QHD-2, QHD-3 and QHD-4 decay from a metastable...

QHD-2, QHD-3 and QHD-4 decay

from a metastable potentialby José Luis Gómez-Muñoz

http://homepage.cem.itesm.mx/lgomez/quantum/

[email protected]

Introduction

Higher Order Quantized Hamilton Dynamics (QHD) is applied to the decay from a metastable potential. QHD gives an

approximation to the Heisenberg Equations of Motion (EOM). The QHD commands used in this document are included in

QUANTUM, which is a free Mathematica add-on that can be downloaded from


This tutorial shows how to use the QUANTUM Mathematica add-on to reproduce the results and graphs from Pahl and Prezhdo

in J. Chem Phys. Vol 116 No. 20, May 2002, Pages 8704-8712

http://homepage.cem.itesm.mx/lgomez/quantum/QHDHigherOrders.pdf

Load the Package

First load the Quantum`QHD` package. Write:

Needs["Quantum`QHD`"]

then press at the same time the keys ˜-Û to evaluate. Mathematica will load the package and print a welcome message:

Needs@"Quantum`QHD`"D

Quantum`QHD`

A Mathematica package for Quantized Hamilton

Dynamics approximation to Heisenberg Equations of Motion

by José Luis Gómez−Muñoz

based on the original idea of Kirill Igumenshchev

This add−on does NOT work properly with the debugger turned on. Therefore

the debugger must NOT be checked in the Evaluation menu of Mathematica.

Execute SetQHDAliases@D in order to use the keyboard to enter QHD objects

SetQHDAliases@D must be executed again in each new notebook that is created

In order to use the keyboard to enter quantum objects write:

SetQHDAliases[ ];

then press at the same time the keys ˜-Û to evaluate. Remember that SetQuantumAliases[ ] must be evaluated again in each

new notebook:

SetQHDAliases@D

ALIASES:

@ESCDon@ESCD ⋅ Quantum concatenation symbol

@ESCDtime@ESCD � Time symbol

@ESCDhb@ESCD — Reduced Planck's constant Hh barL@ESCDii@ESCD � Imaginary I symbol

@ESCDinf@ESCD ∞ Infinity symbol

@ESCD−>@ESCD → Option HRuleL symbol

@ESCDave@ESCD X�\ Quantum average template

@ESCDexpec@ESCD X�\ Quantum average template

@ESCDsymm@ESCD H�⋅�L� Symmetrized quantum product template

@ESCDcomm@ESCD P�,�T− Commutator template

@ESCDpo@ESCD H�L� Power template

@ESCDsu@ESCD �� Subscripted variable template

@ESCDposu@ESCD �� Power of a subscripted variable template

@ESCDfra@ESCD�

�Fraction template

@ESCDeva@ESCD �ê.8�→�,�→�< Evaluation HReplaceAllL template

SetQHDAliases@D must be executed again in each

new notebook that is created, only one time per notebook.

Commutation Relationships and Hamiltonian

This Hamiltoninan and commutation relationships are those used by Pahl and Prezhdo in their paper.

In order to enter the templates and symbols P�, �T−, ��, Â and Ñ you can either use the QHD palette (toolbar) or, if the com-

mand SetQHDAliases[] has been executed in this Mathematica notebook, then press the keys [ESC]comm[ESC], [ESC]su[ESC],

[ESC]ii[ESC] and [ESC]hb[ESC] respectively:

SetQuantumObject@q, pD;Pq, pT− = � ∗ —;

mh =p2

2 m+a2

2q2 +

a3

3q3

p2

2 m+q2 a2

2+q3 a3

3

Heisenberg Equations of Motion

The evolution of the expected value of an observable A in the Heisenberg representation is given by the equation of motion

(EOM):

� —d

dtXA\ = X@A, HD\

2 v8qhdhigher.nb

dtX \ X@ D\

The Heisenberg EOM can be calculated using the QHD-Mathematica command QHDEOM, as shown below. The first argument

specifies the QHD-order, which is set below to ¶ so that no QHD approximation is made in this first example. The second

argument is the observable, which is set to p in the example below. The third argument is the Hamiltonian, remember that it was

stored in variable mh above in this document. Finally options can be included, for example QHDHBarØ1 specifies that reduced

Planck’s constant (Dirac constant) will take the value of 1 (The Ø symbol can be entered by pressing the keys [ESC]->[ESC],

that is [ESC][MINUS][GREATERTHAN][ESC]). The output of QHDEOM can be stored in a variable; it is stored in meom

below:

meom = QHDEOM@∞, p, mhD

98QHDLabel, No closure was applied<, 9Xp\, −a2 Xq\ − a3 Yq2]==

The output of QHDEOM can be shown in a nicer format using QHDForm. The output meom that was stored above is formated

that way below:

QHDForm@meomD

No closure was applied

d Xp\d �

= −a2 Xq\ − a3 Yq2]

