Lab Notes for a Scientific Revolution (Physics)

May 8, 2008

How Precisely can we Measure an Electron’s Heisenberg Uncertainty? (or, How Certain is Uncertainty?)

   In a May 24 post Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?, I set forth the hypothesis that the anomalous magnetic moment first characterized by Schwinger, may in fact be a manifestation of the Heisenberg uncertainty relationship, and in particular, that the excess of the uncertainty over \hbar/2 may in fact originate from the same basis as the excess of the intrinsic spin magnetic moment g-factor g, over the Dirac value of 2.  This hypothesis is most transparently written as \Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} =\left(1+\frac{\alpha}{2\pi } +\ldots \right)\frac{\hbar }{2} , where \alpha is the running electromagnetic coupling for which \alpha \left(\mu \right)\to 1/137.036 at low probe energy \mu .  I also pointed out that a crucial next step was to employ a calculation similar to that shown at http://en.wikipedia.org/wiki/Uncertainty_principle#Wave_mechanics, but for a non-Gaussian wavefunction.

   I have now concluded a full calculation along these lines, of the precise uncertainty associated with a particle wavefunction of the general form \psi (x)=Ne^{-\frac{1}{2} A'x^{2} +B'x-V'\left(x\right)} .  (The primes are a convenience used in calculation where we define A\equiv A'+A'*, etc. when calculating expected values, to take into account the possibility of the wavefunction parameters being imaginary.)  While I refer to V'\left(x\right) as an “intrinsic potential,” it is perhaps better to think about this simply as an unspecified, completely-general polynomial in x, which renders the wavefunction completely general.  I have linked an updated draft of my paper which includes this calculation in full and applies it to the hypothesis set forth above, at Heisenberg Uncertainty and the Schwinger Anomaly. While the calculation is lengthy (and took a fair bit of effort to perform, then cross-check), the essence of what is contained in this paper can be summarized quite simply.  So I shall lay out a brief summary below, using the equation numbers which appear in the above-linked paper.

   The essence of the results demonstrated in this paper is as follows.  Start with the generalized non-Gaussian wavefunction:

\psi (x)=Ne^{-\frac{1}{2} A'x^{2} +B'x-V'\left(x\right)}   (4.1)

Calculate its uncertainty by calculating its Fourier transform \psi (p) (see (6.1)), by calculating each of its variances (\Delta x)^2 (5.4) and (\Delta p)^2 (7.4), and then by multiplying these together and taking the square root to arrive at the uncertainty.  The calculation is lengthy but straightforward, and it leads to the bottom line result:

 \Delta x\Delta p=\frac{\hbar }{2} \sqrt{1-2A'\left(\frac{dV'}{dB'} \right)^{2} +4B'\frac{dV'}{dB'} } =\frac{\hbar }{2} \sqrt{1-4A'V'\frac{d^{2} V'}{dB'^{2} } +4V'} .   (8.5)

   It is important to emphasize that (8.5) is a mathematical result that is totally independent of the hypothesized relationship of the uncertainty to the intrinsic spin.  So, if you ever been dissatisfied with the inequality of the Heisenberg relationship \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar and wondered what the exact uncertainty is for a given wavefunction, you will find this calculated with precision in sections 4 through 8, and the answer is (8.5) above.  The upshot is that (8.5) above is the precise uncertainty for a wavefunction (4.1) with A’, B’ and V’ all real.  We cannot give a position and momentum with precision, but we can give an uncertainty with precision.  The reasons for having A’, B’ and V’ be real are developed in the paper, but suffice it to say that A’, B’ real is necessary to avert a divergent uncertainty, and if V’ were imaginary rather than real, the uncertainty would always be exactly equal to \hbar/2 .

   Now, with the result (8.5) in hand, we return to the original hypothesis which, if it is true, would require that:

\frac{\Delta x\Delta p}{\hbar /2} =\sqrt{1+4B'\frac{dV'}{dB'} -2A'\left(\frac{dV'}{dB'} \right)^{2} } =\sqrt{1+4V'-4A'V'\frac{d^{2} V'}{dB'^{2} } } =\frac{\left|g\right|}{2} =1+\frac{a}{2\pi } +\ldots    (9.1)

   Using the series expansion for \sqrt{1+x} , we then make the connection:

V'\equiv \alpha /4\pi     (9.5)

   Now, it behooves us to return to the wavefunction (4.1), and use (9.5) to write: 

\psi (x)=Ne^{-\frac{1}{2} A'x^{2} +B'x-\frac{\alpha }{4\pi } } ,    (9.6) 

and to rewrite the uncertainty relationship (9.1) as:

