((1/2))))),                  (19)

one finds

(Subscript[C, n]) = ((((2/((2^2) n!))))^((1/2)))(((((ω (Subscript[ω, p])))^((1/4))))/(((((Subscript[ω, p]) + ω))^((1/2)))))((( (((Subscript[ω, p])-ω)/((Subscript[ω, p]) + ω))))^((n/2)))(Subscript[H, n])\((\(\(\@ω\) \((ik\_p + \(mω\_p\) x\_o)\)\)\/\(\(\@m\) \((ω\_p\%2 - ω\^2)\)\^\(1\/2\)\))\)exp(\(\(\(ik\_p\) \(x\_o\) ω\_p\)\/\(ω\_p + ω\) - 
                k\_p\%2\/\(2  m \((ω\_p + ω)\)\) - \(\(mωω\_p\) x\_o\%2\)\/\(2 \((ω\_p + ω)\)\)\)).                      (20)

Finally, using the sum rule on page 786 of Ref.[23],

(Underscript[(∑^∞), m = 0])(((z^m)/((2^2) m!)))(Subscript[H, m])(x) (Subscript[H, m]) (y)= exp ((x^2)+(y^2)-(((x^2) + (y^2) - 2  xyz)/(1 - (z^2)))) (((1-(z^2)) )^(-(1/2))),    (21)

one obtains the wavepackets

(Subscript[ϕ, i])(x, t) =(((Sqrt[2]) (((m (Subscript[ω, p])(ω^2) /π))^(1/
                  4)))/(\(\(|\)\(f \((t)\) \((\( | \^\(1\/2\)\)\((ω\_p + ω)\))\)\^\(1\/2\)\)\)))exp (iχ(t) (x^2)+(1/2) m ω ((x^2)/(\(\(|\)\(f \((t)\)\)\( | \^2\)\)))+(((Subscript[ik, p]) (Subscript[x, o]) (Subscript[ω, p]))/((Subscript[ω, p]) + ω))-((Subscript[k, p]^2)/(2  m ((Subscript[ω, p]) + ω)))-((mω(Subscript[ω, p]) 
                    (Subscript[x, o]^2))/(2 ((Subscript[ω, p]) + ω))))

 ×exp ( -((( m ω (x^2))/(\(\(|\)\(f \((t)\)\)\( | \^2\)\)))-((ω((Subscript[ik, p])+(Subscript[mx, o])(Subscript[ω, p])))/(\(m \((ω\_p\%2 - ω\^2)\)\)))-((2ω x((Subscript[ik, p])+(Subscript[mx, o])(Subscript[ω, p])))/(|f (t)|((((Subscript[ω, p]^2) - (ω^2)))^((1/2)))))((( (((Subscript[ω, p])-ω)/((Subscript[ω, p]) + ω))))^((n/2)))exp(-iωt))/(1-exp(-2i ω t)((Subscript[ω, p]) + ω)((Subscript[ω, p]) - ω)))

 × exp ((( m ω (x^2))/(\(\(|\)\(f \((t)\)\)\( | \^2\)\)))-((ω((Subscript[ik, p])+(Subscript[mx, o])(Subscript[ω, p])))/(\(m \((ω\_p\%2 - ω\^2)\)\))))(((1-(((Subscript[ω, p])-ω)/((Subscript[ω, p]) + ω))exp(-2 iωt)))^(-(1/2))).                          (22)

A step-by-step symbolic manipulation of the above formulation can be carried out by means of the following Mathematica package. In this package, we use the command PowerContract which was first implemented in the PowerTools package by James M. Feagin[24].

<< "SQUEEZED`"

Figure 9.  Squeezed state propagation in a Paul trap.

The time development of the Gaussian wavepackets is plotted in Figure 9. We find that the width of the wavepackets is squeezed and extended periodically. This process can be controlled and manipulated coherently by tuning the voltage (Subscript[ϕ, o])=U+V cos (Subscript[ω, o]) t  applied between the caps and the ring electrode on the Paul trap. We can use this kind of trap for high precision measurement.For example, we can detect the coordinate of the charged particle inside the trap precisely by preparing a highly squeezed wavepacket for the particle.