On the other hand, the command QHDDifferentialEquations formats the output of QHDEOM as a standard Mathematica

equation, as those that can be part of the input of standard Mathematica commands like DSolve and NDSolve. The output meom

that was stored above is formated that way below:

QHDDifferentialEquations@meomD

9Xp\′@�D � −a2 Xq\@�D − a3 Yq2]@�D=

A different symbol for time can be specified. The operator Ø can be entered pressing the keys

[ESC][MINUS][GREATERTHAN][ESC], see the equations below:

QHDDifferentialEquations@meom, QHDSymbolForTime → zD

9Xp\′@zD � −a2 Xq\@zD − a3 Yq2]@zD=

The EOM for Xp\ depends on Xq\ and Yq2] (please remember that YA2] is NOT the same as XA\2). Next, the EOMs of Xq\and Yq2] can generate new expectation values, such as Xp ⋅ q\s = Y p⋅q+q⋅p

2]. Regarding each expectation value as a dynami-

cal variable, we can generate a hierarchy of EOMs, which in general is an infinite hierarchy. We can obtain some of the EOM’s

of that infinite hierarchy using the QHD-Mathematica command QHDHierarchy, which has the same syntax as the command

QHDEOM that was used above. Likewise, the output of QHDHierarchy can be stored in a variable (mhie in the example below)

and shown using the command QHDForm below:

v8qhdhigher.nb 3

mhie = QHDHierarchy@∞, p, mhD;QHDForm@mhieD

Calculations were stopped at order QHDMaxOrder→2

d Xp\d �

= −a2 Xq\ − a3 Yq2]d Xq\d �

=Xp\m

d Xq2\d �

=2 Xpq\

�

m

d Xpq\�

d �=

Xp2\m

− a2 Yq2] − a3 Yq3]d Xp2\d �

= −2 a2 Xpq\�− 2 a3 Ypq2]

�

A EOM hierarchy can be shown in a graph using the command QHDGraphPlot on the output of QHDHierachy. Arrows point

from a first dynamical variable to a second dynamical variable that includes the first one in its EOM; please compare the table

above with the graph below:

QHDGraphPlot@mhieD

Xq\

Xp\

Yq2]

Xpq\�

Yp2]

Yq3]

Ypq2]

�

Calculations were stopped at order QHDMaxOrderØ2

QHD-2 Closure Approximation

4 v8qhdhigher.nb

QHDHierarchy stops calculating infinite hierarchies as specified by its option QHDMaxOrder, which takes a default value of 2.

Some of the dynamical variables (expectation values) in the truncated hierarchy will not have their equation of motion (EOM)

included, as it was shown above by the colors in the ouput of QHDGraphPlot. A closure procedure has to be applied in order to

approximate those dynamical variables in terms of those other variables that do have their EOMs included in the hierarchy. For

instance, the approximation

XABC\ºXAB\XC\+XAC\XB\+XBC\XA\-2XA\XB\XC\is used to approximate the third-order dynamical variables Yq3] and Ypq2]

� in terms of the firts and second order variables

Xp\, Xq\, Yp2], Yq2] and Xp ⋅ q\s = Y p⋅q+q⋅p

2], thus we obtain a second order QHD finite hierarchy of equations, QHD-2.

The QHD-2 hierarchy is obtained as the output of the command QHDHierarchy with the first argument set to 2. Same as above,

the second argument is the first variable in the hierarchy, the third argument is the hamiltonian, and optional arguments can be

given after that. The output of QHDHierarchy is stored in the variable mhier2, and it is shown using the QHDForm command

below:

mhier2 = QHDHierarchy@2, p, mhD;QHDForm@mhier2D

Closure procedure was applied to order 2

d Xp\d �

= −a2 Xq\ − a3 Yq2]d Xq\d �

=Xp\m

d Xq2\d �

=2 Xpq\

�

m

d Xpq\�

d �=

Xp2\m

+ 2 a3 Xq\3 − a2 Yq2] − 3 a3 Xq\ Yq2]d Xp2\d �

= 4 a3 Xp\ Xq\2 − 2 a3 Xp\ Yq2] − 2 a2 Xpq\�− 4 a3 Xq\ Xpq\

�

The closed QHD-2 hierarchy can be shown using the QHDGraphPlot command, compare the table above and the graph below:

v8qhdhigher.nb 5

QHDGraphPlot@mhier2D

Xq\

Xp\

Yq2]

Xpq\�

Yp2]


Solution of the QHD-2 Equations: Decay from Metastable Potential

Next we evaluate the hierarchy when the parameters tkes the numerical values used by Pahl and Prezhdo in their paper, and the

result of that evaluation is stored in the variable mnumhier2:

mnumhier2 = mhier2 ê. :m → 1, a2 → 1, a3 →1

2, — → 1>;

QHDForm@mnumhier2D


d Xp\d �

= −Xq\ −Xq2\2

d Xq\d �

= Xp\d Xq2\d �

= 2 Xpq\�

d Xpq\�

d �= Yp2] + Xq\3 − Yq2] −

3

2Xq\ Yq2]

d Xp2\d �

= 2 Xp\ Xq\2 − Xp\ Yq2] − 2 Xpq\�− 2 Xq\ Xpq\

�

6 v8qhdhigher.nb

The command QHDInitialConditionsTemplate generates an initial conditions template for the hierarchy:

