In this report, we start by defining key aspects of classical Lagrangian mechanics including the principle of least action and how one can use this to derive the Euler-Lagrange equations. Momentum and Conservation laws shall also be introduced, deriving relations between position, momenta and the Lagrangian of a given system. Following this, we develop our study of classical mechanics further using Legendre transforms on the Euler-Lagrange equation and our conservation laws to define Hamiltonian mechanics. In our new notation, we use Poisson brackets when evaluating the rate of change of a classical observable. Next, we cross to quantum mechanics, giving some definitions which shall be used for later discussion. We then state and prove the Ehrenfest theorem, from which we draw our first correspondence between classical and quantum mechanics, most notably between the Poisson bracket and the commutator. Furthermore, the Ehrenfest theorem applied to operators of position and momentum shows a further correspondence with classical results. Finally, we take an example of the simple harmonic oscillator, using both classical and quantum methods to solve for this system and comment on the similarities and differences between the results graphically and qualitatively.

1. Introduction

In life, one would take the shortest route to get to where they are going and therefore expel the least possible amount of energy and the particles in our universe act no different. This principle is described by Lagrangian mechanics, developed by Joseph-Louis Lagrange in the late 18th century. [1] An extension from Newtonian mechanics, Lagrangian methods were created in order to be able to work with more complicated systems, using a new coordinate system which can take account for any system constraints. The key principle involves minimizing the value of an integral in order to derive the equations of motion of a particle. These equations can be used to unlock many new ideas and create another equivalent branch of classical mechanics.