This property has also been widely used in quantum optics[25] and has an application in gravitational-wave detection[26].

5. CONCLUSIONS

This work deepens the understanding of the physics in a Paul trap and provides us with a visual tool for further research and for pedagogical purposes. Based on this work, one can study the coherent control of laser excitation and deexcitation in this quantum system.

ACKNOWLEDGMENTS

I am deeply indebted to Professor Anthony F. Starace for for his  support and critically reading the manuscript. I would like to thank Professor Roy J. Glauber for helpful discussions. I also appreciate the referee's suggestions.This work was supported in part by the U.S. National Science Foundation under Grant No. PHY-9722110.

REFERENCE

[1] F .  DiFilippo et al., Phys. Rev. Lett. 73, 1481(1994). 
[2] M .  G .  Raizen et al., Phys .  Rev .  A 45, \(6493 \((1992)\) . \ 
\([3]\)\ R . \ E . \ 
            Marrs\ et\ \(\(al\)\(.\)\)\), Phys. Rev. Lett. 72, \(4082 \((1994)\) . \ \ 
\([4]\)\ Z . \(-T . \ 
                Lu\)\ et\ \(\(al\)\(.\)\)\), Phys. Rev. Lett. 72, \(3791 \((1994)\) . \ \ 
\([5]\)\ R . \ 
            Spreeuw\ et\ \(\(al\)\(.\)\)\), Phys. Rev. Lett. 72, \(3164 \((1994)\) . \ 
\([6]\)\ M . \ H . \ Anderson\), J .  R .  Ensher, E .  A .  Cornell, Science, 269,\(198\ \((1995)\) . 
\([7]\)\ Susan\ L . \ 
            Fischer\ and\ Christian\ Wolfgang\), Computers in Physics,  10, \(123\ \((1996)\) . 
\([8]\)\ Stephen\ Wolfram\),Mathematica, a System for Doing Mathematics by Computer,\(2  
          nd\ Edition\ \((Addison - Wesley, \ Redwood\ City, \ CA, \ 
              1991)\) . 
\([9]\)\ Wolfgang\ Paul\), Rev. Mod. Phys. 62, \(3 \((1990)\) . \ \ 
\([10]\)\ Lowell\ S . \ 
            Brown\ and\ Gerald\ Gabrielse\), Rev. Mod. Phys. 58, \(233 \((1986)\) . \ 
\([11]\)\ W . \ D . \ McLachlan\), Theory and Application of Mathiue Functions\(\((Oxford\ University\ Press, \ Oxford, \ 
              1947)\) . 
\([12]\)\) D .  J .  Bate, K .  Dholakia, R .  C .  Thompson, and D .  C .  Wilson, J. Mod. Opt. 39, \(305\ \((1992)\) . 
\([13]\)\ F . \ 
            Diedrich\ et\ \(\(al\)\(.\)\)\), Phys. Rev. Lett. 62, \(403 \((1989)\) . \ 
\([14]\)\ M . \ Combescure\), Ann .  Inst .  Herri .  Proincarè 44, \(293 \((1986)\) . \ 
\([15]\)\ Lowell\ S . \ Brown\), Phys. Rev. Lett. 66, \(527 \((1991)\) . \ 
\([16]\)\ Roy\ J . \ Glauber\), in Quantum Measurements in Optics, \(eds . \ P . \ Tombesi\ and\ D . \ F . \ 
            Walls\ \((Plenum\ Press, \ New\ York, \ 1992)\) . \ 
\([17]\) 
          Ya . \ B . \ Zel' dovich\), N .  L .  Manakov, and L .  P .  Rapoport, Usp .  Fiz .  Nauk  117, 569(1975) [Sov. Phys. Usp. 18, 920 (1975)].
[18] N .  L .  Manakov, M .  A .  Preobrazhensii , L .  P .  Rapoport, and A .  F .  Faishtein, Zh. Eksp. Teor. Fiz. 75, 1243 (1978)  [Sov .  Phys .  JETP  48, 626 (1978)].
[19] R .  J .  Glauber, Phys. Rev. Lett. 10, \(84\ \((1963)\) . 
\([20]\)\ Jan\ Grochmalicki\ and\ Maciej\ Lewenstein\), Phys. Rep. 208, \(189\ \((1991)\) . 
\([21]\)\ John\ R . \ Klauder\ and\ Bo - 
          Sture\ Skagerstam\), Coherent States: Applications in Physics and Mathematical Physics \(\((World\ Scientific, \ Singapore, \ 
                  1985)\) . 
\([22]\)\) I .  S .  Gradshteyn and I .  M .  Ryzhik, Table of Integrals, Series and Products,  \(\((Academic\ Press, \ New\ York, \ 
              1980)\) . 
\([23]\)\ P . \ M . \ Morse\ and\ H . \ Feshbach\),Methods of Theoretical Physics, Vols .  2,\(\((McGraw - Hill, \ New\ York, \ 
              1953)\) . 
\([24]\)\ James\ M . \ Feagin\), Quantum Methods with Mathematica, \(\((Springer - Verlag, \ New\ York, \ 
                1994)\) . 
\([25]\)\ C . \ Cohen - Tannoudji\), J .  Dupont - Roc, and G .  Grynberg, Photons and Atoms,Introduction to Quantum Electrodynamics \(\((Wiley, \ New\ York, \ 1989)\) . 
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ABOUT THE AUTHOR

