Difference between revisions of "Dynamically Shifted Oscillator"

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 +
= Varying ξ<sub>0</sub> =
 +
==Lowest Eigenvalues ==
 +
=== ξ<sub>0</sub> < 0 ===
 +
[[File:DSOLowestFiveEigenvaluesVaryingNegativeShifts.png|400px]]
  
 +
=== ξ<sub>0</sub> > 0 ===
 +
[[File:DSOLowestFiveEigenvaluesVaryingPositiveShifts.png|400px]]
  
−
= ξ<sub>0</sub> = 1 =
+
==Lowest Eigenfunctions ==
 +
=== ξ<sub>0</sub> < 0 ===
 +
[[File:DSOLowestEigenvalueEigenfunctionVaryingNegativeShifts.png|400px]]
 +
 
 +
=== ξ<sub>0</sub> > 0 ===
 +
[[File:DSOLowestEigenvalueEigenfunctionVaryingPositiveShifts.png|400px]]
 +
 
 +
= ξ<sub>0</sub> = -1 =
 +
[[File:DSOPotentialShiftMinusOne.png|400px]]
 
== Lowest Eigenvalues ==
 
== Lowest Eigenvalues ==
  
−
===Discretized Hamiltonian===
 
 
{|border="1"
 
{|border="1"
−
! ''N''
+
! Method
 +
! Δ''x''
 
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
 
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
 
|-
 
|-
−
| 1000
+
| Discretized Hamiltonian
−
| 0.3094688 
+
| 0.010
−
| 0.73422037
+
| 0.3095
−
| 1.49997346
+
| 0.7342
−
| 2.19740304
+
| 1.5000
−
| 2.99866356
+
| 2.1974
 +
| 2.9987
 +
|-
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| Euler-Cromer
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| 0.001
 +
| 0.3095
 +
| 0.734
 +
| 1.4997
 +
| 2.197
 +
|
 +
|-
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| Runge-Kutta 45
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| 0.010
 +
|
 +
|
 +
| 1.496
 +
|
 +
| 3.037
 
|}
 
|}
  
−
===Wag the Dog===
+
==Lowest Eigenfunctions==
 +
 
 
{|border="1"
 
{|border="1"
−
!''N''
+
! rowspan="2" | Method
−
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
+
! colspan="5" | ''n''  
 
|-
 
|-
−
| 1000
+
| 0
−
| --
+
| 1
−
| --
+
| 2
−
| 1.496
+
| 3
 +
| 4
 +
|-
 +
| Discretized Hamiltonian
 +
| [[File:DSOWaveFunctionDiscreteMatrix0.png|200px]]
 +
| [[File:DSOWaveFunctionDiscreteMatrix1.png|200px]]
 +
| [[File:DSOWaveFunctionDiscreteMatrix2.png|200px]]
 +
| [[File:DSOWaveFunctionDiscreteMatrix3.png|200px]]
 +
| [[File:DSOWaveFunctionDiscreteMatrix4.png|200px]]
 +
|-  
 +
| Euler-Cromer
 +
| [[File:DSOWaveFunctionEulerCromer0.png|200px]]
 +
| [[File:DSOWaveFunctionEulerCromer1.png|200px]]
 +
| [[File:DSOWaveFunctionEulerCromer2.png|200px]]
 +
| [[File:DSOWaveFunctionEulerCromer3.png|200px]]
 
| --
 
| --
 +
|-
 +
| Power Series
 +
| [[File:DSOWaveFunctionPowerSeries0.png|200px]]
 +
| [[File:DSOWaveFunctionPowerSeries1.png|200px]]
 +
| [[File:DSOWaveFunctionPowerSeries2.png|200px]]
 +
| [[File:DSOWaveFunctionPowerSeries3.png|200px]]
 +
| [[File:DSOWaveFunctionPowerSeries4.png|200px]]
 +
|}
 +
 +
= ξ<sub>0</sub> = +1 =
 +
[[File:DSOPotentialShiftPlusOne.png|400px]]
 +
 +
== Lowest Eigenvalues ==
 +
 +
{|border="1"
 +
! Method
 +
! Δ''x''
 +
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
 +
|-
 +
| Discretized Hamiltonian
 +
| 0.001
 +
| 1.500
 
