Difference between revisions of "Dynamically Shifted Oscillator"

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(→‎Lowest Eigenfunctions)
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−
= ξ<sub>0</sub> = 1 =
+
= ξ<sub>0</sub> = -1 =
 
== 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
−
 
+
−
===Wag the Dog===
+
−
{|border="1"
+
−
!''N''
+
−
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
+
 
|-
 
|-
−
| 1000
+
| Euler-Cromer
−
| --
+
| 0.001
−
| --
+
| 0.3095
−
| 1.496  
+
| 0.734
−
| --
+
| 1.4997
 +
| 2.197
 +
|
 +
|-
 +
| Runge-Kutta 45 (Δ''x'' = 0.010)
 +
| 0.010
 +
|
 +
|
 +
| 1.496
 +
|  
 
| 3.037
 
| 3.037
 
|}
 
|}
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| [[File:DSOWaveFunctionPowerSeries3.png|200px]]
 
| [[File:DSOWaveFunctionPowerSeries3.png|200px]]
 
| [[File:DSOWaveFunctionPowerSeries4.png|200px]]
 
| [[File:DSOWaveFunctionPowerSeries4.png|200px]]
 +
|}
 +
 +
= ξ<sub>0</sub> = +1 =
 +
== Lowest Eigenvalues ==
 +
 +
{|border="1"
 +
! Method
 +
! Δ''x''
 +
! colspan="5" | ''ε''<sub>''i''</sub> (ħω)
 +
|-
 +
| Discretized Hamiltonian
 +
| 0.010
 +
|
 +
|
 +
|
 +
|
 +
|
 +
|-
 +
| Euler-Cromer
 +
| 0.010
 +
| 1.499
 +
| 3.038
 +
| 4.328
 +
| 5.60
 +
|
 +
|-
 +
| Runge-Kutta 45
 +
| 0.010
 +
|
 +
|
 +
|
 +
|
 +
|
 
|}
 
|}

Revision as of 15:47, 2 June 2023


ξ0 = -1

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 (Δx = 0.010) 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

Lowest Eigenvalues

Method Δx εi (ħω)
Discretized Hamiltonian 0.010
Euler-Cromer 0.010 1.499 3.038 4.328 5.60
Runge-Kutta 45 0.010
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