Min-Qi Bao received a PhD in Physics from University of Nebraska-Lincoln. He is working as a senior engineer at Platform Computing Corporation in Toronto,Canada. His interests include developing technical and business models and creating virtual supercomputers through clustering technology.

Min-Qi Bao
P.O. Box 91052 at PHARMA PLUS
2901 Bayview Avenue 
North York, On M2K 2Y6, Canada
Email: mqbao@yahoo.com

ELECTRONIC SUBSCRIPTIONS

Included in the distribution for each electronic subscription is the file paultrap.nb, containing Mathematica code for the material described in this article.

Supplementary code for electronic version

The following routines may be added as .m files to an appropriate directory path and loaded as needed, or evaluated directly.

BeginPackage["pxz`"]
Begin["`Private`"]
\(\(	\ \ \ \)\($DefaultFont\  = \ {"Helvetica-Bold", \ 
            10};\)\)
\(\(\ \ \ \ \ \)\(a\  = \ 0.05\)\(		\)\( (*\ 
        Eq . \ \((2)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \)\(q\  = \ 0.1\)\(			\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \)\(w\  = \ Pi*10^\((\(-9\))\)\)\(		\)\( (*\ 
        Radio\ frequency\ of\ the\ micro - \(\(motion\)\(.\)\)\ *) \)\)
     Do[
          \(Show[
		\ \ \ \ \ \ \ \ \ \ \ \ Plot3D[\((a + 
                      2*q*Cos[w*t*10^8])\)*x*
                  x - \ \((a + 2*q*Cos[w*t*10^8])\)*z*
                  z, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-5\), 5}, {z, \(-5\), 
                5}, \ BoxRatios -> {1, 1, 1}, 
              Boxed -> False, 
\ \ \ \ \ \ \ \ \ \ \ \ \ ViewPoint -> {2, 
                  3.5, 0}, \ Ticks -> None, 
              AxesLabel -> {"x", "z", ""}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ PlotLabel -> "                                   V(x,y=0,z;t)", 
\ \ \ \ \ \ \ \ \ \ \ \ \ PlotRange -> {\(-7\), 7}, 
              DisplayFunction -> 
                Identity], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Graphics3D[{PointSize[0.05], 
                Point[{0, 0, 
                    0.5}]}
\ \ \ \ ], 
\ \ \ \ \ \ \ DisplayFunction -> $DisplayFunction], {t, 0, 19}\)
     ]
    
     End[]
EndPackage[]