| 3.037
 
| 3.037
 +
| 4.329
 +
| 5.604
 +
| 6.818
 +
|-
 +
| Euler-Cromer
 +
| 0.001
 +
| 1.499
 +
| 3.038
 +
| 4.328
 +
| 5.60
 +
|
 +
|-
 +
| Runge-Kutta 45
 +
| 0.010
 +
|
 +
|
 +
|
 +
|
 +
|
 
|}
 
|}
  
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|-
 
|-
 
| Discretized Hamiltonian
 
| Discretized Hamiltonian
−
| [[File:DSOWaveFunction_DiscreteMatrix_n0.png]]
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix0.png|200px]]
−
| [[File:DSOWaveFunction_DiscreteMatrix_n1.png]]
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix1.png|200px]]
−
| [[File:DSOWaveFunction_DiscreteMatrix_n2.png]]
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix2.png|200px]]
−
| [[File:DSOWaveFunction_DiscreteMatrix_n3.png]]
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix3.png|200px]]
−
| [[File:DSOWaveFunction_DiscreteMatrix_n4.png]]
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix4.png|200px]]
 +
|-
 +
| Euler-Cromer
 +
| [[File:DSOWaveFunctionShiftPlusOneEulerCromer0.png|200px]]
 +
| [[File:DSOWaveFunctionShiftPlusOneEulerCromer1.png|200px]]
 +
| [[File:DSOWaveFunctionShiftPlusOneEulerCromer2.png|200px]]
 +
| [[File:DSOWaveFunctionShiftPlusOneEulerCromer3.png|200px]]
 +
| --
 +
|-
 +
| Power Series
 +
|
 +
| -
 +
|
 +
|
 +
|
 
|}
 
|}

Latest revision as of 09:52, 23 July 2023

Varying ξ0

Lowest Eigenvalues

ξ0 < 0

DSOLowestFiveEigenvaluesVaryingNegativeShifts.png

ξ0 > 0

DSOLowestFiveEigenvaluesVaryingPositiveShifts.png

Lowest Eigenfunctions

ξ0 < 0

DSOLowestEigenvalueEigenfunctionVaryingNegativeShifts.png

ξ0 > 0

DSOLowestEigenvalueEigenfunctionVaryingPositiveShifts.png

ξ0 = -1

DSOPotentialShiftMinusOne.png

Lowest Eigenvalues

Method Δx εi (ħω)
Discretized Hamiltonian 0.010 0.3095 0.7342 1.5000 2.1974 2.9987
Euler-Cromer 0.001 0.3095 0.734 1.4997 2.197
Runge-Kutta 45 0.010 1.496 3.037

Lowest Eigenfunctions

Method n
0 1 2 3 4
Discretized Hamiltonian DSOWaveFunctionDiscreteMatrix0.png DSOWaveFunctionDiscreteMatrix1.png DSOWaveFunctionDiscreteMatrix2.png DSOWaveFunctionDiscreteMatrix3.png DSOWaveFunctionDiscreteMatrix4.png
Euler-Cromer DSOWaveFunctionEulerCromer0.png DSOWaveFunctionEulerCromer1.png DSOWaveFunctionEulerCromer2.png DSOWaveFunctionEulerCromer3.png --
Power Series 200px 200px 200px 200px DSOWaveFunctionPowerSeries4.png

ξ0 = +1

DSOPotentialShiftPlusOne.png

Lowest Eigenvalues

Method Δx εi (ħω)
Discretized Hamiltonian 0.001 1.500 3.037 4.329 5.604 6.818
Euler-Cromer 0.001 1.499 3.038 4.328 5.60
Runge-Kutta 45 0.010

Lowest Eigenfunctions

Method n
0 1 2 3 4
Discretized Hamiltonian DSOWaveFunctionShiftPlusOneDiscreteMatrix0.png DSOWaveFunctionShiftPlusOneDiscreteMatrix1.png DSOWaveFunctionShiftPlusOneDiscreteMatrix2.png DSOWaveFunctionShiftPlusOneDiscreteMatrix3.png DSOWaveFunctionShiftPlusOneDiscreteMatrix4.png
Euler-Cromer DSOWaveFunctionShiftPlusOneEulerCromer0.png DSOWaveFunctionShiftPlusOneEulerCromer1.png DSOWaveFunctionShiftPlusOneEulerCromer2.png DSOWaveFunctionShiftPlusOneEulerCromer3.png --
Power Series -
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