Re: [Phys-l] Quantum of action

[John Denker <jsd@av8n.com>]

On 01/07/2012 09:22 AM, Moses Fayngold wrote:

> Recently I stumbled upon an apparently trivial question: what is the
> origin of quantum of action (Plank's constant h). Originally, Plank,
> Einstein, & others were talking about energy quantization Delta E = h
> omega, with h already figuring as a universal constant. Later (1916)
> there appeared the Wilson-Zommerfeld quantum condition (often
> referred to as the Bohr-Zommerfeld condition) Int (p dq) = n h, with
> the integral taken over the period of a conservative system. This has
> a graphical interpretation as the quantization of phase space, which
> has dimensionality of action, and the Int itself is = S, the part of
> classical action. Hence the universally used expression "quantum of
> action", which I myself have used in my QM class without giving it
> more thought. But then I was reminded that general definition of
> action includes the temporal part as well, according to which the
> complete action is dS = pdx - Hdt, where H is the classical
> Hamiltonian. In that case the Bohr-Sommerfeld condition only proves
> the quantization of the quantity S + Int H dt. How come then did
> Plank, Sommerfeld and others conclude that h was a quantum of
> action?
>
> Also, is there a way to derive quantization of action within the
> framework of the new quantum theory, by bringing in the indeterminacy
> relation between time and energy, Delta E Delta t > h/2 pi (apart
> from Delta p Delta x > h/2 pi). If we additionally require Int Hdt =
> m h with integer m, so we would have S= j h with integer j = n - m.
> This also seems reasonable.  But I have never seen in literature any
> explicit mentioning of these complications. Any thoughts about this?
> Or any references to a sources discussing this questions?

I've been thinking some more about that.  Let's discuss the physics 
question first, before turning to the history question.

Executive summary:  In phase space, Planck's constant is the area 
per /basis/ state, for any given basis.  It is not the quantum of
action, and it is not the quantum of energy.  All this is discussed 
in greater detail, with diagrams and movies, at
  http://www.av8n.com/physics/coherent-states.htm


By way of background, remember there is a difference between "states"
and "basis states".  Consider for example a so-called two-state system,
such as the spin of an electron.  There are not just two states!  There
are two /basis/ states in any particular basis;  however, (a) there are 
infinitely many different ways of choosing a basis, and (b) even if you 
have settled on a particular basis, there are uncountably many non-basis 
states.

In a harmonic oscillator, there are countably many basis states in any
particular basis;  however (a) there are uncountably many different ways
of choosing a basis, and (b) even once you have settled on a particular
basis, there are uncountably many non-basis states.  Uncountable is a
whole lot bigger than countable.

So ... If you are talking about the actual states of the actual system,
they are not discrete.  They are not quantized at all.  Energy is not
quantized, and phase space is not quantized.  Real oscillators have some
damping, which means their resonance is not infinitely sharp.  Real
spectroscopic lines have some nonzero width.  Real wavepackets have some
finite extent in time and space.  Planck's constant is not the quantum 
of action, because the action is not quantized.  Planck's constant is
not the quantum of energy, because the energy is not quantized.

What you can say is that Planck's constant is the amount of area per
/basis/ state, in any particular basis.  It is perfectly possible to
collect a set of N states with less than Nh total area, but then the
states will not be linearly independent.

Quantum mechanics was born out of thermodynamics, and the two subjects
remain intimately connected.  The partition function Z is called the 
"sum over states" and indeed Z stands for Zustandssumme.  However, it
is not a sum over all states;  it is a sum over /basis/ states in some
basis of your choosing.  The states are not discrete;  only the basis
states are discrete.

As for the quantized energy levels in a harmonic oscillator, such things
are useful when doing theory, but they do not exist in the real world.
You cannot create such a thing.  They are stationary states, which means
they exist for all time, so it would not be necessary (or possible) to
create such a thing.

> Originally, Plank,
> Einstein, & others were talking about energy quantization Delta E = h
> omega, with h already figuring as a universal constant. 

That's not the correct history.  That's backwards.  The idea of a universal 
constant for measuring the area per /basis/ state in phase space predates 
quantum mechanics.

As for the idea that "all energy is quantized", (a) that came later, and
(b) it's wrong physics anyway.  The energy of real states is not quantized.
Also area in phase space is not quantized.  Planck's constant is not the
quantum of action, and it's also not the quantum of energy.

The best way to think about it is this:  Planck's constant is the area 
per /basis/ state in any given basis.

All this is discussed in greater detail, with diagrams and movies, at
  http://www.av8n.com/physics/coherent-states.htm

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

Pedagogical remarks:

Planck knew everything there was to know about classical mechanics and
thermodynamics -- including statistical mechanics -- /before/ doing 
anything with quantum mechanics.

I mention this because there are some folks, including some on this list,
who continue to insist that "the historical approach" is a good way to
organize and motivate the teaching of physics.  IMHO that seems nonsensical,
and the history of QM is a good way to illustrate the point.

Nowadays, for numerous good reasons, we teach quantum mechanics to kids
who do not know much about canonical mechanics or statistical thermodynamics,
and probably never will.
 -- Following the historical sequence would be bad pedagogy.  It would be
  wildly impractical.
 -- Pretending the practical sequence is historical is wrong history.  It
  is just plain dishonest.

Students first encounter the Hamiltonian in connection with QM.  Does that
mean that Hamilton was a contemporary of Schrödinger?  I don't think so!

Returning to the original subject of this thread:  Did the h in statistical 
thermodynamics come after the h in Schrödinger's equation?  I don't think so!

The pedagogical sequence should be logical and as simple as possible.  The 
actual historical sequence is much less logical.  It is complicated and
difficult and riddled with mis-steps.  Telling simplified "history" stories 
to students is a grave insult to all scientists ... past, present, and 
future.  For students who will be going into science, this causes them 
to seriously underestimate how difficult real science is.

What's worse, this causes _non-scientists_ to seriously underestimate how 
difficult real science is!  In a democracy, this is a disaster, because
non-scientists are the ones paying us to do science.

Physicists do not have a license to teach bogus history, any more than
historians have a license to teach bogus physics.

QM is a positive example, in the sense that most teachers are wise enough 
to follow the logical progression rather than the historical progression.
In contrast, special relativity is all-too-commonly a negative example, 
where teachers introduce the topic in terms of ideas that have been obsolete 
for more than 100 years.  Don't do it.  The modern ideas are simpler and in
every way better:
  http://www.av8n.com/physics/spacetime-welcome.htm