BeginPackage["pxy`"]
      Begin["`Private`"]
      $DefaultFont  =  {"Helvetica-Bold",  
            10};
\(\(\ \ \ \ \ \ \)\(a\  = \ 0.05;\)\(		\)\( (*\ 
        Eq . \ \((2)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \)\(q\  = \ 0.1;\)\(		\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \)\(w\  = \ Pi*10^\(-9\);\)\(		\)\( (*\ 
        Radio\ frequency\ of\ the\ micro - \(\(motion\)\(.\)\)\ *) \)\)
      Do[
        Show[ParametricPlot3D[{5*Sin[u]*Cos[v], 5*Sin[u]*Sin[v],
        (a + 2*q*Cos[w*t*10^8])*(5*Sin[u]*Cos[v])^2 
                   (a + 2*q Cos[w*t*10^8])*(5*Sin[u]*Sin[v])^2},{u,  0,  2  Pi}, 
           {v,  -Pi/2,  Pi/2}, BoxRatios -> {1, 1, 1},
           Boxed -> False,ViewPoint -> {4, 4, 1}, Ticks -> None,
			           AxesLabel -> {"x", "y", ""},
           PlotLabel -> "                                 \
  V(x,y,z=0;t)",
           PlotRange -> {-6, 6}, DisplayFunction -> Identity], 
              Graphics3D[{PointSize[0.05], Point[{0, 0, 0.5}]}  
           ],
     DisplayFunction -> $DisplayFunction],{t, 0, 19 }
     ]
    
     End[]
EndPackage[]

BeginPackage["CLASSIC`"]
		Begin["`Private`"]
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\($DefaultFont\  = \ {"Helvetica-Bold", \ 10};\)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(q\  = \ 0.1\)\(			\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(w\  = \ 
        Pi*10^\(-9\)\)\(		\)\( (*\ 
        Radio\ frequency\ of\ the\ micro - \(\(motion\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(W\  = \ 
        q*\(w/2\)/Sqrt[2]\)\(	\)\( (*\ 
        Eq . \ \((12)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(Plot[
          Cos[W*t*10^9]*\((1 + \((q/2)\)*Cos[w*t*10^9])\)/\((1 + q/2)\), {t, 
            0, 106}, 
\ \ \ \ \ \ \ \ \ \ \ PlotPoints -> 150, \ 
          AxesLabel -> {"t (a.u.)", "x(t)"}];\)\(
\)\(	\ \ \ \ \ \ \)\( (*\ Plotting\ the\ classical\ path\ as\ indicated\ in\ Eq . \ \((10)\)\ *) \)\)
      End[]
EndPackage[]

\(\(BeginPackage["CLASSIC1D`"]\)\(				\)\)
      Begin["`Private`"]
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\($DefaultFont\  = \ {"Helvetica-Bold", \ 10};\)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(q\  = \ 0.1\)\(			\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
           w  =  
        Pi*10^(-9)            
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(W\  = \ 
        q*\(w/2\)/Sqrt[2]\)\(	\)\( (*\ 
        Eq . \ \((12)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \)\(Do[
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ Show[
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plot[\((2*q*
                    Cos[w*t*10^9/6])\)*x*x, \ {x, \ \(-3\), \ 3}, \ 
              PlotRange -> {\(-2\), 
                  2}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AspectRatio -> 1, \ Ticks -> None, \ 
              AxesLabel -> {"x", \ "V(x,t)"}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesOrigin -> {0, 0.6}, \ 
              DisplayFunction -> 
                Identity], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Graphics[{{PointSize[0.03], \ 
                  Point[{Cos[
                          W*t*10^9/
                              6]*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((1 + \((q/2)\)*Cos[w*t*10^9/6])\)/\((1 + 
                              q/2)\), 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6}]}, \ {PointSize[0.001], Point[{\(-3\), 0}], \ 
                  Point[{3, 
                      0}]\ }}
		\ \ \ \ \ \ \ \ \ \ \ ], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> $DisplayFunction], {t, 0, 
            170}
\ \ \ \ \ \ ];\)\(
\)\(	\ \ \ \ \)\( (*\ 
        Plotting\ the\ classical\ path\ as\ indicated\ in\ Eq . \ \((10)\)\ *) \)\)
End[]
EndPackage[]