QHDInitialConditionsTemplate@mnumhier2, 0D

9Xp\@0D � �, Xq\@0D � �, Yq2]@0D � �, XHp ⋅ qL�\@0D � �, Yp2]@0D � �=

Copy-paste the output of the previous command as input in the next one. Fill in the placeholders (É) with the appropiate initial

values. Those initial values can be numbers. On the other hand, in the calculation below they are symbols like po, and then these

symbols are evaluated at the desired numerical values. These initial conditions correspond to a Gaussian wavepacket, J. Chem

Phys. Vol 113 No. 20, May 2002, Pages 8704-8712


The initial conditions are stored in the variable minicond2 below:

minicond2 =

:Xp\@0D � po, Xq\@0D � qo, Yq2]@0D � qo2 +—

2 m ∗ ω, XHp ⋅ qL

�\@0D � po ∗ qo,

Yp2]@0D � po2 +— ∗ m ∗ ω

2> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → 1<

:Xp\@0D � 0, Xq\@0D � 0, Yq2]@0D �1

2, XHp ⋅ qL

�\@0D � 0, Yp2]@0D �

1

2>

The command QHDNDSolve takes as arguments the numerical version of the hierarchy (which was stored in the variable

mnumhier2), the initial conditions (which were stored in minicond2), the initial time and the final time. The output of this

command is the numerical solution of the differential equations, in the form of InterpolatingFunction objects. This output can be

used to plot (graph) the dynamical variables as functions of time, as it will be shown below in this document. The output is stored

in the variable msol2 in the calculation below:

msol2 = QHDNDSolve@mnumhier2, minicond2, 0, 6.4D

99Xp\ → InterpolatingFunction@880., 6.4<<, <>D,Xq\ → InterpolatingFunction@880., 6.4<<, <>D,Yq2] → InterpolatingFunction@880., 6.4<<, <>D,XHp ⋅ qL

�\ → InterpolatingFunction@880., 6.4<<, <>D,

Yp2] → InterpolatingFunction@880., 6.4<<, <>D==

The command command QHDPlot takes as its second argument the output of QHDNDSolve, which was stored in the variable

msol2. The first argument specifies the dynamical variables or expression that we want to plot (graph) as a function of time, see

the plot below:

v8qhdhigher.nb 7

QHDPlot@q, msol2D

0 1 2 3 4 5 6

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

QHDPlot accepts the same Options as the standard Mathematica command Plot. Some of those options are used, and the plot

stored in the variable mgraph2, it will be used later in this document, see the commands below:

mgraph2 =

QHDPlot@q, msol2, PlotStyle → Directive@Thick, DotDashed, Darker@MagentaDDD

0 1 2 3 4 5 6

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\


The QHD-3 hierarchy for this problem can be obtained with the first argument of QHDHierarchy set to 3. This calculation takes

several seconds in a laptop computer, the result is stored in the variable mhier3 below:

8 v8qhdhigher.nb



d Xp\d �

= −a2 Xq\ − a3 Yq2]d Xq\d �

=Xp\m

d Xq2\d �

=2 Xpq\

�

m

d Xpq\�

d �=

Xp2\m

− a2 Yq2] − a3 Yq3]d Xp2\d �

= −2 a2 Xpq\�− 2 a3 Ypq2]

�

d Xq3\d �

=3 Xpq2\

�

m

d Xpq2\�

d �= −6 a3 Xq\4 + 12 a3 Xq\2 Yq2] − 3 a3 Yq2]2 − a2 Yq3] − 4 a3 Xq\ Yq3] +

2 Xp2q\�

m

d Xp2q\�

d �=

Xp3\m

− 12 a3 Xp\ Xq\3 + 12 a3 Xp\ Xq\ Yq2] − 2 a3 Xp\ Yq3] +

12 a3 Xq\2 Xpq\�− 6 a3 Yq2] Xpq\

�− 2 a2 Ypq2]

�− 6 a3 Xq\ Ypq2]

�

d Xp3\d �

=—2 a3

2− 18 a3 Xp\2 Xq\2 + 6 a3 Yp2] Xq\2 + 6 a3 Xp\2 Yq2] − 3 a3 Yp2] Yq2] +

24 a3 Xp\ Xq\ Xpq\�− 6 a3 Xpq\�

2 − 3 a2 Yp2q]�− 6 a3 Xq\ Yp2q]

�− 6 a3 Xp\ Ypq2]

�

The products above are symmetrized, for instance Yp ⋅ q2]s= Z p⋅q2+q2⋅p

2^

The QHD-3 hierarchy can be shown using the QHDGraphPlot command, compare the table above and the graph below:

v8qhdhigher.nb 9


Xq\

Xp\

Yq2]

Xpq\�

Yp2]

Yq3]

