Difference between revisions of "Dynamically Shifted Oscillator"

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(→‎ξ0 = +1)
(→‎Lowest Eigenvalues)
 
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= Varying ξ<sub>0</sub> =
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==Lowest Eigenvalues ==
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=== ξ<sub>0</sub> < 0 ===
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[[File:DSOLowestFiveEigenvaluesVaryingNegativeShifts.png|400px]]
  
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=== ξ<sub>0</sub> > 0 ===
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[[File:DSOLowestFiveEigenvaluesVaryingPositiveShifts.png|400px]]
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==Lowest Eigenfunctions ==
 +
=== ξ<sub>0</sub> < 0 ===
 +
[[File:DSOLowestEigenvalueEigenfunctionVaryingNegativeShifts.png|400px]]
 +
 +
=== ξ<sub>0</sub> > 0 ===
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[[File:DSOLowestEigenvalueEigenfunctionVaryingPositiveShifts.png|400px]]
  
 
= ξ<sub>0</sub> = -1 =
 
= ξ<sub>0</sub> = -1 =
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[[File:DSOPotentialShiftMinusOne.png|400px]]
 
== Lowest Eigenvalues ==
 
== Lowest Eigenvalues ==
  
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= ξ<sub>0</sub> = +1 =
 
= ξ<sub>0</sub> = +1 =
 +
[[File:DSOPotentialShiftPlusOne.png|400px]]
 +
 
== Lowest Eigenvalues ==
 
== Lowest Eigenvalues ==
  
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|-
 
|-
 
| Discretized Hamiltonian
 
| Discretized Hamiltonian
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| 0.010
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| 0.001
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|  
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| 1.500
−
|  
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| 3.037
−
|  
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| 4.329
−
|  
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| 5.604
−
|  
+
| 6.818
 
|-
 
|-
 
| Euler-Cromer  
 
| Euler-Cromer  
−
| 0.010
+
| 0.001
 
| 1.499
 
| 1.499
 
| 3.038
 
| 3.038
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|-
 
|-
 
| Discretized Hamiltonian
 
| Discretized Hamiltonian
−
|
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| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix0.png|200px]]
−
|
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| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix1.png|200px]]
−
|
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix2.png|200px]]
−
|
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| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix3.png|200px]]
−
|
+
| [[File:DSOWaveFunctionShiftPlusOneDiscreteMatrix4.png|200px]]
 
|-  
 
|-  
 
| Euler-Cromer
 
| Euler-Cromer
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| Power Series
 
| Power Series
 
|
 
|
−
|
+
| -
 
|
 
|
 
|
 
|
 
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|}
 
|}

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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