\(\(BeginPackage["CLASSIC3D`"]\)\(				\)\)
     Begin["`Private`"]
\(\(\ \ \ \ \ \ \ \ \)\($DefaultFont\  = \ {"Helvetica-Bold", \ 10};\)\)
           q  =  0.1   
        w  =  
        Pi*10^(-9)            
\(\(\ \ \ \ \ \ \ \ \)\(W\  = \ q*\(w/2\)/Sqrt[2]\)\(	\)\( (*\ 
        Eq . \ \((12)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
        Do[
	         Show[
		            Graphics3D[{PointSize[0.03], Point[{Cos[W*t*10^9/3]*
 (1 + (q/2)*
                                        Cos[w*t*10^9/3])/(1 + q/2),Cos[W*t*10^9/3]*
 (1 + (q/2)*
                                        Cos[w*t*10^9/3])/(1 + q/2),\(Cos[\(-W\)*
                                        t*10^9/
                                        3]*
					\ \ \ \ \ \ \ \((1 - \((q/2)\)*Cos[w*t*10^9/3])\)/\((1 - q/2)\)\)}]}, Ticks -> True, BoxRatios -> {1, 1, 1},
               Boxed -> True, AspectRatio -> 1, \(AxesLabel -> {"x", \ "y", \ "z"}\)], Axes -> True, 
                  FaceGrids -> All, PlotRange -> {{-1.5, 1.5}, {-1.5, 
                        1.5}, {-1.5, 1.5}}], {t, 0, 170, 2}
             ];
                     
        End[]
EndPackage[]

\(\(BeginPackage["wf0`"]\)\(	\)\)
     Begin["`Private`"]
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(q = 0.4\)\(\ 	\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(m = 1836.152\)\(		\)\( (*\ 
        Proton\ mass\ in\ a . u . \ \ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(w = 
        Pi*10^\((\(-9\))\)\)\(		\)\( (*\ 
        Radio\ frequency\ of\ the\ micro - \(\(motion\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(W\  = \ 
        w*\(q/2\)/Sqrt[2]\)\(	\)\( (*\ 
        Eq . \ \((12)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(f[t_]\  = \ 
        Exp[I*W*\(t/2.419\)/10^\(-9\)]*\((1 + \((q/2)\)*
                Cos[w*\(t/2.419\)/
                      10^\(-9\)])\)\)\(
\)\(\ \ \ \ \ 			\ \)\( (*\ 
        Eq . \ \((15)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(cf[t_]\  = \ 
        Exp[\(-I\)*
              W*\(t/2.419\)/10^\(-9\)]*\((1 + \((q/2)\)*
                Cos[w*\(t/2.419\)/
                      10^\(-9\)])\)\)\(
\)\(			\ \ \ \ \ \ \)\( (*\ 
        The\ complex\ conjugate\ of\ Eq . \ \((15)\) . \ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(X[t_]\  = \ \((m/4)\)*q*w*
          Sin[w\ \(t/2.419\)/10^\(-9\)]/\((1 + \((q/2)\)*
                  Cos[w\ \(t/2.419\)/
                        10^\(-9\)])\)\)\(
\)\(			\ \ \ \ \ \ \)\( (*\ 
        Eq . \ \((14)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(WF[n_, \ x_, \ t_]\  := \ 
        Exp[\(-I\)\ n*
              W*\(t/2.419\)/10^\(-9\)]*\((\((\((m*w/Pi)\)^\((1/4)\))\)/
              Sqrt[\((2^n)\)*
\ \ \ \ \ \ \ \ \ \ \ \ \ Factorial[n]])\)*
          HermiteH[n, \ 
            Sqrt[m*W]*x/Abs[f[t]]]*
\ \ \ \ \ \ \ \ \ \ \ \ \ Exp[
            I*X[t]*x*x]*
          Exp[\(-m\)*W*
                x*\(\(x/Abs[f[t]]\)/Abs[f[t]]\)/2]/
\ \ 			\ \ \ \ \ Sqrt[
              Abs[f[t]]]\)\(\ \ \ \)\( (*\ 
        Eq . \ \((13)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
             Do[
                    Show[
		                 \(Plot[
                0.015 + 
                  30*WF[0, x, t]*Conjugate[WF[0, x, t]], {x, \(-10000\), 
                  10000}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotRange -> {\(-0.05\), 0.07}, \ 
                AxesOrigin -> {0, 
                    0.015}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Ticks -> None, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel -> {"x", ""}, DisplayFunction -> 
                  Identity], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plot[\(-Cos[w*\(t/2.419\)/10^\(-9\)]\)*x*
                  x*15*10^\(-10\), 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-10000\), 
                  10000}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> Identity], 
              PlotLabel -> "The Probability Density 
                        of the Ground State
   of a
                        Trapped Charged Particle (n=0)", \ 
              TextStyle -> {FontFamily -> "Times", 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ FontSize -> 
                    10}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> $DisplayFunction\)
	          ],
                {t, 1, 36}
          ]
	      (* 
          Ploting The Probability Density of the Ground State of a *) 
	          