Ypq2]�

Yp2q]�

Yp3]


Solution of the QHD-3 Equations



10 v8qhdhigher.nb


2, — → 1>;

QHDForm@mnumhier3D


d Xp\d �

= −Xq\ −Xq2\2

d Xq\d �

= Xp\d Xq2\d �

= 2 Xpq\�

d Xpq\�

d �= Yp2] − Yq2] −

Xq3\2

d Xp2\d �

= −2 Xpq\�− Ypq2]

�

d Xq3\d �

= 3 Ypq2]�

d Xpq2\�

d �= −3 Xq\4 + 6 Xq\2 Yq2] −

3 Xq2\22

− Yq3] − 2 Xq\ Yq3] + 2 Yp2q]�

d Xp2q\�

d �= Yp3] − 6 Xp\ Xq\3 + 6 Xp\ Xq\ Yq2] −

Xp\ Yq3] + 6 Xq\2 Xpq\�− 3 Yq2] Xpq\

�− 2 Ypq2]

�− 3 Xq\ Ypq2]

�

d Xp3\d �

=1

4− 9 Xp\2 Xq\2 + 3 Yp2] Xq\2 + 3 Xp\2 Yq2] −

3

2Yp2] Yq2] +

12 Xp\ Xq\ Xpq\�− 3 Xpq\

�

2 − 3 Yp2q]�− 3 Xq\ Yp2q]

�− 3 Xp\ Ypq2]

�



9Xp\@0D � �, Xq\@0D � �, Yq2]@0D � �, XHp ⋅ qL�\@0D � �, Yp2]@0D � �,

Yq3]@0D � �, YIp ⋅ q2M�]@0D � �, YIp2 ⋅ qM

�]@0D � �, Yp3]@0D � �=







v8qhdhigher.nb 11

minicond3 =


2 m ∗ ω,

XHp ⋅ qL�\@0D � po ∗ qo, Yp2]@0D � po2 +

— ∗ m ∗ ω

2, Yq3]@0D � qo3 +

3

2qo

—

m ∗ ω,

YIp ⋅ q2M�]@0D � po ∗ qo2 +

1

2po

—

m ∗ ω, YIp2 ⋅ qM

�]@0D � po2 ∗ qo +

1

2qo ∗ — ∗ m ∗ ω,

Yp3]@0D � po3 +3

2po ∗ — ∗ m ∗ ω> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → 1<

:Xp\@0D � 0, Xq\@0D � 0, Yq2]@0D �1

2, XHp ⋅ qL

�\@0D � 0, Yp2]@0D �

1

2,

Yq3]@0D � 0, YIp ⋅ q2M�]@0D � 0, YIp2 ⋅ qM

�]@0D � 0, Yp3]@0D � 0>






msol3 = QHDNDSolve@mnumhier3, minicond3, 0, 4.6D

99Xp\ → InterpolatingFunction@880., 4.6<<, <>D,Xq\ → InterpolatingFunction@880., 4.6<<, <>D,Yq2] → InterpolatingFunction@880., 4.6<<, <>D,XHp ⋅ qL

�\ → InterpolatingFunction@880., 4.6<<, <>D,

Yp2] → InterpolatingFunction@880., 4.6<<, <>D,Yq3] → InterpolatingFunction@880., 4.6<<, <>D,YIp ⋅ q2M

�] → InterpolatingFunction@880., 4.6<<, <>D,

YIp2 ⋅ qM�] → InterpolatingFunction@880., 4.6<<, <>D,

Yp3] → InterpolatingFunction@880., 4.6<<, <>D==



the plot below:

12 v8qhdhigher.nb

QHDPlot@q, msol3D

0 1 2 3 4

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

QHDPlot accepts the same Options as the standard Mathematica command Plot. Some of those options are used, and the plot

stored in the variable mgraph3, it will be used later in this document, see the commands below:

mgraph3 = QHDPlot@q, msol3, PlotStyle → Directive@Thick, Dashed, Darker@BrownDDD

0 1 2 3 4

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

The standard Mathematica command Show can be used to compare mgraph3, which was stored above, and mgraph2, which was

stored in a previous section. The paper by Pahl and Prezhdo shows that QHD-3 is a better approximation to Quantum Mechanics

than QHD-2, see the graph below:

v8qhdhigher.nb 13

Show@mgraph2, mgraph3D

0 1 2 3 4 5 6

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\


The QHD-4 hierarchy for this problem can be obtained with the first argument of QHDHierarchy set to 4. This calculation takes

several minutes in a laptop computer; the result is stored in the variable mhier4 below:

14 v8qhdhigher.nb



d Xp\d �

= −a2 Xq\ − a3 Yq2]d Xq\d �

=Xp\m

d Xq2\d �

=2 Xpq\

�

m

d Xpq\�

d �=

Xp2\m

− a2 Yq2] − a3 Yq3]d Xp2\d �

= −2 a2 Xpq\�− 2 a3 Ypq2]

