Undergrad Quantum Mechanics in Spacetime: Is it Discrete?

[jaketodd]

A long time ago, someone told me that Einstein thought spacetime is "smooth and continuous."

Quantum mechanics is a proven fact. Quantum mechanics exists in spacetime, though.

So can we at least conclude that spacetime is not continuous, and instead discrete? ...quanta of space and time?

Without a discrete foundation (spacetime) for particles and energy to exist within, how can we have quantum mechanics?

Thanks,

Jake

 

[phinds]

I suggest, as with most such basic questions, you do a forum search. This subject has been beaten to death here on PF

 

[Ibix]

So can we at least conclude that spacetime is not continuous, and instead discrete? ...quanta of space and time?

The fact that quantum field theory (what you are calling "a proven fact") is a relativistic quantum theory would suggest that the answer is "no".

A quantum theory of gravity might revise that answer, but it might not. My (very limited) understanding is that some of our current candidate theories do (although not in any naive pixel-grid kind of way) and some don't.

 

[Dale]

Without a discrete foundation (spacetime) for particles and energy to exist within, how can we have quantum mechanics?

Quantum mechanics does not require discrete space or time. The wave function is continuous over space and time. With continuous spacetime you wind up with discrete energy levels, which is the quantized aspect.

 

[jaketodd]

Quantum mechanics does not require discrete space or time. The wave function is continuous over space and time. With continuous spacetime you wind up with discrete energy levels, which is the quantized aspect.

Thanks Dale, but don't discrete energy levels require a discrete fabric, in which to exist? I'm building from the bottom up; discrete spacetime, leading to discreetness in everything that exists within it. It sounds like you're assuming continuity, and somehow, mysteriously, arriving at discrete energy levels...sounds like you're playing both sides...no offense. Thanks again, please clear this up.

Jake

 

[Nugatory]

Thanks Dale, but don't discrete energy levels require a discrete fabric, in which to exist?

No, they do not, and the easiest way to see that a discrete fabric is not required is to look at the quantum mechanical solutions for simple bound systems: the energy level are quantized but neither time nor position are, This is stuff you'll encounter in the first few ("baby steps, let's walk before we run") weeks of an introductory QM course.

I'm building from the bottom up; discrete spacetime, leading to discreetness in everything that exists within it.

Building WHAT from the bottom up? Quantum mechanics has been built from the bottom up over the past century with convincing experimental evidence supporting every step. Starting over again at the bottom only makes sense if you have reason to believe that there is something wrong with the existing structure, and you cannot make that argument if you have no idea what's actually in that existing structure.
(And please do be mindful of the Physics Forums rule about personal theories)

It sounds like you're assuming continuity, and somehow, mysteriously, arriving at discrete energy levels

Not "assuming" - it's right there in the math, which is why I mentioned looking at the solutions for simple bound systems above. The appearance of discrete energy levels is only "mysterious" if you haven't tried solving Schrödinger's equation for such systems. If you haven't done the math you haven't done anything, and if you've done the math the discrete energy levels are no more "mysterious" than the appearance of the term $\sqrt{b^2-4ac}$ in the quadratic formula.

 

[Dale]

Thanks Dale, but don't discrete energy levels require a discrete fabric, in which to exist?

No. I am not sure why you would think that. There is nothing in the math to suggest it.

It sounds like you're assuming continuity, and somehow, mysteriously, arriving at discrete energy levels...sounds like you're playing both sides..

It isn’t me, that is just the way the math works out. Nothing mysterious about it, the derivations are straightforward and available for anyone to read.

Even in QM not all systems have quantized energy levels. For example a free particle is not quantized. The discreetness is not something that is added by hand or otherwise assumed. It is something that arises naturally from the math and even then only for certain systems.

You may want to start here:

https://en.m.wikipedia.org/wiki/Finite_potential_well

 

[bhobba]

Even in QM not all systems have quantized energy levels. For example a free particle is not quantized. The discreetness is not something that is added by hand or otherwise assumed. It is something that arises naturally from the math and even then only for certain systems.

The why of it is quite deep - see the following interesting videos that examines it and other 'strange' issues like summing divergent series:


Thanks
Bill

 

[PeroK]

Thanks Dale, but don't discrete energy levels require a discrete fabric, in which to exist? I'm building from the bottom up; discrete spacetime, leading to discreetness in everything that exists within it. It sounds like you're assuming continuity, and somehow, mysteriously, arriving at discrete energy levels...sounds like you're playing both sides...no offense. Thanks again, please clear this up.

Jake

In QM, the energy levels of a free particle form a continuous spectrum. I.e. they are not discrete. If the particle is in a potential well or a harmonic oscillator, then the energy levels are discrete.

There's a certain analogy with a guitar string, say, which allows standing waves of discrete natural frequencies.

 

[Herman Trivilino]

don't discrete energy levels require a discrete fabric, in which to exist?

You don't in general have discrete energy levels. When you study quantum mechanics you learn the conditions under which energy is and isn't quantized.

You don't need quantized spacetime to explain every observation ever made or every experiment ever conducted. We have no evidence to indicate that space or time is quantized.