      End[]
EndPackage[]

BeginPackage["wf1`"]
        Begin["`Private`"]
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(q = 0.4\)\(	\)\( (*\ 
        Eq . \ \((3)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
                m = 
        1836.152       
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(w = 
        Pi*10^\((\(-9\))\)\)\(	\)\( (*\ 
        Radio\ frequency\ of\ the\ micro - motion\ in\ a . u . \ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(W\  = \ 
        w*\(q/2\)/Sqrt[2]\)\(	\)\( (*\ 
        Eq . \ \((12)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(f[t_]\  = \ 
        Exp[I*W*\(t/2.419\)/10^\(-9\)]*\((1 + \((q/2)\)*
                Cos[w*\(t/2.419\)/
                      10^\(-9\)]\ )\)\)\(
\)\(			\ \ \ \ \)\( (*\ 
        Eq . \ \((15)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(cf[t_]\  = \ 
        Exp[\(-I\)*
              W*\(t/2.419\)/10^\(-9\)]*\((1 + \((q/2)\)*
                Cos[w*\(t/2.419\)/
                      10^\(-9\)]\ )\)\)\(
\)\(			\ \ \ \ \)\( (*\ 
        The\ complex\ conjugate\ of\ Eq . \ \((15)\) . \ *) \)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(X[t_]\  = \ \((m/4)\)*q*
          w*Sin[w\ \(t/2.419\)/10^\(-9\)]/\((1 + \((q/2)\)*
                  Cos[w\ \(t/2.419\)/
                        10^\(-9\)]\ )\)\)\(
\)\(			\ \ \ \ \)\( (*\ 
        Eq . \ \((14)\)\ in\ the\ \(\(text\)\(.\)\)\ *) \)\)
            \(WF[n_, \ x_, \ t_]\  := \ 
          Exp[\(-I\)\ n*
                W*\(t/2.419\)/10^\(-9\)]*\((\((\((m*w/Pi)\)^\((1/4)\))\)/
                Sqrt[\((2^n)\)*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Factorial[
                      n]])\)*HermiteH[n, \ 
              Sqrt[m*W]*x/Abs[f[t]]]*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Exp[
              I*X[t]*x*x]*
            Exp[\(-m\)*W*
                  x*\(\(x/Abs[f[t]]\)/Abs[f[t]]\)/
                    2]/
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sqrt[Abs[f[t]]]\)     
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(Do[
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Show[
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plot[0.03 + 30*WF[1, x, t]*Conjugate[WF[1, x, t]], {x, \(-10000\), 
              10000}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotRange -> {\(-0.065\), 0.085}, \ 
            AxesOrigin -> {0, 
                0.03}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Ticks -> 
              None, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel -> {"x", ""}, 
            DisplayFunction -> 
              Identity], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plot[\(-2\)*Cos[w*\(t/2.419\)/10^\(-9\)]*x*
              
              x*15*10^\(-10\), 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-10000\), 
              10000}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> Identity], 
          PlotLabel -> "The Probability Density of the First
                         Excited State