�

d Xq3\d �

=3 Xpq2\

�

m

d Xpq2\�

d �= −a2 Yq3] − a3 Yq4] +

2 Xp2q\�

m

d Xq4\d �

=4 Xpq3\

�

m

d Xp2q\�

d �=

Xp3\m

− 2 a2 Ypq2]�− 2 a3 Ypq3]

�

d Xpq3\�

d �=

3 —2

2 m+ 24 a3 Xq\5 − 60 a3 Xq\3 Yq2] + 30 a3 Xq\ Yq2]2 +

20 a3 Xq\2 Yq3] − 10 a3 Yq2] Yq3] − a2 Yq4] − 5 a3 Xq\ Yq4] +3 Xp2q2\

�

m

d Xp3\d �

= −—2 a3 − 3 a2 Yp2q]�− 3 a3 Yp2q2]

�

d Xp2q2\�

d �= 48 a3 Xp\ Xq\4 − 72 a3 Xp\ Xq\2 Yq2] + 12 a3 Xp\ Yq2]2 + 16 a3 Xp\ Xq\ Yq3] −

2 a3 Xp\ Yq4] − 48 a3 Xq\3 Xpq\�+ 48 a3 Xq\ Yq2] Xpq\

�− 8 a3 Yq3] Xpq\

�+

2 Xp3q\�

m+ 24 a3 Xq\2 Ypq2]

�− 12 a3 Yq2] Ypq2]

�− 2 a2 Ypq3]

�− 8 a3 Xq\ Ypq3]

�

d Xp3q\�

d �= −

3

2—2 a2 +

Xp4\m

− 4 —2 a3 Xq\ + 72 a3 Xp\2 Xq\3 −18 a3 Yp2] Xq\3 − 54 a3 Xp\2 Xq\ Yq2] + 18 a3 Yp2] Xq\ Yq2] + 6 a3 Xp\2 Yq3] −

3 a3 Yp2] Yq3] − 108 a3 Xp\ Xq\2 Xpq\�+ 36 a3 Xp\ Yq2] Xpq\

�+

36 a3 Xq\ Xpq\�

2 + 18 a3 Xq\2 Yp2q]�− 9 a3 Yq2] Yp2q]

�+ 36 a3 Xp\ Xq\ Ypq2]

�−

18 a3 Xpq\�Ypq2]

�− 3 a2 Yp2q2]

�− 9 a3 Xq\ Yp2q2]

�− 6 a3 Xp\ Ypq3]

�

d Xp4\d �

= −4 —2 a3 Xp\ + 96 a3 Xp\3 Xq\2 − 72 a3 Xp\ Yp2] Xq\2 + 8 a3 Yp3] Xq\2 −24 a3 Xp\3 Yq2] + 24 a3 Xp\ Yp2] Yq2] − 4 a3 Yp3] Yq2] − 144 a3 Xp\2 Xq\ Xpq\

�+

48 a3 Yp2] Xq\ Xpq\�+ 48 a3 Xp\ Xpq\

�

2 + 48 a3 Xp\ Xq\ Yp2q]�− 24 a3 Xpq\

�Yp2q]

�−

4 a2 Yp3q]�− 8 a3 Xq\ Yp3q]

�+ 24 a3 Xp\2 Ypq2]

�− 12 a3 Yp2] Ypq2]

�− 12 a3 Xp\ Yp2q2]

�

The products above are symmetrized, for instance Yp ⋅ q2]s= Z p⋅q2+q2⋅p

2^

The QHD-4 hierarchy can be shown using the QHDGraphPlot command, compare the table above and the graph below:

v8qhdhigher.nb 15


Xq\

Xp\

Yq2]

Xpq\�

Yp2]

Yq3]

Ypq2]�

Yq4]

Yp2q]�

Ypq3]�

Yp3] Yp2q2]�

Yp3q]�

Yp4]


Solution of the QHD-4 Equations



16 v8qhdhigher.nb


2, — → 1>;

QHDForm@mnumhier4D


d Xp\d �

= −Xq\ −Xq2\2

d Xq\d �

= Xp\d Xq2\d �

= 2 Xpq\�

d Xpq\�

d �= Yp2] − Yq2] −

Xq3\2

d Xp2\d �

= −2 Xpq\�− Ypq2]

�

d Xq3\d �

= 3 Ypq2]�

d Xpq2\�

d �= −Yq3] −

Xq4\2

+ 2 Yp2q]�

d Xq4\d �

= 4 Ypq3]�

d Xp2q\�

d �= Yp3] − 2 Ypq2]

�− Ypq3]

�

d Xpq3\�

d �=

3

2+ 12 Xq\5 − 30 Xq\3 Yq2] + 15 Xq\ Yq2]2 +

10 Xq\2 Yq3] − 5 Yq2] Yq3] − Yq4] −5

2Xq\ Yq4] + 3 Yp2q2]

�

d Xp3\d �

= −1

2− 3 Yp2q]

�−

3

2Yp2q2]

