|Sep25-09, 06:23 AM||#18|
Heisenberg Uncertainty Principle
Sounds a bit like the chicken and the egg story. If you accept that the HUP is nothing more than the statement that two operators do not commute, than you can ask the question: what is more "fundamental"? The existence of non-commuting operators, or the describtion of states as elements of a Hilbert space?
As far as I know, canonical quantization comes down to both the introduction of commutation relations on some pair of conjugate variables plus the statement that states are now elements of some Hilbert space.
(Please note that you are not required to choose position and momentum as the non-commuting conjugate pair. In second quantization, for example, this role is played by the field operators, or equivalently, the ladder operators. The momentum and position are then no longer operators, but nothing more than labels!)
|Sep25-09, 08:22 AM||#19|
(From the historical viewpoint,)
Unless the quantum mechanics (based on the Schrodinger equation) is denied,
we must believe HUP. (Even if HUP is strange to us.)
Unfortunately, This is the fact.
I think that the turning point was when Hylleraas calculated the helium ground state energy by the variational methods based on the Schrodinger equation in 1928-1930.
(the error was only 0.01eV. The experimental value is -79.005eV)
On the other hand, Bohr model could not explain about the helium.
(But when we say this fact correctly, we must say that Bohr model could not explain about the helium without the computational method )
To get more information, search on Google by the words "bohr helium ground state energy".
|Sep25-09, 10:24 AM||#20|
Next, you perceive a contradiction in the HUP relative to theory. So did EPR* in their famous 1935 paper. They showed clearly an contradiction in the theory. Or so they thought. Actually, they made an unwarranted assumption which led to the apparent contradiction. That is what I referred to in my earlier post. That assumption is realism, i.e. the simultaneous reality of non-commuting observables. Turns out that is not only unwarranted, it also essentially conflicts with the HUP. Realism is not a requirement of QM.
QM incorporates the HUP as being fundamental, so I fail to see how any reasonable example is going to conflict with QM theory, given experiments to date. You will need to come up with a prediction that is different than theory. They tried that in EPR, and experiments (such as Aspect) supported QM over EPR.
*Einstein was the E in EPR, I assume you know that. So keep in mind that you are in good company with your questions. I do believe you would benefit from a thorough understanding of EPR, as it is all about the HUP and logic. But you will also want to study how entanglement allows the HUP to be probed in ways that EPR could never imagine.
|Sep25-09, 09:11 PM||#21|
These are different.
The mathematics only requires that the form of the interaction what ever that may be, does not allow one to perform simultaneous measurements of commuting properties to an accuracy greater than that given by the Heisenburg relationship.
However, some people claim it is a Principle of Nature, i.e. Heisenberg Uncertainty Principle of Nature.
They believe mathematics is physics, i.e. if it can be done mathematically and gives the correct numerical result it must represent the real nature of Nature. So beware.
|Oct7-09, 03:55 PM||#22|
I think as someone mentioned earlier in the thread, the uncertainty principle is usually used statistically on ensembles, however, I have seen some texts where it strongly implies it can be used for single particle measurements , the justification normally being that it can be used to explain the phenomena of the 2 slit experiment when only one particle is passing through , however I have also seen texts where this has been disputed.
Regardless I think it can be considered in the following way, firstly in the extreme case where either position or momentum can be considered to be measured to infinite accuracy , the inequality cannot be seen to apply, as deriving a value of the other variance would involve dividing by zero, which is not allowed mathematically.
Secondly we consider where one of the variables is not measured to be infinitely accurately, but vanishingly close, in this case , we would have to consider what we meant by that , as I would think that we cannot measure a position displacement more accurately than the plank scale, similarly for a momentum displacement , in order to measure it incredibly accurately we would need to have a device that could display an incredibly large amounts of digits (near to infinite as possible) after the decimal place, which we would then have to read, which would take a very long time if at all possible.
Finally even if we discount the previous two points, standard quantum mechanics is only really applicable for particles travelling at considerably less than c , as it is a non- relativistic approximation. In order to really examine the extreme cases you are considering you would need to use quantum field theory , as this is compatible with special relativity.
So, summing up, as the uncertainty relation can be derived from Schrodingers wave equation, I would regard it as a low energy, non - relativistic approximation , that nontheless can be used in the majority of applications but that does not necessarily apply exactly under the extreme conditions you are using as thought experiments. I hope this helps , please let me know if I can clarify further.
|Oct7-09, 04:09 PM||#23|
Also , it is important to point out that what the uncertainty relations mean can be coloured by whichever ontological perspective you are viewing from, for example in some interpretations of standard quantum mechanics the wave equation represents what are actual "potential" measurement probabilities that we can expect to achieve depending on what experiment we carry out and it is meaningless to ascribe definite properties of non-commutating variables in between measurements, and in other interpretations , it just represent all the potential INFORMATION (or lack of) it is possible for us to know , and in other interpretations it represents the relational potential observables between quantum object and observer (measurement device). There are now many competing interpretations, suggest you read about :
Cramers transactional interpretation
Copenhagen Interpretations (there are a few)
Relational Quantum Mechanics
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