3
$\begingroup$

I was wondering why traditional calculus is used for studying quantum physics considering that there seems to be no continuum at quantum scale but discrete. For example, differential equations consider magnitudes as continuous, but is it OK to consider space and time continuous at quantum scale?

Sorry if I'm asking a stupid question. It just came to my mind while watching a documentary (not a PhD nor physics student here!).

$\endgroup$
1
5
$\begingroup$

is it OK to consider space and time continuous at quantum scale?

While quantum objects have physical quantities such as angular momentum, energy, spin etc., that can be discrete, this does not in any way mean that the space and time where these objects exist is quantized.

Space and time are treated as continuous quantities in standard quantum mechanics. This however, is not a statement or affirmation that the underlying spacetime is continuous. That is, it is possible that the underlying spacetime is quantized, but is irrelevant to the fact that standard quantum mechanics is constructed on continuous spacetime manifolds. There are quantum theories where the spacetime is treated as discrete, a good example being loop quantum gravity.

why traditional calculus is used for studying quantum physics considering that there seems to be no continuum at quantum scale but discrete

Physical quantities like position, momentum, energy etc., are described by eigenvalue problems of the form, for example $$H\psi=E\psi$$ that can contain linear differential operators, for example $$H\rightarrow i\hbar\frac{\partial}{\partial t}$$

even though the eigenvalues $E$ can be part of a discrete or continuous spectrum. That is, we can and do use traditional calculus to describe the evolution of quantum states, even if these states belong to a space describing discrete physical quantities.

It is not a necessary condition in quantum mechanics that the physical quantities corresponding to states, are continuous quantities themselves.

$\endgroup$
2
$\begingroup$

In many parts of quantum physics, the mathematics that is most often used is linear algebra. That is, the state of a quantum particle/system is described using a vector (actually a ray) in a so-called Hilbert space.

The "stuff that we can measure" (called observables) e.g. position, velocity, angular momentum etc. is represented by linear operators that act on the states in the Hilbert space. The possible outcomes of a measurement are given by the eigenvalues of these linear operators. In general, these eigenvalues form spectra that have both continuous and discrete parts! A typical misconseption of quantum mechanics is that everything is discrete, but that is not the case.

In introductory courses on quantum mechanics (and of course in many applications) we want a representation of the state of a particle/system expressed in terms of some basic observable, often position. It just so happens that position is an operator with continuous eigenvalue spectrum. Hence when we express the state "in the position basis", we get a continuous function, called the wave function (in position space). The equation that governs the time evolution of this (continuous) wave function is then a differential equation that we call the Schrödinger equation.

Another aspect entirely is whether treating space(time) as something continuous is justified. Based on the experimental evidence available to us, there is nothing that points to space(time) itself being discrete. Sabine Hossenfelder has a good video on the topic here.

$\endgroup$

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