The wave function's argument is position, but can the position of a particle be predicted from the wave function? Specifically, can the wave function allow you to find the equations of motion for a particle, like in classical mechanics?
Answer
I'm assuming non-relativistic quantum mechanics. A quantity close to what you're describing is the propagator. We can define it in two common ways, namely,
$$K(x,t;x',t') = \langle x | \hat U(t,t') | x' \rangle$$
where $\hat U$ is the time evolution operator, or via the path integral,
$$K(x,t;x',t') = \int \mathcal D[x(t)] \, e^{iS[x(t)]}.$$
The propagator gives us the amplitude to go from one spatial point to another in a time $t'-t$. It does not require us to specify which path, only the start and end points. In fact, the propagator in the path integral is computed precisely by summing over all possible paths, both those that are allowed classically, and solutions which do not satisfy equations of motion.
We can construct something analogous to an equation of motion in classical mechanics using the wave function. If we define the Wigner function as,
$$P(x,p) = \frac{1}{\pi \hbar}\int_{-\infty}^{+\infty} \psi^\dagger(x+y)\psi(x-y)e^{2ipy/\hbar} \, dy$$
then it satisfies an evolution equation,
$$\frac{\partial P}{\partial t} = - \{ \{ P, H\}\}$$
using the Moyal bracket. One can solve this equation, and the Wigner approach is broader to include a correspondence with operators to compute expectation values.
Like quantum mechanics however, the Wigner approach is still probabilistic and $P(x,p)$ is a quasi-probability distribution.
No comments:
Post a Comment