�

d Xp2q2\�

d �= 24 Xp\ Xq\4 − 36 Xp\ Xq\2 Yq2] + 6 Xp\ Yq2]2 +

8 Xp\ Xq\ Yq3] − Xp\ Yq4] − 24 Xq\3 Xpq\�+ 24 Xq\ Yq2] Xpq\

�− 4 Yq3] Xpq\

�+

2 Yp3q]�+ 12 Xq\2 Ypq2]

�− 6 Yq2] Ypq2]

�− 2 Ypq3]

�− 4 Xq\ Ypq3]

�

d Xp3q\�

d �= −

3

2+ Yp4] − 2 Xq\ + 36 Xp\2 Xq\3 − 9 Yp2] Xq\3 − 27 Xp\2 Xq\ Yq2] +

9 Yp2] Xq\ Yq2] + 3 Xp\2 Yq3] −3

2Yp2] Yq3] − 54 Xp\ Xq\2 Xpq\

�+

18 Xp\ Yq2] Xpq\�+ 18 Xq\ Xpq\

�

2 + 9 Xq\2 Yp2q]�−

9

2Yq2] Yp2q]

�+

18 Xp\ Xq\ Ypq2]�− 9 Xpq\

�Ypq2]

�− 3 Yp2q2]

�−

9

2Xq\ Yp2q2]

�− 3 Xp\ Ypq3]

�

d Xp4\d �

= −2 Xp\ + 48 Xp\3 Xq\2 − 36 Xp\ Yp2] Xq\2 + 4 Yp3] Xq\2 −12 Xp\3 Yq2] + 12 Xp\ Yp2] Yq2] − 2 Yp3] Yq2] − 72 Xp\2 Xq\ Xpq\

�+

24 Yp2] Xq\ Xpq\�+ 24 Xp\ Xpq\

�

2 + 24 Xp\ Xq\ Yp2q]�− 12 Xpq\

�Yp2q]

�−

4 Yp3q]�− 4 Xq\ Yp3q]

�+ 12 Xp\2 Ypq2]

�− 6 Yp2] Ypq2]

�− 6 Xp\ Yp2q2]

�


v8qhdhigher.nb 17


9Xp\@0D � �, Xq\@0D � �, Yq2]@0D � �, XHp ⋅ qL�\@0D � �, Yp2]@0D � �, Yq3]@0D � �,

YIp ⋅ q2M�]@0D � �, Yq4]@0D � �, YIp2 ⋅ qM

�]@0D � �, YIp ⋅ q3M

�]@0D � �,

Yp3]@0D � �, YIp2 ⋅ q2M�]@0D � �, YIp3 ⋅ qM

�]@0D � �, Yp4]@0D � �=







minicond4 =


2 m ∗ ω,


— ∗ m ∗ ω

2,

Yq3]@0D � qo3 +3

2qo

—

m ∗ ω, YIp ⋅ q2M

�]@0D � po ∗ qo2 +

1

2po

—

m ∗ ω,

Yq4]@0D � qo4 + 3 qo2—

m ∗ ω+3

4

—

m ∗ ω

2

, YIp2 ⋅ qM�]@0D � po2 ∗ qo +

1

2qo ∗ — ∗ m ∗ ω,

YIp ⋅ q3M�]@0D � po ∗ qo3 +

3

2po ∗ qo ∗

—

m ∗ ω, Yp3]@0D � po3 +

3

2po ∗ — ∗ m ∗ ω,

YIp2 ⋅ q2M�]@0D � po2 ∗ qo2 +

1

2po2

—

m ∗ ω+qo2 ∗ — ∗ m ∗ ω

2−—2

4,

YIp3 ⋅ qM�]@0D � po3 ∗ qo +

3

2po ∗ qo ∗ — ∗ m ∗ ω,

Yp4]@0D � po4 + 3 po2 ∗ — ∗ m ∗ ω +3

4H— ∗ m ∗ ωL2> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → 1<

:Xp\@0D � 0, Xq\@0D � 0, Yq2]@0D �1

2, XHp ⋅ qL

�\@0D � 0, Yp2]@0D �

1

2, Yq3]@0D � 0,

YIp ⋅ q2M�]@0D � 0, Yq4]@0D �

3

4, YIp2 ⋅ qM

�]@0D � 0, YIp ⋅ q3M

�]@0D � 0,

Yp3]@0D � 0, YIp2 ⋅ q2M�]@0D � −

1

4, YIp3 ⋅ qM

�]@0D � 0, Yp4]@0D �

3

4>






18 v8qhdhigher.nb

msol4 = QHDNDSolve@mnumhier4, minicond4, 0, 4D

99Xp\ → InterpolatingFunction@880., 4.<<, <>D,Xq\ → InterpolatingFunction@880., 4.<<, <>D,Yq2] → InterpolatingFunction@880., 4.<<, <>D,XHp ⋅ qL

�\ → InterpolatingFunction@880., 4.<<, <>D,

Yp2] → InterpolatingFunction@880., 4.<<, <>D,Yq3] → InterpolatingFunction@880., 4.<<, <>D,YIp ⋅ q2M

