محفوظ في:
| المؤلف الرئيسي: | |
|---|---|
| التنسيق: | Recurso digital |
| اللغة: | |
| منشور في: |
Zenodo
2025
|
| الوصول للمادة أونلاين: | https://doi.org/10.5281/zenodo.16929175 |
| الوسوم: |
إضافة وسم
لا توجد وسوم, كن أول من يضع وسما على هذه التسجيلة!
|
جدول المحتويات:
- <p dir="ltr"> In (1), it is argued that there is an inconsistency due to the fact that a time-independent bound state wavefunction W(x) has zeros. In particular, (1) argues that if a particle is at the left hand side or right hand side of the system, it must pass to the other side and so the probability W(x)W(x) cannot be 0 at any point, otherwise the particle cannot get from one side to another. </p> <p dir="ltr"> Here, we argue that the use of probability to solve a problem implies an inherent uncertainty of some kind. It might be suggested that this inherent uncertainty is simply due to a lack of information and that using probability is a quick way to solve the problem without worrying about excessive details. We suggest, however, that even in Newtonian mechanics, it seems that there is a probabilistic feature present in two body collisions which conserve momentum (and energy if they are elastic). Given two colliding particles in an elastic collision with energies e1 and e2, what is the outcome? Based on energy , one might argue that any ei,ej pair such that e1+e2=ei+ej is suitable and that there is no reason to think that any pair carries more weight than the other. In other words, trying to determine which ei,ej pair is created in a collision seems like asking for too much information and accepting the above statistical assumption is fine, even at the Newtonian particle level. </p> <p dir="ltr"> We have argued that applying these ideas to free particles leads to exp(-iEt)exp(ipx) (one dimension) as a complex probability as there is no energy distribution as in an ideal gas. Each free particle has the same modulus of probability such that any ei,ej pair or pxi, pxj pair (one dimension) has the same probability if they have the same sum. This, however, leads to a stochastic situation in x and t which one might not have expected. In other words, there is an inherent uncertainty present associated with more than ei,ej or pxi, pxj (one dimension) pairs having the same weight. Given that exp(-iEt)exp(ipx) is a probability, one might consider time independent problems and try to solve them probabilistically. This involves introducing uncertainty into the problem which may actually be linked to the solution. In particular. If one considers a two slit system, a particle passes through one slit or the other, but one does not know which and writes and OR (sum) of exp(i p dot r1) + exp(i p dot r2) where r1 is a vector from slit 1 to a point on the screen and r2, from slit to the same point. The result (which involves superposition) requires that one give up the notion of knowing through which slit the particle passes. </p> <p dir="ltr"> Similarly, solving for a bound state (time independent) means giving up knowing in which direction a particle moves, as we discuss. Thus, we argue that one cannot say that a particle moves from one side of the system to the other as in (1) and so must pass through a 0 point of the wavefunction W(x). The wavefunction W(x) which has the zero is created by giving up information on the direction of motion, just like one gives up information on knowing through which slit a particle passes. Thus, we argue that there is no inconsistency based on the ideas of probability of W(x) having 0 points.</p> <p> </p>