Query on Heisenberg Uncertainty, Dirac Spinors, and Fermions "At Rest"

[Jay R. Yablon]

Hello to all:

I'd like to get some advice on a problem I am presently considering.

Let's consider a Dirac spinor u(p), and for a concrete example, let's
take the u^(1) for an spin up electron, where N is the usual
normalization factor:

/ 1 \
| 0 |
u^(1)=N | p_z / E+m | (1)
\ p_+/E+M /

To place this "at rest," we set p=0 and E=m, so that:

/ 1 \
| 0 |
u^(1)=N | 0 | (2)
\ 0 /

But, because of the uncertainty principle:

delta t delta E >= (1/2) hbar (3a)
delta x delta p >= (1/2) hbar (3b)

there is really no such thing as a "zero" momentum p=0, or E=m, only the
expected values <p>=0, <E>=m. How, therefore, might one write u^(1) for
an electron "at rest," in light of uncertainty?

Specifically, can we ever really have the state (2) in light of
Heisenberg, or, does one need to take the p and E in (1) and use in
their place, the expected values <p> and <E>, and / or the Root mean
square deviations delta p and delta E? For example, might one use, at
rest:

E --> m +/- delta E (4a)
p --> 0 +/- delta p (4b)

or some such expression other than p=0, or E=m.

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

[Jay R. Yablon]

"Jay R. Yablon" <jyablon@nycap.rr.com> wrote in message
news:67jog5F2ooi2mU1@mid.individual.net...
> Hello to all:
>
> I'd like to get some advice on a problem I am presently considering.
>
> Let's consider a Dirac spinor u(p), and for a concrete example, let's
> take the u^(1) for an spin up electron, where N is the usual
> normalization factor:
>
> / 1 \
> | 0 |
> u^(1)=N | p_z / E+m | (1)
> \ p_+/E+M /
>
> To place this "at rest," we set p=0 and E=m, so that:
>
> / 1 \
> | 0 |
> u^(1)=N | 0 | (2)
> \ 0 /
>
> But, because of the uncertainty principle:
>
> delta t delta E >= (1/2) hbar (3a)
> delta x delta p >= (1/2) hbar (3b)
>
> there is really no such thing as a "zero" momentum p=0, or E=m, only
> the expected values <p>=0, <E>=m. How, therefore, might one write
> u^(1) for an electron "at rest," in light of uncertainty?
>
> Specifically, can we ever really have the state (2) in light of
> Heisenberg, or, does one need to take the p and E in (1) and use in
> their place, the expected values <p> and <E>, and / or the Root mean
> square deviations delta p and delta E? For example, might one use, at
> rest:
>
> E --> m +/- delta E (4a)
> p --> 0 +/- delta p (4b)
>
> or some such expression other than p=0, or E=m.
>
> 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
By the way, I think the first step is to recognize that E and p make
their way into the Dirac spinors via solving the Dirac eqaution:

(gamma^u p_u - m) psi = 0 (5)

and this from:

(i gamma^u d_u - m) psi = 0 (6)

wherein:

i d_u psi = p_u psi. (7)

So, perhaps if the wavefunction is modified, and leaving (6) as is, this
will end up changing (7) and (5) and then causing the terms in the Dirac
spinors to change as well.

Is this on the right track?

Thanks again,

Jay.