�] → InterpolatingFunction@880., 4.<<, <>D,

Yq4] → InterpolatingFunction@880., 4.<<, <>D,YIp2 ⋅ qM


YIp ⋅ q3M�] → InterpolatingFunction@880., 4.<<, <>D,

Yp3] → InterpolatingFunction@880., 4.<<, <>D,YIp2 ⋅ q2M


YIp3 ⋅ qM�] → InterpolatingFunction@880., 4.<<, <>D,

Yp4] → InterpolatingFunction@880., 4.<<, <>D==



the plot below:

QHDPlot@q, msol4D

0 1 2 3 4

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\



the plot below:

v8qhdhigher.nb 19

mgraph4 = QHDPlot@q, msol4, PlotStyle → Darker@OrangeDD

0 1 2 3 4

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

The standard Mathematica command Show can be used to compare mgraph4, which was stored above, with mgraph3 and

mgraph2, which were calculated in previous sections. The paper by Pahl and Prezhdo shows that QHD-4 is a better approxima-

tion to Quantum Mechanics than QHD-3 and QHD-2, see the graph below:

Show@mgraph2, mgraph3, mgraph4D

0 1 2 3 4 5 6

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

Standard Mathematica options can be used so that the graph looks like the corresponding graph in Pahl and Prezhdo paper, see

the result below:

20 v8qhdhigher.nb

Show@mgraph2, mgraph3, mgraph4,

Axes → True, AxesOrigin → 80, −2<, PlotLabel →

"Decay from Metastable Potential\n Compare with Pahl and Prezhdo\n

J.Chem.Phys., Vol 116, No. 20, May 2002\n Pages 8704−8712 Fig 3"D

0 1 2 3 4 5 6

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

�

Xq\

Decay from Metastable Potential

Compare with Pahl and Prezhdo

J.Chem.Phys., Vol 116, No. 20, May 2002

Pages 8704-8712 Fig 3

Saving the QHD hierarchies for future use

The calculation QHD hierarchies is time consuming, therefore it is a good idea to save them for future use. The simplest way to

do it is using the standard Mathematica command Put. The QHD-2 hierarchy is stored in the file qhd2.m, see the command

below:

Put@mhier2, "qhd2.m"D

The QHD-3 hierarchy is stored in the file qhd3.m, see the command below:


The QHD-4 hierarchy is stored in the file qhd4.m, see the command below:


The command FileNames[“*.m”] can be used to see the files that were created by the command Put (together with any other

preexisting file with .m extension), see the command below:

v8qhdhigher.nb 21

FileNames@"∗.m"D

8hier4.m, HierarchyWater.m, qhd2.m, qhd3.m, qhd4.m, schrodingersolbarrier.m<

Dependence of Decay Time on Ñ

Next commands have been written as they would be used in a different Mathematica session, using the command Get to load the

hierarchy that was saved above (instead of calculating it again). Notice that the commutation relation and the Hamiltonian are

defined again, and its important to use those that correspond to the hierarchy that is loaded. Next commands calculate and graph

the dependence of decay time on Ñ, the result is stored in the variable data2, which will be used later to reproduce a graph of the

paper by Pahl and Prezhdo, see the graph below:

22 v8qhdhigher.nb

Needs@"Quantum`QHD`"D;SetQuantumObject@q, pD;Pq, pT− = � ∗ —;

mh =p2

2 m+a2

2q2 +

a3

3q3;

hhier2 = Get@"qhd2.m"D;iniguess = 15;

data2 = TableB

hnumhier2 = hhier2 ê. :m → 1, a2 → 1, a3 →1

2, — → h>;

hinicond2 =



�\@0D � po ∗ qo,

Yp2]@0D � po2 +— ∗ m ∗ ω

2> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → h<;

hsol2 = QHDNDSolve@hnumhier2, hinicond2, 0, iniguess + 0.5D;hqf2 = QHDFunction@q, hsol2D;htime = Hht ê. FindRoot@hqf2@htD � −2, 8ht, iniguess<,

AccuracyGoal → 3, PrecisionGoal → 3, MaxIterations → 1000DL;iniguess = htime;

8h, htime<,8h, 0.85, 2, 0.025<F;

ListLinePlot@data2, AxesLabel −> 8Style@—, LargeD, "crossing time τ"<D

1.2 1.4 1.6 1.8 2.0 Ñ

5

6

7

8

9

crossing time t

Below we have the same calculation as above, but with QHD-3 instead of QHD-2, using the “qhd3.m” file, and storing the result

in data3:

v8qhdhigher.nb 23


mh =p2

2 m+a2

2q2 +

a3

3q3;


data3 = TableB


2, — → h>;

hinicond3 = :Xp\@0D � po, Xq\@0D � qo, Yq2]@0D � qo2 +—

2 m ∗ ω,


— ∗ m ∗ ω

2, Yq3]@0D � qo3 +

3

2qo

—

m ∗ ω,

YIp ⋅ q2M�]@0D � po ∗ qo2 +

1

2po

—

m ∗ ω, YIp2 ⋅ qM

�]@0D � po2 ∗ qo +

1

2qo ∗ — ∗ m ∗ ω,

Yp3]@0D � po3 +3

2po ∗ — ∗ m ∗ ω> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → h<;



