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


El mié., 5 dic. 2018 a las 20:43, John Denker via Phys-l (<>) escribió:

On 12/5/18 4:04 PM, Diego Saravia wrote:

Is there a general demostration along these lines about area in clasical
phase space and dimension of quantum states (number of base states) or a
published one?

Pretty much any book on statistical mechanics touches on
this. They tend to not spend much time on it. Sometimes
they just assume it in passing rather than explaining it.
Often they assume you know things like:
-- The classical Liouville theorem says area in phase space
is conserved. There are lots of accessible writeups on
-- The corresponding QM result says that the equations
of motion are unitary, and therefore area in phase space
is conserved.
-- There is a concept of "state" i.e. microstate. To do
thermodynamics you need to be able to enumerate a set of
basis states. There's an influential book by some guy
named Boltzmann (1898) that really pounds on this idea.
However, I wouldn't call this "accessible" ... I find
it nearly impenetrable. But lots of good stuff is there
if you know what to look for.
-- In cases where it is easy to enumerate the quantum
states, e.g. harmonic oscillator or particle in a box,
it is easy to show that the area per basis state is h
(not hbar). Unitary evolution means that any other
basis will have the same property. There's an influential
published paper by some guy named Planck (1903) that
makes this point.

When Boltzmann's book came out, Planck was one of the
few guys smart enough to read it and make sense of it.
And wow, he really made good use of it.

Possibly useful:

Also Feynman's _Statistical Mechanics_ book talks about
the fundamentals. The ideas are correct and well explained,
but it was not carefully edited, so beware of typos in the

Feynman and Hibbs _Quantum Mechanics and Path Integrals_
will change the way you think about everything (including
thermodynamics) but it is not an easy read.
Forum for Physics Educators

Diego Saravia