What would a hard boundary mean in physics on the quantum-mechanical level?

Question

In this question I asked about the possibilities of what a boundary might be like, with emphasis on the storytelling.

Now I'd like to investigate what a hard boundary would mean in quantum mechanics, in a more “hard SF” manner.

Imagine a bubble of walled-off spacetime that occurred in the lab so it could be examined up close an personal. Whether the inside is in stasis, destroyed, censored, or whatever is not important here. The interesting thing is that a boundary exists and quantum-mechanical wave functions are prohibited from entering the region contained by the boundary.

Imagine, perhaps, that it's an energy well of arbitrary height. Or, I'm intrigued by @Beta’s remark, “The mirror is 100% legal; all fields and space-time curvature are symmetrical at the boundary, or equivalently the boundary condition allows no normal components of anything.” Or, it just somehow prevents a wavefunction collapse from ever choosing that position.

The phenomena resulting from this should be benign. It needs to interact with normal matter! The bubble won’t just fall through the Earth like a neutrino, or fly off at the speed of light. Rather, it needs to be massive so it can stay put in the room, follow the standard path through spacetime like a massive object, and be held and pushed by normal matter. E.g. it could be placed in a stand and stay put in the lab, even as the Earth turns under it.

I'm thinking that something like Pauli Exclusion could be made to work: the electrons of matter would feel the excluded bubble nearby and distort the shape of the wave function, requiring higher energy. Making it a hard solid object is the main issue!

Second, what might be happening very close to the edge? If it's pushed and must move like a billiard ball, but it is not a point-mass, the force must somehow be communicated around the entire space. If it's infinity rigid there would be problems with motion due to relativistic effects.

Anybody up to communing with the Hamiltonian?

Answer 1

I'm a physicist, but an experimental one. This means I don't deal with hypothetical stuff like this very often, and when I do, I feel the need to figure out a real-world example or test for it. Also, this post contains a fair amount of math and is a little lengthy, so this is your warning. The bullet points at the bottom show my summary and conclusions from this.

I'm afraid I have no idea how to develop such a thing; an object or region of space that is an infinitely high energy wall to particles. I know normal matter and fields can be energy barriers, but I don't think any of them can be infinite in height. (They can be, however, practically infinite, which is something else entirely!) That second question, though, is something I can tell you about.

I'm going to assume some nice symmetry (like a sphere) so that the math is easy, but we also get to really focus on what it means to have an infinite plateau. I'm also going to assume, since this is a macroscopic object, that I don't have to worry about electron tunneling (that changes the problem setup and answer).

As I understand it, your potential energy is a discontinuous, piece-wise function which looks like:

$U(x) = 0$ outside the object ($x\geq0$)

$U(x) = \infty$ inside and on the border of the object ($x<0$)

Therefore I'm going to use the one dimensional time independent Schrödinger Wave equation.

$$\hat{H}\Psi = E\Psi$$ $$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x) }{{\partial x}^2}+U(x)\Psi(x) = E\Psi(x)$$ Inside the object (on on the border) $\Psi(x) = 0$ is the only solution. That's boring, and we don't really care about that, until we start applying boundary conditions for the other part. Now, just outside the object is where things could get interesting: $$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x) }{{\partial x}^2}+ 0\Psi(x) = E\Psi(x)$$ $$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x) }{{\partial x}^2} = E\Psi(x)$$ Since the wavefunction must be 0 at the boundary (x=0), I'm going to say: $$\Psi(x) = A\sin{(\alpha x)}$$ If we solve for $\alpha$, we get an equation like: $$\alpha = \sqrt{{\frac{2mE}{\hbar^2}}}$$

and after normalizing it, your wave function looks like:

$$\Psi(x) = \Bigg(\frac{2\sqrt[4]{{\frac{2mE}{\hbar^2}}}}{\sqrt{2\sqrt{{\frac{2mE}{\hbar^2}}} x - 2\sin(2\sqrt{{\frac{2mE}{\hbar^2}}} x)}}\Bigg) \sin{\Big(\sqrt{{\frac{2mE}{\hbar^2}}} x\Big)}$$

Which is pretty ugly. However, there are some takeaways from this:

• $\Psi^2$, which shows the odds of finding a particle interacting with the plateau at any particular point in space, is a decaying sine wave. (See here for a generalized, not-entirely-accurate plot where $\alpha = 1$.) This means that what happens near the edge also happens in set distances away from the edge.
• VERY close to the edge and at some particular distances, the wavefunction goes to zero. This means near the very edge of this object and at some particular distances, there is a "true vacuum." (This also makes me think of band structures, but it's not the same!)
• It doesn't really matter what the energy of the particle is; the energy just acts as a scaling factor in this equation. All particles approaching this object behave in the same way.
• A particle near this behaves like it's in an infinite well, but also a little bit like a free particle. It can have any energy level, but still goes to zero at some points in space.
• This also assumes there is only one particle interacting with the object. Multiple particles, especially ones that interact with each other, can change this wavefunction. This isn't entirely useless, though! It can act as a guide for our intuition for *real world*$^{TM}$ situations.