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Hamilton Equations, Commutator, and Energy Conservation^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Classical Hamiltonian of an Oscillator

## 3. Quantum Hamiltonian

#### 3.1. A Word on the Derivative with Respect to the Operator

## 4. Quantum Hamilton Equations from Heisenberg Equations

#### Sum Separable Hamiltonians

## 5. Quantum Hamilton Equations and Energy Conservation

#### 5.1. Sum Separable Hamiltonian

#### 5.2. Quantum Hamilton Equations from Energy Conservation

## 6. Fundamental Commutator from Energy Conservation

## 7. Conclusions

- 1
- The fundamental commutator can be derived from Heisenberg equations of motion and the quantum Hamilton equations.
- 2
- The quantum Hamilton equations can be derived by energy conservation argument after the calibration of the clock.
- 3
- The quantum Hamilton equations can also be derived from Heisenberg equations of motion, together with the fundamental commutator. Also, the quantum Hamilton equations thus derived are shown to be energy conserving.
- 4
- The important fundamental postulate of quantum theory is that first put forth by Schrödinger, for the quantum state Equation (13). One can derive the Heisenberg equations of motion from the quantum state equation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EOM | Equations of Motion |

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## Share and Cite

**MDPI and ACS Style**

Chew, W.C.; Liu, A.Y.; Salazar-Lazaro, C.; Na, D.-Y.; Sha, W.E.I.
Hamilton Equations, Commutator, and Energy Conservation. *Quantum Rep.* **2019**, *1*, 295-303.
https://doi.org/10.3390/quantum1020027

**AMA Style**

Chew WC, Liu AY, Salazar-Lazaro C, Na D-Y, Sha WEI.
Hamilton Equations, Commutator, and Energy Conservation. *Quantum Reports*. 2019; 1(2):295-303.
https://doi.org/10.3390/quantum1020027

**Chicago/Turabian Style**

Chew, Weng Cho, Aiyin Y. Liu, Carlos Salazar-Lazaro, Dong-Yeop Na, and Wei E. I. Sha.
2019. "Hamilton Equations, Commutator, and Energy Conservation" *Quantum Reports* 1, no. 2: 295-303.
https://doi.org/10.3390/quantum1020027