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



Planck's constant is also called _the quantum of action_ .
Action means area in phase space.

I cringe whenever somebody says that quantum mechanics
says energy is quantized in units of ℏ ... because
energy is not quantized, and ℏ doesn't even have units
of energy. It has units of action i.e. momentum times
distance aka energy times time.

Maybe sometimes you can turn ℏ into an energy, if there
is some definite frequency involved, if the system is
harmonic ... but what if it isn't? A particle in a box
is grossly anharmonic. A 1/r potential is grossly
anharmonic. A lot of things in this world are anharmonic.

As I said before, I see two perfectly good questions here:
1) How do we understand the zero-point energy, and
2) How do we understand the uncertainty principle.

I'm not at all convinced they are the same question.
Just because the uncertainty principle is a thing and
ground-state energy is a thing doesn't mean one explains
the other.

On 11/2/18 9:42 AM, Carl Mungan wrote:

I’m a bit surprised that you so quickly dismiss HUP.

-- I like the uncertainty principle.
-- I like phase space.
-- 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.

I worry when people mention the uncertainty principle
and energy in the same breath. Such statements are not
necessarily wrong, but they might be, and they're usually
not where the pedagogical emphasis should be. The
uncertainty principle doesn't naturally talk about energy.
It talks about area in phase space.

Also, the uncertainty principle talks about all states, not
just the ground state.

This is super-important. Being able to count states is what
allows you to make the connection between the modern (post-1898)
concept of entropy in terms of probability and the earlier
notions of entropy in terms of heat and temperature.

Here's how I visualize the uncertainty principle. The model
is not entirely faithful to reality, but it's close:
https://www.av8n.com/physics/coherent-states.htm#fig-glauber-movie-squeezed

The state has a certain length and width, which are constrained
by the uncertainty principle. The length and width have got
nothing to do with the energy of the state. The energy is
approximately 100 in the appropriate units, and is not quantized.
You could change it to some non-integer like π^4 and the behavior
would look the same.

If you thought that energy was quantized, at least you were in
good company. Einstein made the same mistake. He even won a
prize for saying so. This probably caused the progress of
physics to be set back 50 years.
-- Planck got it right in 1903, and warned everybody not
to over-interpret his formulas.
-- Einstein didn't listen, and engaged in some serious
over-interpretation starting in 1905.
-- Glauber mostly straightened things out many decades
later.

This thread was started by the observation that a rigid rotor
doesn't have any zero-point energy. That's a tremendously
valuable clue. This applies to orbital angular momentum, which
you can think about in semi-classical terms ... and also to
spin, which is completely non-classical. I'm not even sure
that spin has a phase space in the usual sense. But you can
count states, and each state has one unit of ℏ, so all the
important stuff still works.