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# System prepended metadata

title: Large Mass Harmonic Oscillator on Eucledian time

---

## Test note

$$
\sum_i^N K(x_i)M^i 
$$

Inline math?: $\sum_i^N K(x_i) M^i$

![](https://i.imgur.com/ScgvAoP.png)
# Large Mass Harmonic Oscillator on Eucledian time

Info:
* Eucledian time points: 3
* N = 80
* Matrix size of N^3
* Action: $S = \frac12 \sum_i \Delta_\tau \sigma x_i^2 $
* $\sigma = 1$
* Runtime ~10 minutes
* Fokker-Planck equation
  * Should be positive eigenvalues of the equation:
  * $F_{\textrm{FP}} P(\textbf{x};\tau) = -\lambda P(\textbf{x};\tau)$

## Eigenvalues:

|   k  |    $||Ax-kx||/||kx||$ |
| :-------------: | :-------------: | 
|  -1.843422+0.001579i     |     0.998479   |
|  -2.616751-0.107733i     |     0.965833   |
|  -2.729703+0.136284i    |       1.01945   |
|  -3.640527+0.073839i     |      1.03712   |
|  -3.781522-1.430333i     |     0.762849   |
|  -3.894783+1.512824i     |     0.745834   |
|  -5.352922-2.580163i     |     0.538589   |
|  -5.508457+2.678970i     |     0.437366   |
|  -7.712095+3.674526i     |      0.42868   |
|  -10.555826-4.278824i    |      0.434935   |
|  -10.950730+4.342183i    |      0.438883   |
  -19.321643-0.247089i    |      0.817175   |
|  -45.441350+0.332406i    |        0.5352   |
|  -80.623264+0.052920i    |      0.376649   |
|  -110.389972-0.384503i   |       0.273537   |
|  -146.793279+0.052263i   |       0.168638   |
|  -173.062205+0.048124i   |      0.0891449   |
|  -186.049725+0.026397i    |     0.0361323   |
 ---------------------- --------------------

## Ground state
Under is the ground state for the 3 eucledian time points
| $x_0$ | $x_1$ | $x_2$ |
|:-------------:|:-------------:|:-----:|
|![](Figures/LM_HO_F_GS_x0.png) | ![](Figures/LM_HO_F_GS_x1.png) | ![](Figures/LM_HO_F_GS_x2.png) |


## First exited states
Under is a plot of the first 3 exited states in euledian time $x_0 = 0$.
| $1$ | $2$ | $3$ |
| ------------- |:-------------:|:-----:|
|![](Figures/LM_HO_F_1S_x0.png) | ![](Figures/LM_HO_F_2S_x0.png) | ![](Figures/LM_HO_F_3S_x0.png) |