# Could almost nothing really be something? Fourier series.

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## Main Question or Discussion Point

Could almost nothing really be something? Fourier series.

Say we have a large box. Say we have some function defined in this box that is square integrable. Say this function is small except for some small region in the box. This function could be represented as an infinite Fourier series, an infinite sum of functions that add to nearly zero for most of the box but constructively sum for some small region.

In a similar way we can add waves in quantum mechanics such that probability is small but for some localized region in a box. Could the Universe be such that when we have a region with low probability of finding a particle that there really exist this infinite set of waves that just happen to sum to zero at that spot at that time?

Could almost nothing "really" be something?

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Could the Universe be such that when we have a region with low probability of finding a particle that there really exist this infinite set of waves that just happen to sum to zero at that spot at that time?
Keep on studying QM, since what you are describing is basically a Fourier transform. The part about the basis waves being "really there" deserves a word of caution: this is identical to the question of whether the components of a 3D vector in a particular basis are "really there", and in any case science does not answer these questions. Having said that, my answer is that the vector is more real than its components, and so I think that the state of nothing is real and the fact that you choose to express nothing in a basis with infinitely many non-zero components does not make those non-zero components real.

Matterwave
Gold Member
I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).

I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).
That's pretty interesting, I didn't even know that. Does that mean that a 3-dimensional vector is/can be described by 3 Dirac-delta functions? I've glanced at the delta function but I'm not too familiar with its applicability. I see no reason that this wouldn't also work for abstract vectors and 'states', but I must ask if this is the case.

Gold Member
I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).

I'm thinking of the Gaussian function that can be near zero but for some small region.:

f(x) = a*e^(-((x-b)^2 /2*c^2)))

By the definition of the Gaussian we must let the box get very large?

It's interesting to note that the Feynman Path Integral of Quantum Mechanics for at least a free particle can easily be derived from the Dirac Delta function as noted here:

http://hook.sirus.com/users/mjake/delta_physics.htm [Broken]

This makes me wonder if all of physics could be derived from the Dirac Delta function.

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ZapperZ
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