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Re: [Phys-L] uncertainty principle .... was: ground-state energies



Let's talk some more about what the uncertainty principle
does and doesn't say ... especially as applied to a particle
in a box. As previously mentioned:

-- I like the uncertainty principle.
-- I like phase space.
-- I like unitary evolution.
-- I like Liouville's theorem.
-- I like symplectic integrators.
-- I like Feistel ciphers networks.
-- I like the optical brightness theorem.
-- I like the second law of thermodynamics
++ I like the fact that those are all essentially the same thing.
If you invented a process that violated the uncertainty
principle, it would also violate the second law.

The absolutely essential first step is to look at the phase
space of the particle in a box:
https://www.av8n.com/physics/img48/phase-space-particle-box.png

As always, the axes in phase space are position and momentum.

The basis states for the "k-basis" are shown by the colorful
bars in the diagram. The progression of colors indicates how
the phase of the wavefunction changes as a function of x; no
attempt is made to portray the magnitude.

As usual, L represents the size of the box, and n is the quantum
number that indicates which k-basis state we are talking about.
We can associate a wave number k with each state:
k_n = n k_1
where
k_1 = π/L

That is to say, the states are equally spaced in wavenumber.
Unlike (say) a harmonic oscillator, they are *not* equally
spaced in energy.

We can also define a momentum-like quantity
p[rms] = ℏk
for each state, but the "[rms]" is a reminder that we have
to be careful.

Let capital P denote the momentum *operator*:
P = ℏ d/dx

and it would be natural to define "the" momentum p to
be the expectation value of that operator:

p = ⟨P⟩ = ∫ sin(kx) (ℏ d/dx) sin(kx) dx / norm

but this p is zero for every state in the k-basis, as you can
easily verify by inspection. Integral of sine times cosine.
Therefore we can write
Δp Δx = 0 [states per_se]
which looks like the uncertainty principle, except that
the RHS is zero.

We are more likely to be interested in the RMS momentum:

p[rms] = √⟨P^2⟩ = √∫ sin(kx) (ℏ d/dx)(ℏ d/dx) sin(kx) dx / norm
= ℏk (as before)

Let's be clear: There is zero uncertainty about p[rms] in any
of the k-basis states; p[rms]_n = ℏ k_n exactly for all n.
The k-basis states have zero width in the diagram; the
colored bars are shown with some nonzero width just to make
them visible. Therefore we can write
Δp[rms] Δx = 0 [states per_se]
which looks like the uncertainty principle, except that
once again the RHS is zero. This time the momentum is
nonzero, but the delta momentum is still zero.

More interestingly, we can "associate" some area in phase
space with these k-basis states, in particular the region
bounded by one state and the next (in the p-direction) and
bounded by the sides of the box (in the x-direction). These
are visible in the diagram, with labels of the form Δp ∧ Δx.
The equations of motion are unitary, which guarantees that
over time any region in phase space will maintain the same
area, even if it changes shape. Conservation of phase space
is a really big deal.

It is absolutely routine in classical physics, when applying
Liouville's theorem, to find that the relevant area is not
the area "of" any particular state, but rather the area
bounded by some ensemble of states.
https://www.av8n.com/physics/liouville-intro.htm

We can easily calculate the "associated" area per state:
Δp[rms] Δx = ℏπ = h/2 ["associated" areas]
which kinda sorta looks like the usual statement of the
uncertainty principle. HOWEVER it must be emphasized that
a) This is the area between states, absolutely not the
area of the state itself, and
b) The RHS of this equation is larger than the RHS of
the usual uncertainty principle by a factor of 2π.
So if you use this to estimate the ground-state energy,
you're going to be off by one-and-a-half orders of
magnitude.

===============

There's another way of looking at this that might be more
in the spirit of the uncertainty principle. Each k-basis
state is a standing wave. It can be decomposed into two
running waves, one in the +dx direction and one in the -dx
direction. So if you measure "the" momentum by taking a
snapshot of the P operator, *not* taking the expectation
value, you would get two values, ±ℏk, if you arranged the
experimental details just right. So maybe there really is
some uncertainty in the p direction, namely 2nℏπ/L. That
gives us
Δp[snap] Δx = nh [snapshot]
Note that the RHS here is even worse than before by a factor
of 2n, so it's off by a factor of 4πn relative to the usual
statement of the uncertainty principle. Ouch.

================

It must be emphasized that the k-basis is not the only possible
basis ... and (!) basis states are not the only possible states.

The particle-in-a-box is not some irrelevant backwater example.
When the box is sufficiently large, the particle-in-a-box
becomes a free particle. The minimum-uncertainty states that
appear in the usual statement of the uncertainty principle are
superpositions of particle-in-a-box states.

================

As previously mentioned: Don't believe everything you read
in books. Here is a fairly typical discussion:

https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-Dimensional_box

This is an important result that tells us:

1. The energy of a particle is quantized and
2. The lowest possible energy of a particle is NOT zero. This is called
the zero-point energy and means the particle can never be at rest
because it always has some kinetic energy.

This is also consistent with the Heisenberg Uncertainty Principle: if
the particle had zero energy, we would know where it was in both
space and time.

How do I not love this? Let me count the ways.

*) Phase space is position and momentum. The units of ℏ are p∧x
(not x∧t). We should be talking about phase space (whereas
«both space and time» sounds like Minkowsi special relativity).

*) The energy of the particle is not quantized. Although the
k-basis states have definite evenly-spaced k values, and definite
but unevenly-spaced energy values, this is not the only basis
... and (!) basis states are not the only states.

*) Saying the two things are "consistent" is pure pettifoggery.
Just because the uncertainty principle is a thing and zero-point
energy is a thing doesn't mean that one explains the other ...
quantitatively or even qualitatively.

So the book is about as wrong as it possibly could be.