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Re: Various Questions



Regarding Jim Green's various questions:

1) Various texts give different results for Dx Dp: should this be h, hbar
or hbar/2 -- what is with this variation????? How did Heisenberg write it???

It depends on just what you mean by the symbols: Dx Dp. If they refer to
a Fermi-like quantum estimate of the product of ill-defined uncertainties
or spread in position & momentum for a bound or otherwise localized
ground(ish) or low-lying state, then 'hbar' is good enough. If it refers
to the product of the *standard deviation* of the parent distribution for
a hypothetically performed ideal position measurement with the *standard
deviation* of the parent distribution for a *different* hypothetical
choice of experiment where the momentum of the state was ideally measured
rather than the position, then the rigorous *lower bound* on the product
of these standard deviations is 'hbar/2'. If it refers to the 'effective
area' occupied in classical phase space corresponding to each quantum
state in an orthogonal quantum basis of the Hilbert space of quantum
states, then the product is 'h'.

2) For the First Law, what symbol is preferred these days for DQ and DW:
Delta, delta, dbar, or some other Byzantine symbol?

For infinitesimal quantities I prefer to use a 'd' with a slash through
it, i.e. dbar, for Q and W indicating that they are infinitesimal
quantities, yet are not necessarily differentials of any function. For
the internal energy and other thermodynamic functions, however, I prefer
to write their differentials as dE, dS, dH, etc. For finite integrals
over infinitesimal quantities I prefer use a capital delta representing
the change in the quantity if the infinitesimal integrated was the exact
differential of a function. In this case the delta indicates the net
change in the function between the endpoints of the domain of
integration. If the infinitesimal being integrated is not an exact
differential then the integral does not represent the difference in two
function values and using a delta would be misleading. In this case I
just write the quantity's symbol (i.e. 'Q' or 'W') without the prepending
slashed-d , and this indicates the finite integrated quantity.

In some appropriately *restricted* cases a quantity that normally would
be an inexact infinitesimal or its integral over some path may, in fact,
turn out to actually represent an actual function (or its differential)
in which case I would drop the slash from the 'd' in the differential and
*maybe* write the integrated change with a capital delta. Examples of
these special cases include calculating the EMF between two points in a
region where there are no time-varying magnetic fields, the heat absorbed
under quasi-static and boundary-rigid conditions, the work done under
adiabatic conditions, etc.

3) How would one refer to the two parts of DxDp phase space? Real?
Momentum? Momentum is just as real as Dx isn't it? Any suggestions for
customary usage?

I don't understand what you are asking here. Certainly since both
position and momentum are represented by Hermitian operators their
experimental values are always real (rather than possibly complex). The
classical phase space for them R^2 is real-valued as well. What are you
asking?

Just something to say -- it is raining and I can't work my garden.

Do you normally work in your garden at night?

David Bowman
dbowman@georgetowncollege.edu