Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

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



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

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

I don't "dismiss HUP". I respect it very highly.
I show my respect by not saying things about it
that aren't true.

I’m pretty sure all intro QM books “explain” (or at least “make
plausible”) zero-point energy as being due to HUP. Based on the
rotational case, I now find that “plausibility” argument to be
suspect, but I want further discussion from those who teach intro QM
before I decide to throw it out.

Well, you pretty much invalidated that plausibility argument
in the opening lines of the original post. Don't believe
everything you read in books.

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.

Most people think of quantum mechanics as an offshoot of classical
canonical dynamics (Hamiltonians and Lagrangians, etc.) ... but the
way I think of it, and the way some guy named Planck thought of it,
QM is an offshoot of thermodynamics.

The Heisenberg uncertainty principle and the second law are to my
mind the same thing. Planck was one of the few people on earth
smart enough to read Boltzmann's book and make sense of it. I draw
a straight line from that book to QM. If you're as smart as Planck
it's a rather short straight line, just a couple of years. He drew
pictures of the phase space of a harmonic oscillator. Entropy involves
counting areas in phase space and taking logarithms. Boltzmann thought
the unit of area was arbitrary, so entropy was defined only up to
an overall additive constant. But if you draw successive states,
each with the same predefined area, then !boom! the rest is history.
You might as well call the unit of area "h". There are simple phase
space diagrams in Planck's 1903 paper that make my eyes bug out every
time I see them.

It's ridiculously easy to draw the phase space for a particle in
a box, with axes of x and k. Standing-wave basis states are
convenient (*). They are uncertain in the x-direction by one
box-size, and they are equally spaced in the k-direction by
units of 1/x ... so we get cells of equal area.

(*) Of course this is not the only possible basis ... and basis
states are not the only possible states!

Connecting the particle-in-box to the uncertainty principle is easy.
So far so good.

====

The phase space for a harmonic oscillator is harder, but you can
draw successive equal-area states. They get thinner in the x and
p directions as N increases, but the incremental area is constant.
OTOH this basis -- the widely-used photon-number basis -- is a
nightmare for applying the uncertainty principle. The principle
normally applies to a pair of conjugate variables, dynamically
conjugate in the sense of classical physics. If you know the
Lagrangian, you can pick *almost* any variable you like, and
the Lagrangian will tell you what is the conjugate momentum.
Or vice versa. But this doesn't work for the N variable. There
is no ∂L/∂(N dot). Phase has the right dimensions to be sorta
kinda maybe akin to conjugate to N, but it's not really. As a
related matter, time is not conjugate to energy. You can derive
Heisenberg-ish tradeoff relationships involving N and phase, or
E and t, but these have to be derived separately and verrry
carefully; they should not be considered corollaries of the
normal uncertainty principle.

A lot of introductory textbooks fib about this. There's not
much I can do about that.

====

The phase space for a rigid rotor is not just complicated, it's
deceptive. The Lagrangian "looks" like the Lagrangian for a
free particle, and classically you can do things like writing
∂L/∂(θ dot) to find the conjugate momentum, but it's tricky.
For one thing, the boundary conditions are periodic, but it's
even worse than that, because the definition of dot product
is different. The dot product defines what we mean by length
and angle. You can't even define a θ variable that is both
continuous and periodic.

The correspondence principle remains valid and useful. It
tells us that in the classical limit, when we consider states
that have large action (i.e. large area in phase space), not
near the origin in phase space, it must be possible to construct
states that behave almost classically. This includes massive
pointy things that point in a particular direction, with a
particular classical value of θ. HOWEVER you can't run the
correspondence principle backwards. Just because something
is true classically doesn't mean it is true for all quantum
states.

The nastiness of the θ variable explains why it's hard to ask
what is "the" θ of this-or-that given state, but and instead
easier to ask what is the projection of the state onto the
conventional spectroscopic basis (1s, 2p, 3d, et cetera).
If you really want something that points in the +x direction
you can construct an sp hybrid. The sp wavefunction is
continuous and periodic as a function of θ, whereas θ itself
is not.

I'm not sure this answers the question, but this is how I
think about angles for quantum rotors and related systems.

===

Suppose your rotor is something simple like positronium.
You can still switch out of polar coordinates and ask about
the Cartesian rectangular coordinates of the electron and
the positron. You can directly apply the uncertainty
principle to those coordinates. It's probably not what
you want, but at least it's doable.