Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-obesrvation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.

Consider a particle in a box in quantum mechanics in n spatial dimensions.. If you do this, then Schrödinger's equation for a pure energy Eigenstate becomes (inside the box):

$$E\psi = - \frac{\hbar^{2}}{2m}\nabla^{2}\psi$$

And where you force $\psi$ to be zero everywhere outside the box, and on the boundary of the box. Using a bunch of PDE machinery involving separation of variables, we find that the unique solution to this equation is a an infinite sum of terms that look like

$$\psi=A\Pi_{i=1}^{n}\sin\left(\frac{m_{i}\pi x_{i}}{L_{i}}\right)$$

where all of the $m$ are integers, and the $\Pi$ represents a product with one $\sin$ term for each dimension in our space${}^{1}$. Plugging this back into Schrödinger's equation tells us that the energy of this state is

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{n} \frac{m_{i}^{2}}{L_{i}^{2}}\right)$$

Now, let's assume that in the first $d$ dimensions, our box has a large width $L$, while in the last $n-d$ dimensions, our box has a small width $\ell$. Then, we can split this sum into

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{d} \frac{m_{i}^{2}}{L^{2}}+\sum_{i=d+1}^{n} \frac{m_{i}^{2}}{\ell^{2}}\right)$$

So, now we can see what's happening--if $L \gg \ell$, there is a much greater energy cost associated with moving in the more constrained or smaller $n-d$ directions than there is in moving in the less constrained $d$ dimensions--the smallest transitions cost an energy proportional to the inverse square of the size of the dimension. By making these dimensions small enough, we can guarantee that no experiment humans have done has even approached the energy threshold required to induce this transition, meaning that the portion of a particle's wavefunction associated with these extra dimensions is constrained to stay the way they are, making them unobservable.

${}^{1}$So, if n=2, a typical state would look something like $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$

Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-observation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.

Consider a particle in a box in quantum mechanics of $n$ spatial dimensions. If you do this, then Schrödinger's equation for a pure energy Eigenstate becomes (inside the box):

$$E\psi = - \frac{\hbar^{2}}{2m}\nabla^{2}\psi$$

And where you force $\psi$ to be zero everywhere outside the box, and on the boundary of the box. Using a bunch of PDE machinery involving separation of variables, we find that the unique solution to this equation is a an infinite sum of terms that look like

$$\psi=A\prod_{i=1}^{n}\sin\left(\frac{m_{i}\pi x_{i}}{L_{i}}\right)$$

where all of the $m$ are integers, and the $\Pi$ represents a product with one sine term for each dimension in our space${}^{1}$. Plugging this back into Schrödinger's equation tells us that the energy of this state is

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{n} \frac{m_{i}^{2}}{L_{i}^{2}}\right)$$

Now, let's assume that in the first $d$ dimensions, our box has a large width $L$, while in the last $n-d$ dimensions, our box has a small width $\ell$. Then, we can split this sum into

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{d} \frac{m_{i}^{2}}{L^{2}}+\sum_{i=d+1}^{n} \frac{m_{i}^{2}}{\ell^{2}}\right)$$

So, now we can see what's happening — if $L \gg \ell$, there is a much greater energy cost associated with moving in the more constrained or smaller $n-d$ directions than there is in moving in the less constrained $d$ dimensions — the smallest transitions cost an energy proportional to the inverse square of the size of the dimension. By making these dimensions small enough, we can guarantee that no experiment humans have done has even approached the energy threshold required to induce this transition, meaning that the portion of a particle's wavefunction associated with these extra dimensions is constrained to stay the way they are, making them unobservable.

${}^{1}$So, if $n=2$, a typical state would look something like $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$

Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-obesrvation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.

Consider a particle in a box in quantum mechanics in n spatial dimensions.. If you do this, then Schrödinger's equation for a pure state becomes (inside the box):

$$E\psi = - \frac{\hbar^{2}}{2m}\nabla^{2}\psi$$

And where you force $\psi$ to be zero everywhere outside the box, and on the boundary of the box. Using a bunch of PDE machinery involving separation of variables, we find that the unique solution to this equation is a an infinite sum of terms that look like

$$\psi=A\Pi_{i=1}^{n}\sin\left(\frac{m_{i}\pi x_{i}}{L_{i}}\right)$$

where all of the $m$ are integers, and the $\Pi$ represents a product with one $\sin$ term for each dimension in our space${}^{1}$. Plugging this back into Schrödinger's equation tells us that the energy of this state is

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{n} \frac{m_{i}^{2}}{L_{i}^{2}}\right)$$

Now, let's assume that in the first $d$ dimensions, our box has a large width $L$, while in the last $n-d$ dimensions, our box has a small width $\ell$. Then, we can split this sum into

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{d} \frac{m_{i}^{2}}{L^{2}}+\sum_{i=d+1}^{n} \frac{m_{i}^{2}}{\ell^{2}}\right)$$

So, now we can see what's happening--if $L \gg \ell$, there is a much greater energy cost associated with moving in the more constrained or smaller $n-d$ directions than there is in moving in the less constrained $d$ dimensions--the smallest transitions cost an energy proportional to the inverse square of the size of the dimension. By making these dimensions small enough, we can guarantee that no experiment humans have done has even approached the energy threshold required to induce this transition, meaning that the portion of a particle's wavefunction associated with these extra dimensions is constrained to stay the way they are, making them unobservable.

${}^{1}$So, if n=2, a typical state would look something like $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$

Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-obesrvation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.

Consider a particle in a box in quantum mechanics in n spatial dimensions.. If you do this, then Schrödinger's equation for a pure energy Eigenstate becomes (inside the box):

$$E\psi = - \frac{\hbar^{2}}{2m}\nabla^{2}\psi$$

And where you force $\psi$ to be zero everywhere outside the box, and on the boundary of the box. Using a bunch of PDE machinery involving separation of variables, we find that the unique solution to this equation is a an infinite sum of terms that look like

$$\psi=A\Pi_{i=1}^{n}\sin\left(\frac{m_{i}\pi x_{i}}{L_{i}}\right)$$

where all of the $m$ are integers, and the $\Pi$ represents a product with one $\sin$ term for each dimension in our space${}^{1}$. Plugging this back into Schrödinger's equation tells us that the energy of this state is

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{n} \frac{m_{i}^{2}}{L_{i}^{2}}\right)$$

Now, let's assume that in the first $d$ dimensions, our box has a large width $L$, while in the last $n-d$ dimensions, our box has a small width $\ell$. Then, we can split this sum into

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{d} \frac{m_{i}^{2}}{L^{2}}+\sum_{i=d+1}^{n} \frac{m_{i}^{2}}{\ell^{2}}\right)$$

So, now we can see what's happening--if $L \gg \ell$, there is a much greater energy cost associated with moving in the more constrained or smaller $n-d$ directions than there is in moving in the less constrained $d$ dimensions--the smallest transitions cost an energy proportional to the inverse square of the size of the dimension. By making these dimensions small enough, we can guarantee that no experiment humans have done has even approached the energy threshold required to induce this transition, meaning that the portion of a particle's wavefunction associated with these extra dimensions is constrained to stay the way they are, making them unobservable.

${}^{1}$So, if n=2, a typical state would look something like $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$