\frac{\Delta x\Delta p}{\hbar /2} =\sqrt{1+\frac{1}{\pi } B'\frac{d\alpha }{dB'} -\frac{1}{8\pi ^{2} } A'\left(\frac{d\alpha }{dB'} \right)^{2} } =\sqrt{1+\frac{\alpha }{\pi } -A'\frac{\alpha }{4\pi ^{2} } \frac{d^{2} \alpha }{dB'^{2} } } =\frac{\left|g\right|}{2} =1+\frac{\alpha }{2\pi } +\ldots (9.7)

   Now, let’s get directly to the point: an electron with the wavefunction (9.6), with A' and B' real, will have the uncertainty relationship (9.7), period.  For \alpha =1/137.036, the leading uncertainty term \sqrt{1+\frac{\alpha }{\pi } } =1.00116073607, while the leading anomaly term 1+\frac{\alpha }{2\pi } =1.00116140973.  These two terms differ by just under 7 parts in 10^{-7} .  Therefore, we can state the following:

   TheoremFor a wavefunction \psi (x)=Ne^{-\frac{1}{2} A'x^{2} +B'x-\frac{\alpha }{4\pi } } with A' and B' real, the uncertainty ratio \frac{\Delta x\Delta p}{\hbar /2} , to leading order in \alpha , differs from the intrinsic Schwinger g-factor g/2 by less than 7 parts in 10^{-7} .

   We have stated this as a theorem, because this is a simple statement of fact, and involves no interpretation or hypothesis whatsoever.  However, in order to sustain the broader hypothesis

\Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} \ge \frac{\hbar }{2} ,   (3.4)

we do need to engage in some interpretation.

   First, we define (9.6) as the intrinsic wavefunction of a ground state electron with no orbital angular momentum and no applied external potential.  And, we define (9.7) as the intrinsic uncertainty of this intrinsic wavefunction.  Not every electron will have this wavefunction or this uncertainty or this g-factor, but this wavefunction becomes the baseline electron wavefunction from which any variation is due to extrinsic factors, such as possessing orbital angular momentum or being placed into an external potential, for example, that of a proton.  Thus, our hypothesis (3.4) is a hypothesis about the intrinsic uncertainty associated with the intrinsic wavefunction, and it says that:

   Reformulated HypothesisThe intrinsic uncertainty associated with the intrinsic electron wavefunction is identical with the intrinsic g-factor of the anomalous magnetic moment.

   The final section 10 of this draft paper linked above, is in progress at this time.  What I am presently trying to do, is make some sense of what appears to be a “new” type of g-factor \left|g_{{\rm ext}} \right|, emanating from an extrinsic potential (polynomial) V_{{\rm ext}}  in the wavefunction:

\psi (x)=Ne^{-\frac{1}{2} A'x^{2} +B'x-V_{int} \left(x\right)-V_{{\rm ext}} \left(x\right)} =Ne^{-\frac{1}{2} A'x^{2} +B'x-\frac{\alpha }{4\pi } -V_{{\rm ext}} \left(x\right)}    (10.1)

This new g-factor is defined in (10.2), and is isolated in (10.3) as such:

\begin{array}{l} {\frac{\left|g_{{\rm ext}} \right|}{2} =\sqrt{1+\frac{1}{\pi } B'\frac{d\alpha +4\pi dV_{{\rm ext}} }{dB'} -\frac{1}{8\pi ^{2} } A'\left(\frac{d\alpha +4\pi dV_{{\rm ext}} }{dB'} \right)^{2} } -\sqrt{1+\frac{1}{\pi } B'\frac{d\alpha }{dB'} -\frac{1}{8\pi ^{2} } A'\left(\frac{d\alpha }{dB'} \right)^{2} } } \\ {\quad \quad =\sqrt{1+\frac{\alpha +4\pi V_{{\rm ext}} }{\pi } -A'\frac{\alpha +4\pi V_{{\rm ext}} }{4\pi ^{2} } \frac{d^{2} \alpha +4\pi d^{2} V_{{\rm ext}} }{dB'^{2} } } -\sqrt{1+\frac{\alpha }{\pi } -A'\frac{\alpha }{4\pi ^{2} } \frac{d^{2} \alpha }{dB'^{2} } } } \end{array}.   (10.3)

In section 10, I have provided my “first impression” of where this new g-factor may fit in, in relation to the Paschen-Back effect, but would be interested in the thoughts of the reader regarding what to make of the above g-factor (10.3) and where it might fit into the “scheme of things.” 

Thanks for listening, and for your thoughts.

Jay.

Advertisements

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Create a free website or blog at WordPress.com.

%d bloggers like this: