## Abstract

In many texts, the transition from classical mechanics to quantum mechanics is achieved by substituting the action for the phase angle. The paper presents a different approach to show some connections between classical and quantum mechanics for a single particle for an audience at graduate and postgraduate levels. Firstly, it is shown that a wave equation of action can be derived under the free particle condition and the Legendre transform. The wave-like solutions of the action, Hamiltonian and momentum of the free particle are presented. Using the discrete approximation, the equation of motion of a single particle, in scalar potential field, is obtained in a similar form to Schrö dinger's equation. The rest of the paper discusses the propagation, superposition of the wave-like dynamic variables and their connections to quantum mechanics. The superposition of the variables of a particle is generally distinct from the superposition of classical waves (e.g. acoustics). The quantum superposition provides a self-consistent interpretation of the wave-like solutions of the variables. Connections between the classical and quantum relations for corresponding variables are observed from the one-to-one comparisons.

Original language | English |
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Article number | 015401 |

Journal | European Journal of Physics |

Volume | 38 |

Issue number | 1 |

Early online date | 25 Oct 2016 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

## Keywords

- classical mechanics
- connections
- particle
- quantum mechanics
- wave-like variable