Download Ideas On Containment of Physical Information Within the

Ideas On Containment of Physical Information Within the Boundary: The Quantum Particle in a Box
with Infinite Potential Walls
Disclaimer: The conclusions I reach here are my own. I do not attest to be saying anything new. I
welcome any corrections or feedback on them.
The information of the quantum particle in a box is contained in the boundary of the box. The energy of
the particle is proportional to the square of the principle quantum number and depends inversely as the
square of its dimension (see attached diagram: E = (n2 ℏ2π2/2mA)). Increase the dimension of the box
and the energy of the single-particle system decreases proportionately. But what does this have to do
with information?
Consider the particle to be in a 2-dimensional box (the 2nd diagram attached) with a moveable partition.
Statistical mechanics tells is that as the partition is moved to the left, the entropy of the single-particle
system increases as the number of degrees of freedom of the particle has increased (by “degrees of
freedom” I mean that the particle can be in many many new positions that it previously did not have
access to). If the partition is originally at the halfway point of the box, then the entropy increases by
ln(2).
The Boltzman Entropy is S = kblog(Ω), where Ω is the number of microstates in phase space that the
particle has access to, consistent with the constraints of the system (for example, total number of
particle, total energy, etc). When those constraints are relaxed (for example by doubling the size of the
box), the entropy increases accordingly.
The information of the quantum particle in the box is closely related to its entropy. Entropy measures
our ignorance of the details of the microscopic system. Therefore, when we move the partition in the
2d case, or change the length A in the 1-d case, we are generating entropy and increasing our ignorance
of the precisely where the particle is located. The relationship between entropy and information can be
illustrated by a simple example:
Suppose we have a box of gas, all contained a tiny region of the box, say the upper right corner. Now let
it evolve in time. Of course it will disperse and diffuse throughout the box. Our ignorance of the precise
location of each gas particle has greatly decreased. The latter situation is a situation with high entropy;
the former lower. If we were to measure the box at some later time and found all of the particle back in
their original positions in the corner, then we would be VERY surprised. That particular configuration the gas all back in the original corner - would contain a lot of information.
In another sense, information is something which makes distinctions. Take a piece of paper and draw a
line down the middle. Demarcating the left from the right, you have set up a situation by which a choice
is possible (before the sheet of paper was a uniform whole, the sheet). “Making a choice” reduces the
uncertainty of which “half” you are talking about (choice is an essential ingredient). Drawing the line
generates entropy; choosing transfers information…information contained in the demarcating line.
The information of the particle in the 1D box resides in the boundary: what and where the boundary is
identified as the degrees of freedom of the particle. What is more, the information exists beyond the
boundary and inheres in the entire apparatus which is behind the boundary as well. That this seems to
be true is exemplified by the case when the V ≠ ∞, but V = V0. Then quantum tunneling is possible. The
width of the wall gives rise to the tunneling degrees of freedom. It is not just that the walls control the
degrees of freedom: I think that the walls themselves are the degrees of freedom. The walls are the
things which demarcate and make the space.