Re: [Phys-L] [SPAM] Re: heat content

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding John D's answer to Anthony L's question:

> On 02/16/2014 06:03 PM, Anthony Lapinski wrote:
> > I always heard that at absolute zero, molecules have minimum
> > (not zero) motion. However, there is misinformation about this
> > online. So what happens at 0 K -- do molecules have motion or
> > not?
>
> Absolutely they do have motion, even at absolute zero.
>
> The fact that a hydrogen atom has nonzero size is most
> easily explained in terms of the zero-point motion of
> the electron.

I think a fairly compelling argument can be made for the contrary answer.  But I believe which answer is actully correct tends to hinge on the technical details of just what one considers to be the definition of the notion of 'motion'.  The idea that the quantum fluctuations in the possible results of a position or a momentum measurement of a quantum system in a prior *stationary* energy eigenstate (such as the ground state having a nonzero zero point energy greater than the classical ground state energy) constitutes a type of zero point 'motion' hinges on the fact that a multiplicity of possible distinct position values are obtainable from a position measurement, and the existence of multiplepossible nonzero momentum values from a momentum measurement for a system in a stationary state whose quantum Hamiltonian commutes with neither dynamical observable.  But it should be noted that such a concept of 'motion' is only a posteriori *after* such a measurement or measurements is/are made, in which case the post measurement state is *not* the initial state that was to be interrogated.

Now my way of thinking about the notion of 'motion' requires that at least *something* (e.g.  position, orientation, shape, spin, etc.) about the state be *changing* with time.  If there is no change with time, then there is no motion.  A stationary state is not called that for nothing.  A truly stationary state left to itself is completely time *independent* for *all* of the system's physical properties in that the probability distribution for the outcome of any measurement of *any* (and all) physical property(ies) is completely time *independent* as long as the measurement (or measurement-like interaction) never actually occurs.  But we also know that actually performing a measurement is definitely *not* leaving the system to itself, because the measurement interferes with the system, unless, of course, the measurement(-like interaction) only measures the system's energy or some other observable(s) that commutes with it.

So if one's notion of 'motion' (such as mine) requires that at least something about the undisturbed system be time *dependent* then a system in it quantum ground state (or indeed any truly stationary state) has no motion at all, zero point or otherwise.  But if one's definition of motion allows for time dependences *after* the system is interfered with (by making some measurement or having the system suffering a similar interaction with the system's environment) *then*, using such a notion of motion, the system *does* have a zero point *motion* (besides its nonzero zero point energy).

> I strongly recommend the following exercise:  you can
> estimate the Bohr radius using little more than dimensional
> analysis plus electrostatics plus the most basic quantum
> mechanical idea, namely p = ℏ k.  Treat the atom as a 
> particle in a box.
>  KE =  p^2 / 2m  for a particle in a box of size 2 a0
>  PE = − qq / r   for a separation r on the order of a0.
>
> If this takes more than half a sheet of paper you're doing
> it wrong.
>
> Bottom line:  If we didn't have any KE in the lowest 
> particle-in-a-box state we wouldn't have atoms.  This 
> would be a bad thing.  It is well known that atoms are 
> required to make physicists.

Again, one could argue that the system *doesn't* have a KE because the state is not an eigenstate of the KE operator.  Rather, it has a *multiplicity* of *potential* KE values as outcomes of a *potential* measurement of the KE observable *if* one decided to interfere with the system so as to perform such a measurement (and force the system to pick a value to register in the measurement apparatus).

> =================
>
> As another angle on the same idea:  As I wrote in the
> cryptography forum yesterday:  There is no such thing 
> as purely zero-point fluctuations as distinct from purely 
> thermal fluctuations;  those are just two asymptotes on 
> the *same* graph:
>   http://www.av8n.com/physics/oscillator.htm
>
>
> ===========================
>
> On 02/16/2014 06:09 PM, Chuck Britton wrote:
>
> > > As I understand it - Zero Point Energy is motion without 
> > > Entropy. 
>
> That's true.
>
> > > (totally ordered)
>
> Gaack!  I wouldn't have said that.  Please do not confuse 
> entropy with disorder.
>
> *) Entropy is a property of the macrostate.  It is defined
> as the ensemble average of the surprisal.
>
> *) Disorder, to the extent it can be defined at all, is
> a property of the microstate.
>  http://www.av8n.com/physics/thermo/entropy.html#sec-s-vs-disorder
>
> In the present case, the zero-point fluctuations *are* random.
> They are disorderly ... even they they are associated with
> zero entropy.

Yes and no.  At least two different kinds of entropy are at play here.  The entropy associated with the identity of the quantum state of the system is indeed zero.  This is also the entropy the system's thermodynamics cares about.  But the entropy associated with the uncertainty in the outcome of a measurement of some physical property whose quantum operator does not have the system's state as an eigenstate is not zero.  The more random or uncertain the distribution of possible measurement results, the greater this other kind of entropy is.  IOW, the entropies of the distributions of values of the quantities that have these fluctuations  are *greater* than zero.  Otherwise there would not be any random fluctuations in the first place, (they would just be fixed certain values for those quantities/properties).

> This is related to the fact that despite what you've been
> taught all along, energy is not quantized.  Planck's constant
> doesn't even have /dimensions/ of energy.  It is better to
> think of ℏ as the _quantum of action_ i.e. the quantum of
> area in phase space ... and even then, it is more a unit
> of measurement than a strictly quantized quantity.
>
> A zero-entropy state is spread out over one unit of area in
> phase space ... not zero units.  If you want the picture that
> goes with this, along with some more discussion, please see
>  http://www.av8n.com/physics/coherent-states.htm

The effective mean area in phase space occupied for a quantum state is 1 h-unit *if* the (zero entropy quantum) state is fairly close to a coherent or classical-like state.  But if an unsuitable choice of state is made (such as for states that are delocalized and greatly extended over both x-space and p-space) then the occupied area can become quite ill-defined and possibly effectively much larger, *unless* one chooses to count the occupied area in phase space in a funny way, for instance, by possibly doing something like subtracting off the area (weighted by the absolute value of the Wigner function) occupied by the state where the Wigner function is negative from the weighted area occupied where the Wigner function is positive valued and taking the net difference of the positive and negative weighted areas as the effective overall occupied area.

A simple example of a pure quantum (zero entropy) state that occupies a large amount of phase space where about half of the 'occupied' area is positive and half is negative is an ordinary SHO state with a large excitation n-number.  In appropriately scaled x & p units the Wigner function of such an energy eigenstate is rotationally/circularly symmetric in (x, p) phase space, and the value of the Wigner function does not begin to decay rapidly with increasing classical energy until it begins to significantly exceed the exact quantum energy of the state.  Inside that region (i.e. for lower classical energies) the Wigner function is like a target with alternating concentric circular bands of positive and negative regional values.  The sign alternates in the radial direction going through n nodes before finally decaying away at energies large compared to the exact quantum energy.  The exact value of the Wigner function for such a state is given by:

W_n(x,p) = ((-1)^n)*exp(-u)*L_n(2*u)/([pi]*h-bar)

where u is the classical energy divided by the quantum ground state energy, i.e.
u = (p^2/m + m*[omega]^2*x^2)/(h-bar*[omega]) and
L_n() is the nth order Laguerre polynomial.

David Bowman