8h, htime<,8h, 0.6, 2, 0.025<F;


0.8 1.0 1.2 1.4 1.6 1.8 2.0 Ñ

4

5

6

7

crossing time t

Below we have the same calculation as above, but with QHD-4 instead of QHD-3 or QHD-2, using the “qhd4.m” file, and storing

the result in data4:

24 v8qhdhigher.nb


mh =p2

2 m+a2

2q2 +

a3

3q3;


data4 = TableB


2, — → h>;

hinicond4 = :Xp\@0D � po, Xq\@0D � qo,

Yq2]@0D � qo2 +—


�\@0D � po ∗ qo, Yp2]@0D � po2 +

— ∗ m ∗ ω

2,

Yq3]@0D � qo3 +3

2qo

—

m ∗ ω, YIp ⋅ q2M

�]@0D � po ∗ qo2 +

1

2po

—

m ∗ ω,

Yq4]@0D � qo4 + 3 qo2—

m ∗ ω+3

4

—

m ∗ ω

2

, YIp2 ⋅ qM�]@0D � po2 ∗ qo +

1

2qo ∗ — ∗ m ∗ ω,

YIp ⋅ q3M�]@0D � po ∗ qo3 +

3

2po ∗ qo ∗

—

m ∗ ω, Yp3]@0D � po3 +

3

2po ∗ — ∗ m ∗ ω,

YIp2 ⋅ q2M�]@0D � po2 ∗ qo2 +

1

2po2

—

m ∗ ω+qo2 ∗ — ∗ m ∗ ω

2−—2

4,

YIp3 ⋅ qM�]@0D � po3 ∗ qo +

3

2po ∗ qo ∗ — ∗ m ∗ ω, Yp4]@0D �

po4 + 3 po2 ∗ — ∗ m ∗ ω +3

4H— ∗ m ∗ ωL2> ê. 8qo → 0, po → 0, ω → 1, m → 1, — → h<;



8h, htime<,8h, 0.6, 2, 0.025<F;


v8qhdhigher.nb 25

0.8 1.0 1.2 1.4 1.6 1.8 2.0 Ñ

3.5

4.0

4.5

5.0

5.5

crossing time t

Standard Mathematica options can be used so that the graph looks like the corresponding graph in Pahl and Prezhdo paper, see

the result below:

ListLinePlot@8data2, data3, data4<, Frame → True,

FrameLabel → 8Style@—, LargeD, "crossing time τ"<,PlotRange → 880.6, 2<, 82, 10<<, PlotLabel →

"Decay from Metastable Potential\n Compare with Pahl and Prezhdo\n

J.Chem.Phys., Vol 116, No. 20, May 2002\n Pages 8704−8712 Fig 2"D

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.02

4

6

8

10

Ñ

cro

ssin

gti

met




Pages 8704-8712 Fig 2

Dependence of Decay Time on po

Next commands have been written as they would be used in a different Mathematica session, using the command Get to load the


26 v8qhdhigher.nb


defined again, and its important to use those that correspond to the hierarchy that is loaded. Next commands calculate and graph

the dependence of decay time on p0, the initial momentum, the result is used to reproduce part of a graph of the paper by Pahl

and Prezhdo, see the graph below:


mh =p2

2 m+a2

2q2 +

a3

3q3;

hhier2 = Get@"qhd2.m"D;


2, — → 1>;

iniguess = 1;

pdata2 = TableBhinicond2 =



�\@0D � po ∗ qo,

Yp2]@0D � po2 +— ∗ m ∗ ω

2> ê. 8qo → 0, ω → 1, m → 1, — → 1<;



8po, htime<,8po, −2, 5, 0.05<F;

ListLinePlot@pdata2, Frame → True, Axes → False,

FrameLabel → 8Style@p0, LargeD, "crossing time τ"<,PlotLabel → "Decay from Metastable

Potential\nCompare with Pahl and Prezhdo\nJ.Chem.Phys.,

Vol 116, No. 20, May 2002\nPages 8704−8712 Fig 4HbL"D

v8qhdhigher.nb 27

-2 -1 0 1 2 3 4 5

1

2

3

4

5

6

p0

cro

ssin

gti

met




Pages 8704-8712 Fig 4HbL

This tutorial shows how to use the QUANTUM Mathematica add-on to reproduce the results and graphs from Pahl and Prezhdo

in J. Chem Phys. Vol 116 No. 20, May 2002, Pages 8704-8712


by José Luis Gómez-Muñoz


[email protected]

28 v8qhdhigher.nb

QHD-2, QHD-3 and QHD-4 decay from a metastable...

Documents

Transcript of QHD-2, QHD-3 and QHD-4 decay from a metastable...