0
$\begingroup$

Suppose you have a simple 1D harmonic oscillator given by the Lagrangian, say, $$L(x,\dot x) = \dot x^2 - \frac{1}{10}x^2.$$

Semi-discretizations of the Euler-Lagrange equations, where spatial variables are treated as continuous and time is discretized into finite-sized steps, is extremely well studied and forms the basis of scientific computing. I was playing around with purely discrete formulations of Hamilton's principle: where $x$ also is allowed to attain only values on the integers.

In this setting a trajectory $\gamma$ is described by a sequence of $\{x_0, x_1, \ldots, x_T\}\in\mathbb{Z}^T$ and the action can be discretized as $$S(\gamma) = \sum_{i=1}^{T-1}\left[(x_{i+1}-x_i)^2 - \frac{x_{i+1}^2}{10}\right].$$

Now Hamilton's principle would look like the following: a trajectory $\gamma$ is physical if $$S(\{x_0, \ldots, x_i, \ldots, x_T\}) \leq S(\{x_0, \ldots, x_i\pm 1, \ldots, x_T\})\quad \forall i\in \{1,\ldots,T-1\},$$ i.e, if you cannot improve the action of the trajectory by moving any position along the trajectory to either of its neighbors on $\mathbb{Z}$.

Euler-Lagrange equations don't really exist since nothing in sight is differentiable, but one can still turn the above boundary-value problem into an initial value problem by declaring the physical continuation of the trajectory $\{x_0, \ldots, x_i\}$ to be the point $x_{i+1}$ so that the augmented trajectory $\{x_0, \ldots, x_{i+1}\}$ still satisfies Hamilton's principle.

There may be multiple such $x_{i+1}$. In that case, we choose one valid $x_{i+1}$ uniformly at random.

If I simulate this process for the harmonic oscillator (with the initial conditions $x_0=x_1=-100$) I get the following, showing the PDF of $x$ animating over time:

movie

This doesn't quite look like time evolution of Schroedinger's equation, but it's certainly suggestive.

Is there some way to formalize all of this? Can quantum mechanics be formulated as arising from discretization of classic mechanics onto a lattice?

EDIT: Fixed movie link

$\endgroup$
3
  • $\begingroup$ Are you familiar with Feynman's path integral formulation? Sections 4 and 5 here respectively motivate the classical action and Schrödinger equation: hitoshi.berkeley.edu/221a/pathintegral.pdf This isn't quite what you were looking for, which is using classical mechanics to motivate QM, but it's somewhat similar to what you hoped for (although the discretisation is of time, not of space, resulting in a particle having a discrete set of positions at the discrete times). $\endgroup$
    – J.G.
    Apr 20, 2017 at 8:18
  • $\begingroup$ @J.G. oh this is very helpful, thanks! It looks like what I wrote above integrates over only some paths "near" the classic path (due to the discretization of space allowing several possible paths) whereas QM requires integrating over all paths. $\endgroup$
    – user2617
    Apr 20, 2017 at 8:46
  • $\begingroup$ Yes, there is a groaning industry of quantum mechanics around the clock (=1d periodic lattice.) Try Floratos-leontaris, Vourdas, etc... $\endgroup$ Jul 12, 2017 at 21:40

1 Answer 1

0
$\begingroup$

Well, quantum mechanics should not arise from discretization of classical mechanics on a lattice. Space is not discreet in QM. What could happen of course is that a discreetized approximation to QM arises from a discreetized version of CM but most likely you need to add more. For example we also know that the wave function has complex values. This cannot arise from CM with only a discreetization method.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.