                         of a Trapped Charged Particle (n=1)", \ 
          TextStyle -> {FontFamily -> "Times", 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ FontSize -> 
                10}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction -> $DisplayFunction
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 1, 
          36}
\ \ \ \ \ \ \ \ \ \ \ \ ]\)\(
\)\(	\ \ \ \ \ \)\( (*\ 
        Ploting\ The\ Probability\ Density\ of\ the\ First\ Excited\ State\ of\ a\ *) \)\(
\)\(	\ \ \ \ \ \ \ \ \ \)\( (*\ 
        Trapped\ Charged\ Particle\ \((n = 
              1)\)\ over\ a\ period\ of\ \(\(time\)\(.\)\)\ \ \ \ *) \)\)
        End[]  
EndPackage[]

BeginPackage["squeezed`"]
Begin["`Private`"]
           \(WF[n_, \ 
            x_]\  := \ \((\((\((m*W/Pi)\)^\((1/
                        4)\))\)/\((2^\((n/
                          2)\)*\((\((\(n!\))\)^\((1/
                            2)\))\))\))\)*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ HermiteH[n, \ Sqrt[m*W]*x]*
            Exp[\(-m\)*W*x*x/2]\)
                
          InitialW[x_]  :=  ((m*Wp/Pi)^(1/4))*
            Exp[I*kp*x - m*Wp*((x - x0)^2)/2]
                 
          \(PowerContract[expr_]\  := \ 
          expr\  //. \ {\ 
              m_^q_\ n_^q_\  :> \ \((m\ n)\)^
                    q\  /; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(! IntegerQ[m]\)\  && \ \(! IntegerQ[n]\), \ 
              m_^q_\ n_^p_\  :> \ \((m/n)\)^
                    q\  /; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ q\  >= \ 
                      0\  && \ p\  == \ \(-q\)\  && \ \(! IntegerQ[
                        m]\)\  && \ \(! IntegerQ[n]\)\ }\)
           
           \(cintegrand\  = \ 
          WF[n, \ x]\ InitialW[x]\ \  /. 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E^
                  r_\  :> \ E^
                  Map[Factor, Collect[r, x]]\  // \ \ PowerContract\)
           
            \(integrateRule\  = \ 
          E^\((c_\  + \ b_ \((x - y_)\)^2)\)\ HermiteH[n, 
                a_\ x]\  :> 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E^c\ Sqrt[
                Pi/\((\(-b\))\)]\ Sqrt[1 + a^2/b]^
                n\ *
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ HermiteH[n, 
                a\ y/Sqrt[Together[1 + a^2/b]]]\)
        
             \(completeSquare[expr_, x_]\  := \ 
          expr\  /. \ a_. \ x^2\  + \ b_. \ x\  + \ 
                c_. \  :> 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a \((x\  + \ b/\((2  a)\))\)^2\  + \ 
                Together[b^2/\((\(-4\) a)\)\  + \ c]\)
         
             \(c[
            n_]\  = \ \(\(\(completeSquare[cintegrand, x]\  /. 
                    integrateRule\  // 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PowerExpand\)\  // PowerContract\)\  // \ 
              MapAll[Together, #] &\)\  // 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ MapAll[Collect[#, \(n!\)] &, #] &\) 
                   
             \(WF[n_, \ x_, \ 
            t_]\  := \ \((\((\((m*W/Pi)\)^\((1/
                        4)\))\)/\((2^\((n/
                          2)\)*\((\((\(n!\))\)^\((1/
                            2)\))\))\))\)*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Exp[\(-I\)*n*W*t]*
            HermiteH[n, \ 
              Sqrt[m*W]*
                x/Abs[f[
                      t]]]*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Exp[I*X[t]*x*x]*
            Exp[\(-m\)*W*
                  x*\(\(x/Abs[f[t]]\)/Abs[f[t]]\)/
                    2]/
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sqrt[Abs[f[t]]]\)
         
