# Quantization and Bifurcation beyond Square-Integrable Wavefunctions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Hamilton-Jacobi Formalism

## 3. Time Average along a Complex Quantum Trajectory

## 4. General Quantization Laws without SI Condition

## 5. Energy Quantization beyond SI Wavefunctions

- $n=2m$:$${\psi}_{m}\left(x\right)={C}_{1}{e}^{-{x}^{2}/2}F\left(-m,1/2,{x}^{2}\right)/\mathsf{\Gamma}\left(1/2-m\right)={C}_{1}{e}^{-{x}^{2}/2}{H}_{2m}\left(x\right).$$
- $n=2m+1$:$${\psi}_{m}\left(x\right)={C}_{1}{e}^{-{x}^{2}/2}F\left(-m,3/2,{x}^{2}\right)/\mathsf{\Gamma}\left(-1/2-m\right)={C}_{1}{e}^{-{x}^{2}/2}{H}_{2m+1}\left(x\right).$$

## 6. Quantum Bifurcation beyond SI Wavefunctions

## 7. Spin Degree of Freedom beyond SI Wavefunctions

## 8. Conclusions

- Universal quantization laws: The total energy $E={E}_{k}+Q\left(x\right)+V\left(x\right)$ derived from the time-independent Schrödinger equation is shown to be conserved, but not quantized. Regardless of the confining potential $V\left(x\right)$, quantization always occurs in the kinetic energy ${\langle {E}_{k}\rangle}_{T}$ and the quantum potential ${\langle Q\rangle}_{T}$, whose values can only change by an integer multiple of $\hslash \omega /2$.
- Renewed meaning of the energy eigenvalues: The energy eigenvalues ${E}_{n}$ derived conventionally from the SI condition are shown to be the special energies at which the quantization levels of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$ experience a step jump.
- The origin of energy quantization: Energy quantization in a confined system originates from the discrete change of the numbers of zero of ${\psi}_{E}\left(x\right)$ and ${\psi}_{E}^{\prime}\left(x\right)$, whose values determine the quantization levels of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$.
- Concurrence of Quantization and bifurcation: Bifurcations of equilibrium center points and singular saddle points of the quantum dynamics are shown to be synchronous, respectively, with the quantization process of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$, as the total energy $E$ increases monotonically.
- Undivided SI and NSI wavefunctions: probability interpretation isolates SI wavefunctions from NSI wavefunctions; however, under the quantum H-J formalism, SI and NSI wavefunctions are indivisible with continuously connected velocity field and quantum trajectories.
- The role of NSI wavefunctions in energy quantization: Both SI and NSI wavefunctions contribute to the energy quantization. SI wavefunctions help to locate the bifurcation points at which ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$ have a step jump, while NSI wavefunctions form the flat parts of the stair-like distribution of the quantized energies.
- The role of NSI wavefunctions in spin: The second-order Schrödinger equation generally contains two independent solutions with opposite rotation on the complex plane. The inclusion of both solutions allows the Schrödinger equation to describe the spatial motion as well as the spin motion.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The stair-like distributions of the numbers of zero of ${\psi}_{E}\left(x\right)$ and ${\psi}_{E}^{\prime}\left(x\right)$, as the total energy $E$ changes continuously

**Figure 2.**The step changes of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$ occur at the SI wavefunctions ${\psi}_{E}$ with $E=n+1/2$, as the total energy $E$ changes continuously in a harmonic oscillator. The flat parts of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$ are constituted by the NSI ${\psi}_{E}$ with $E\ne n+1/2$.

**Figure 3.**The distribution and movement of the centers and saddles over the horizontal $x$ axis, as the total energy $E$ changes continuously along the vertical axis.

**Figure 4.**The wavefunctions corresponding to $E=0.1$, $E=0.49$, and $E=0.51$ are NSI, but their quantum trajectories are bound and closely connected to the eigen trajectories of $E=0.5$, which are concentric circles around the equilibrium point at the origin.

**Figure 5.**Velocity fields and quantum trajectories in the energy interval $0.5<E\le 1.5$ show the quantum bifurcation that the number of equilibrium points jumps from one to two as energy across $E=0.5$. Part (

**a**) is the enlargement of Figure 4d near the origin to illustrate the split of the single equilibrium point at the origin into a pair of equilibrium points as $E$ increases from $0.5$ to $0.51$. In this energy interval, there are two equilibrium points and one singular point at the origin, corresponding to the energy levels ${Z}_{{\psi}^{\prime}}=2$ and ${Z}_{\psi}=1$ as shown in Figure 1.

**Figure 6.**Velocity fields and quantum trajectories in the energy interval $1.5<E\le 2.5$ show the quantum bifurcation that the number of equilibrium points jumps from two to three as energy across $E=1.5$. Part (

**a**,

**b**) are the enlargements of the velocity field near the origin to illustrate the emergence of a new equilibrium point (a center). In this energy interval, there are three equilibrium points and two singular points, corresponding to the energy levels ${Z}_{{\psi}^{\prime}}=3$ and ${Z}_{\psi}=2$ as shown in Figure 1. Part (

**d**) plots the eigen trajectories for $E=2.5$ to show the three equilibrium points at ${x}_{eq}=0,\pm \sqrt{5/2}$ and two singular points at ${x}_{s}=\pm \sqrt{1/2}$.

