Some Thoughts about the Uncertainty Principle

Here are some thoughts about a way to make the uncertainty principle more intuitive.

In classical mechanics, action is the integral over time of L [ i.e.T-V]. But let’s call integral of H [ that is  L+ 2V or simply T+V] over time as “total action” . I think that it is total action that is quantized in integral quantities of h [Planck’s constant] but I am not sure how V fits in.  I think the rest energy must be included in all this.

Do all measurements of E require some finite time and all measurements of require some finite distance? If so, can they be thought of as measurements of total action?

In some ways E= h nu is getting it backwards, or at least upside down. What does frequency mean for a particle? On the other hand looking at it as

E tau = h where tau =1/nu is more intuitive. Tau is simply the characteristic time [or period] it takes for a particle of energy E to accumulate one h worth of increase in total action.

Similarly p lambda = h indicates that a particle of momentum p must travel a distance lambda to accumulate one h of increase in total action.

If total action is truly quantized, then there is no measurable change until total action changes by h.

 Heisenberg’s uncertainty principal can be explained as follows: Suppose one measures the change in total action over a time interval, t,  less than tau. Say sometime during that interval, t,  the total action changes by one h.  The only things you know are that sometime during t total action changed by h. From this you conclude that E could be as high as h/t. On the other hand if you think about it a bit you realize that your interval t could have started after a large fraction of tau had passed since the last increase in total action by h, so that most of the accumulation of energy time leading to an addition of h in total action occurred prior to your beginning your measurement. Therefore the energy could be much lower( near zero?). Thus you are uncertain about E by h/t.

Now suppose your measurement takes place over a period, t’,  several times tau, lets say  t’ = 7 tau for example. Depending on exactly when you begin your measurement vis a vis  when a period begins [ i.e. the time when the last increase in total action by h occurred prior to the measurement]  you will measure either 6 h or 7h as the change in total action. So you know the energy is between 6h/t’ and 7h/t’  but since t’ is 7 x t , you have reduced your uncertainty by a factor of 7.

Similarly, for momentum if you measure the particle’s total action over a length , L, shorter than lambda , you may detect a change in total action of h. Then you can conclude that the momentum may be as high as h/L. However, most of the accumulation of momentum times distance since the last change in total action may have occurred in the particle path before your measurement, so p could be much lower (near zero?) so your uncertainty in p is h/L. IF you measure over L’ = 7 lambda  you will measure between 6h and 7h change in total action, and the uncertainty becomes h/L’ or 1/7 of h/L.