2.5. Summary

In this chapter, you have hopefully learned to appreciate the close relation between classical and quantum mechanics. In fact, the mathematical structure of both theories is more similar than you may have thought after your introductory courses on quantum mechanics. Let us list some of the relations we have found.

1. Both classical and quantum mechanics allow us -- in principle -- to follow the evolution of a mechanical system by solving a first-order differential equation in the time. In both, the Hamiltonian is the agent governing this evolution.

   In classical mechanics, the time evolution equations read $$\begin{aligned} \frac{\partial p_j}{\partial t} &= - \frac{\partial H}{\partial q_j}; \\ \frac{\partial q_j}{\partial t} &= \frac{\partial H}{\partial p_j}. \end{aligned}$$ Writing $z_j = (p_j, q_j)$, we may write this in the form $$\frac{\partial z_j}{\partial t} = J \nabla_j H,$$ where $\nabla_j = (\partial/\partial p_j, \partial/ \partial q_j)$ and $$J = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right).$$

   In quantum mechanics, the Hamilton equations are replaced by the time-dependent Schrödinger equation: $${\rm i}\hslash \frac{\partial}{\partial t} \left| \psi(t) \right\rangle= \hat{H} \left| \psi(t) \right\rangle.$$

2. In quantum mechanics, measurements disrupt the time evolution and bring stochastic elements into the theory.

3. The Poisson bracket of classical mechanics is replaced by the commutator in quantum mechanics: $$\left\{ A, B \right\} \rightarrow -{\rm i}\hslash \left[ \hat{A}, \hat{B} \right].$$

4. The path integral allows us to work out the quantum mechanical time evolution as an infinite sum over paths, where each path is weighted by the factor $$\exp\left( {\rm i}S_{\text{path}}/\hslash\right),$$ where $S_{\text{path}}$ is the classical action of the path: $$S_{\text{path}} = \int_{\text{path}} L(q_j, \dot{q}_j) dt,$$ with $L$ the classical Lagrangian of the path.

5. The smallness of $\hslash$ ensures that the paths with a value of the action deviating not more than ${\mathcal O} (\hslash)$ from the stationary value(s) of the action, yield a major contribution to the path integral (time evolution). For $\hslash \rightarrow 0$, only the stationary (i.e. the classical) path(s) survive(s).

6. The time-dependent Schrödinger equation can be derived from the path integral formalism.