             \(psi[n_, x_, 
            t_]\  = \ \(\(c[n]\ WF[n, x, 
                      t]\  /. \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1/\((Sqrt[n_]\ Sqrt[f_\ n_])\)\  -> \ 
                    1/\((n\ Sqrt[
                            f])\)\)\  /. 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sqrt[f_/m]\  :> \ 
                  Sqrt[ExpandAll[f]/
                      m]\)\  /. 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E^
                  r_\  :> \ E^Expand[r]\  // \ PowerExpand\) 
                   
             \(sumRule\  = \ \((\ 
                f_\ E^c_\ Sqrt[2]\ z^n\ 2^\(-n\)/\(n!\)\ HermiteH[n, 
                      x_]\ HermiteH[n, 
                      y_]\  /. 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z\  -> \ E^a_\ Sqrt[
                        zp_]\ )\)\  -> 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ 
                f\ E^c\ Sqrt[
                      2]\ E^\((\((\(-x^2\)\  - \ y^2\  + \ 
                                2\ x\ y\ z)\)/\((1 - 
                                z^2)\))\)/
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sqrt[1 - z^2]\ E^\((x^2\  + \ y^2)\)\  /. \ 
                  z\  -> \ E^a\ Sqrt[zp]\ )\)\  // \ PowerExpand\)
                     
             \(psi[x_, 
            t_]\  = \ \(psi[n, x, 
                  t]\  /. \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\(-W_\) + Wp_)\)^\((n/2)\)\ \((W_ + Wp_)\)^\((\(-1\)/2 - 
                          n/2)\)\  :> \ \((\((\(-W\) + Wp)\)/\((W + 
                              Wp)\))\)^\((n/2)\)\ \((W + Wp)\)^\((\(-1\)/
                          2)\)\)\  /. \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sumRule\  // \ \(PowerExpand\  /. \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sqrt[a_]\  :> \ 
                  PowerExpand[
                    Sqrt[Factor[
                        a]]]\)\  /. 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1/Sqrt[a_]\  :> \ 1/Sqrt[Together[a]]\)  
                  
            w =  W*2*Sqrt[2]/q	  
            f[t_]  =  
          Exp[I*W*t/2.419/
                  10^-9]*(1 + (q/2)*
                    Cos[w*t/2.419/10^-9])/(1 + (q/2))
		    	  
            X[t_]  =  (m/4)*q*w*
            Sin[w t/2.419/10^-9]/(1 + (q/2)*
                    Cos[w t/2.419/10^-9])
			     
            $DefaultFont  =  {"Helvetica-Bold",  10};
Needs["Graphics`Graphics3D`"]
\(\(Show[
\ \ \ \ StackGraphics[
\ \ \ \ \ \ \ \ \ \ \ \ Table[
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plot[
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Evaluate[Conjugate[psi[x, t/1.5]]*
                      psi[x, t/
                          1.5]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  /. \ {q -> 0.4, m -> 1836, 
                      W -> 10^\((\(-4\))\)*\(0.4/2\)/
                            Sqrt[2], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ kp -> \ 500, \ Wp -> 0.40*10^\((\(-4\))\)*\(0.4/2\)/Sqrt[2], 
                      x0 -> \(-70\)}], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-100\), 
                  100}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ DisplayFunction\  -> \ 
                  Identity
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 900, 
                0, \(-90\)}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\ \ \ \ \ \ \ \ ], 
\ \ \ \ \ \ \ \ BoxRatios\  -> \ {1, 1, 0.5}, \ Boxed\  -> \ False, 
          PlotRange -> 
            All, 
\ \ \ \ \ \ \ \ Axes\  -> \ {Automatic, Automatic, 
              None}, \ 
          AxesLabel\  -> \ {"x", "t", ""}
];\)\(
\)\( (*\ 
        Plotting\ the\ squeezed\ states\ over\ a\ period\ of\ \(\(time\)\(.\)\)\ *) \)\)
End[]
EndPackage[]