**Figure 7.**A sequence of pitchfork bifurcation curves shows the variation of equilibrium points ${x}_{eq}\left(E\right)$ with respect to the total energy $E$. The number of ${x}_{eq}\left(E\right)$ at each $E$, denoted by the blue dots, is equal to the energy level ${Z}_{{\psi}^{\prime}}$ as plotted in Figure 1. The branches of the ${x}_{eq}\left(E\right)$ curves start sequentially at $E=0,3/2,7/2,\cdots $, and bifurcate sequentially at $E=1/2,5/2,9/2,\cdots $. Except for the bifurcation points (the blue dots), the entire sequential bifurcation diagram is formed by the NSI wavefunctions ${\psi}_{E}\left(x\right)$ with $E\ne n+1/2$.

**Figure 8.**A sequence of pitchfork bifurcation curves shows the variation of singular points ${x}_{s}\left(E\right)$ with respect to the total energy $E$. The number of ${x}_{s}\left(E\right)$ at each $E$, denoted by the red dots, is equal to the energy level ${Z}_{\psi}$ as plotted in Figure 1. The branches of the ${x}_{s}\left(E\right)$ curves start sequentially at $E=1/2,5/2,9/2,\cdots $, and bifurcate sequentially at $E=3/2,7/2,11/2,\cdots $. Except for the bifurcation points (the red dots), the entire sequential bifurcation diagram is formed by the NSI wavefunctions ${\psi}_{E}\left(x\right)$ with $E\ne n+1/2$.

**Figure 9.**The dual-rotation solutions to the Schrödinger equation with $E=1/2$: (

**a**) quantum trajectory of the wavefunction ${\psi}_{CCW}\left(x\right)$; (

**b**) quantum trajectory of the wavefunction ${\psi}_{CW}\left(x\right)$; (

**c**) comparison between $|{\psi}_{CCW}|$ and $|{\psi}_{CW}|$, and (

**d**) comparison between total potential ${V}_{T}\left({\psi}_{CCW}\right)$ and ${V}_{T}\left({\psi}_{CW}\right)$.

**Figure 10.**The comparisons between CCW solution ${\psi}_{CCW}\left(x\right)$ and CW solution ${\psi}_{CW}\left(x\right)$ regarding the magnitudes of $\psi $ and the total potential ${V}_{T}\left(\psi \right)$ for $E=3/2$ and $E=5/2$, respectively.

**Table 1.**The step changes of ${\langle {E}_{k}\rangle}_{T}$ and ${\langle {E}_{k}+Q\rangle}_{T}$ are synchronous with the changes of saddles and centers.

Quantized Items | $\mathbf{0}<\mathit{E}<\frac{\mathbf{1}}{\mathbf{2}}$ | $\mathit{E}=\frac{\mathbf{1}}{\mathbf{2}}$ | $\frac{\mathbf{1}}{\mathbf{2}}<\mathit{E}<\frac{\mathbf{3}}{\mathbf{2}}$ | $\mathit{E}=\frac{\mathbf{3}}{\mathbf{2}}$ | $\frac{\mathbf{3}}{\mathbf{2}}<\mathit{E}<\frac{\mathbf{5}}{\mathbf{2}}$ | $\mathit{E}=\frac{\mathbf{5}}{\mathbf{2}}$ | $\frac{\mathbf{5}}{\mathbf{2}}<\mathit{E}<\frac{\mathbf{7}}{\mathbf{2}}$ |
---|---|---|---|---|---|---|---|

Wavefunctions | NSI | SI | NSI | SI | NSI | SI | NSI |

Zeros of ${\mathit{\psi}}_{\mathit{E}}\left(\mathit{x}\right)$ | 0 | 0 | 1 | 1 | 2 | 2 | 3 |

Number of saddles | 0 | 0 | 1 | 1 | 2 | 2 | 3 |

Levels of ${\langle {\mathit{E}}_{\mathit{k}}\rangle}_{\mathit{T}}$ | 0 | 0 | 1/2 | 1/2 | 1 | 1 | 3/2 |

Zeros of ${\mathit{\psi}}_{\mathit{E}}^{\prime}\left(\mathit{x}\right)$ | 1 | 1 | 2 | 2 | 3 | 3 | 4 |

Number of centers | 1 | 1 | 2 | 2 | 3 | 3 | 4 |

Levels of ${\langle {\mathit{E}}_{\mathit{k}}+\mathit{Q}\rangle}_{\mathit{T}}$ | 1/2 | 1/2 | 1 | 1 | 3/2 | 3/2 | 2 |

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Yang, C.-D.; Kuo, C.-H. Quantization and Bifurcation beyond Square-Integrable Wavefunctions. *Entropy* **2018**, *20*, 327.
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Yang C-D, Kuo C-H. Quantization and Bifurcation beyond Square-Integrable Wavefunctions. *Entropy*. 2018; 20(5):327.
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**Chicago/Turabian Style**

Yang, Ciann-Dong, and Chung-Hsuan Kuo. 2018. "Quantization and Bifurcation beyond Square-Integrable Wavefunctions" *Entropy* 20, no. 5: 327.
https://doi.org/10.3390/e20050327