Reconciling several views of the electron wavefunction

[Jay R. Yablon]

Dear Friends:

This is to follow up on several recent threads regarding the electron
wavefunction and uncertainty related thereto. Let me lay out a few
viewpoints, which I am seeking to reconcile, to see what your thoughts
are.

1) Solutions to Dirac's equation are obtained using the plane
wavefunction:

psi = u(p) exp[ip^u x^u] (1)

where u(p) is a four component spinor. I have heard suggestions that
one a) should not or b) cannot or c) need not use wavefunctions of any
order higher the above, which is linear in x^u.

2) It is often stated / noted that the uncertainty inequality:

delta x^u delta p^v >= (1/2) hbar (2)

is exactly equal to (1/2) hbar when the wavefunction is of the Gaussian
form (or variants that can be arrived at by "completing the square"):

psi = u(p) exp[-(1/2)A x^u x_u] (3)

see, for example, the calculation at
http://en.wikipedia.org/wiki/Uncertainty_principle#Matrix_mechanics, and
is greater than (1/2)hbar for a wavefunction other than a Gaussian.

3) Although the uncertainty inequality (2) applies equally to the time
(0) and space (k) components, that is:

delta t delta E >= (1/2) hbar
delta x,y,z delta p_x,p_y,p_z >= (1/2) hbar (4)

I am told that the canonical commutation relationship:

[x^u,p^v] = i g^uv u,v=0,1,2,3 (5)

actually should only be applied for u,v=j,k=1,2,3.

Now, some questions:

Q1) What is the uncertainty of a real electron in the ground state? Is
it equal to (1/2)hbar or is it greater than (1/2)hbar? If the
wavefunction is of the form (1), this is not a Gaussian, so I suppose
this would means that the inequality in (2) applies. But, why would
http://en.wikipedia.org/wiki/Uncertainty_principle#Matrix_mechanics be
exploring Gaussian wavefunctions to begin with, other than as a matter
of mathematical but not physical interest, if the solutions to Dirac's
equation are always taken to be plane waves?

Put in different terms, is there any circumstance under which a real,
physical electron could have a Gaussian or near-Gaussian wavefunction
and therefore have an uncertainty equal to or nearly equal to (1/2)hbar?
If so, how does one reconcile this with taking solutions to Dirac's
equation using only plane waves (1), which are complex sinusoids
cos(x)+i.sin(x) that are spread all over?

Q2) Now let's turn to the Robertson-Schrödinger relation, e.g., at
http://en.wikipedia.org/wiki/Uncertainty_principle#Robertson-Schrödinger_relation.
The uncertainty relationship (2), (4) appears to be a consequence of the
canonical commutation relationship (5) via the Robertson portion of
Robertson-Schrödinger. But the Heisenberg inequalities (2), (4) apply
equally to time / energy as well as the space / momentum components, and
(5) is said to apply only to space / momentum components. When this
limiting statement is made about (5) not apply to time / energy, are
there some other assumptions behind it? That is, are there situations
in which (5) does apply, with full four dimensional covariance, and are
the situations where (5) is restricted to space / momentum components a
special case and / or a case that involves particular assumptions? If
so, what are these?

Thanks,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm