Recent changes to Demos.OscillatorChain.TDSE

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 15:35:53 -0000

--- v22
+++ v23
@@ -42,7 +42,7 @@

 In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets. In general, it can be expected that the vibrationally displaced states require larger basis sets than those used for [our calculations of the vibrational ground states](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/).

-Note also the behavior of the total energies: In our classical simulations, the initial displacement of one particle by 20 dimensionless units corresponds to an energy of exactly 6E-4 which is well conserved during our 50 time steps.  In contrast, the quantum simulations display a total energy of 5.796E-3 which is also  well conserved during our 50 time steps. The difference of 5.196E-3 is essentially attributed to the zero point energy which was found to be 5.032E-3, see [here](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/). 
+Note also the behavior of the total energies: In our classical simulations of the cyclic hexamer, the initial displacement of one particle by 20 dimensionless units corresponds to an energy of exactly 6E-4 which is well conserved during our 50 time steps.  In contrast, the quantum simulations of the same hexamer yield a total energy of 5.796E-3 which is also  well conserved during our 50 time steps. The difference of 5.196E-3 is essentially attributed to the zero point energy which was found to be 5.032E-3, see  [our calculations of the vibrational ground states](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/).

 Scaling behavior
 -------

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 12:57:04 -0000

--- v21
+++ v22
@@ -40,7 +40,7 @@
 Comparison
 ----------------

-In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets.In general, the vibrational displacements may require larger basis sets than those used for [our calculations of the vibrational ground states](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/)
+In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets. In general, it can be expected that the vibrationally displaced states require larger basis sets than those used for [our calculations of the vibrational ground states](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/).

 Note also the behavior of the total energies: In our classical simulations, the initial displacement of one particle by 20 dimensionless units corresponds to an energy of exactly 6E-4 which is well conserved during our 50 time steps.  In contrast, the quantum simulations display a total energy of 5.796E-3 which is also  well conserved during our 50 time steps. The difference of 5.196E-3 is essentially attributed to the zero point energy which was found to be 5.032E-3, see [here](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/).

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 12:53:46 -0000

--- v20
+++ v21
@@ -40,9 +40,9 @@
 Comparison
 ----------------

-In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets and/or going to shorter time steps.
+In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets.In general, the vibrational displacements may require larger basis sets than those used for [our calculations of the vibrational ground states](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/)

-Note, however, that the total energies ...
+Note also the behavior of the total energies: In our classical simulations, the initial displacement of one particle by 20 dimensionless units corresponds to an energy of exactly 6E-4 which is well conserved during our 50 time steps.  In contrast, the quantum simulations display a total energy of 5.796E-3 which is also  well conserved during our 50 time steps. The difference of 5.196E-3 is essentially attributed to the zero point energy which was found to be 5.032E-3, see [here](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.TISE/). 

 Scaling behavior
 -------

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 12:10:27 -0000

--- v19
+++ v20
@@ -37,6 +37,13 @@
 * [Input script](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.py)
 * [Output file](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.txt)

+Comparison
+----------------
+
+In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets and/or going to shorter time steps.
+
+Note, however, that the total energies ...
+
 Scaling behavior
 -------

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 10:26:58 -0000

--- v18
+++ v19
@@ -1,4 +1,4 @@
- Wavepacket dynamics of a chain of oscillators
+ Wavepacket dynamics of chains of oscillators
 =============

 For the case of homogeneous chains of harmonic oscillators, we study the quantum-mechanical propagation of a vibrational excitation initially localized at the central site of the chain. These simulations shall serve here as a testing ground for the various propagation schemes implemented in WavePacket and/or in WaveTrain.

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 12 Jan 2023 10:22:00 -0000

--- v17
+++ v18
@@ -1,12 +1,12 @@
  Wavepacket dynamics of a chain of oscillators
 =============

-For the case of a homogeneous chain of harmonic oscillators, we study the quantum-mechanical propagation of a vibrational excitation initially localized at the central site of the chain. These simulations shall serve here as a testing ground for the various propagation schemes implemented in WavePacket and/or in WaveTrain. 
+For the case of homogeneous chains of harmonic oscillators, we study the quantum-mechanical propagation of a vibrational excitation initially localized at the central site of the chain. These simulations shall serve here as a testing ground for the various propagation schemes implemented in WavePacket and/or in WaveTrain. 

 Quantum simulations
 ----------------------

-For the example of a cyclic homogeneous hexamer (N=6) we initialize our propagations with one particle in a coherent state (displacement of 20 dimensionless units) and all other particles in their respective vibrational ground states (zero displacements). 
+For the example of a cyclic homogeneous hexamer (N=6) we initialize our propagations with one particle (at|near the center) in a coherent state (displacement of 20 dimensionless units) and all other particles in their respective vibrational ground states (zero displacements). 

 

 [[img src=http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/wave_reduced_1d.jpg width=500 align=right]]
@@ -31,7 +31,7 @@
 

 [[img src=http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic_000050.png width=500 align=right]]

-For the quantum dynamics of the chain of oiscillators investigated here, reference data for assessing our conventional or tensor train-based results can be easily generated on the basis of the quantum-classical correspondence. According to the Ehrenfest theorem, the quantum-mechanical expectation values of positions and momenta coincide with results from classical trajectories, as long as the vibrational Hamiltonian is a polynomial of order not higher than two. This is indeed the case for our [vibrational model Hamiltonian](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.Main).
+For the quantum dynamics of the chains of oscillators investigated here, reference data for assessing our conventional or tensor train-based results can be easily generated on the basis of the quantum-classical correspondence. According to the Ehrenfest theorem, the quantum-mechanical expectation values of positions and momenta coincide with results from classical trajectories, as long as the vibrational Hamiltonian is a polynomial of order not higher than two. This is indeed the case for our [vibrational model Hamiltonian](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.Main).

 * [Animated simulation](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.mp4)
 * [Input script](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.py)

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 11 Jan 2023 16:02:03 -0000

--- v16
+++ v17
@@ -26,9 +26,8 @@
 * [Input script](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/tdse_cyclic.py)
 * [Output file](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/tdse_cyclic.txt)

-Classical simulation
+Classical simulations
 ---------------
-
 

 [[img src=http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic_000050.png width=500 align=right]]

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 11 Jan 2023 16:00:21 -0000

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 11 Jan 2023 15:58:39 -0000

--- v14
+++ v15
@@ -32,7 +32,7 @@
 

 [[img src=http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic_000050.png width=500 align=right]]

-bla bla bla
+For the quantum dynamics of the chain of oiscillators investigated here, reference data for assessing our conventional or tensor train-based results can be easily generated on the basis of the quantum-classical correspondence. According to the Ehrenfest theorem, the quantum-mechanical expectation values of positions and momenta coincide with results from classical trajectories, as long as the vibrational Hamiltonian is a polynomial of order not higher than two. This is indeed the case for our [vibrational model Hamiltonian](https://sourceforge.net/p/wavepacket/wiki/Demos.OscillatorChain.Main).

 * [Animated simulation](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.mp4)
 * [Input script](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.py)

Demos.OscillatorChain.TDSE modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 11 Jan 2023 15:42:00 -0000

--- v13
+++ v14
@@ -29,7 +29,14 @@
 Classical simulation
 ---------------

-to be done
+

+[[img src=http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic_000050.png width=500 align=right]]
+
+bla bla bla
+
+* [Animated simulation](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.mp4)
+* [Input script](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.py)
+* [Output file](http://wavepacket.sourceforge.net/Demos/OscillatorChain/TDSE/Cyclic/ceom_cyclic.txt)

 Scaling behavior
 -------