In (1), which deals with entropic uncertainty relationships, it is pointed out that x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values. We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.
In (1), which deals with entropic uncertainty relationships, it is pointed out that x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values. We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.
We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
If one defines a conditional probability P(p/x) = a(p) exp(ipx) / W(x) where W(x)=wavefunction then Sum over p P(p/x) = 1, but P(p/x) is complex and its real and imaginary parts may have positive and negative values at different x points. Nevertheless one may establish an average energy conservation equation with the same En (energy of the nth level) at each x which mimics classical mechanics for all n values. This equation is identical to the time-independent Schrodinger equation. If one treats P(p/x) as a distribution and obtains an average En that holds at each x point like temperature holds at each x for a statistical mechanical gas, one should be able to calculate the variance at x as well i.e. Sum over p P(p/x) [ pp/2m + V(x) -En] [ pp/2m +V(x) -En] = Var(E(x)). We examine this value for a particle in a box with infinite potential walls and for a harmonic oscillator. One may then average Var(En(x). We also consider implications on entropy.
Note Nov. 12, 2022 In (1) the discrete energy is e=bn b=constant and n=0,1,2,3... not n=1,2,3 .. as listed below. Ideas of entropy appeared before quantum mechanics was developed. In particular Planck introduced discrete energy levels: en where e= constant and n=1,2,3 to describe allowed energies of an oscillator representing allowed light in a blackbody as shown in (1). In such a case average energy and entropy may be calculated using formulas from Boltzmann. In (1) it is shown that if continuous energy is used entropy is undefined as T=temperature→0 while entropy→0 as T→0 if there are discrete energy levels. As noted this problem applies to light and uses an oscillator model, but leads to the notion of discrete energy levels. In this note we suggest using a particle in a box instead and consider only two levels for simplicity. We still use Boltzmann’s probability factor exp(- energy(i)/T) where i=1,2. In such a case the two probabilities are 1-exp(-e(i)/T) and exp(-e(i)/T). If there is a gap between e(1) and e(2) then as T→0 the particle is in state e(1) i.e. has an entropy of 0. If there is no gap then one has a problem as e(2)-e(1)-->0, but T→0 as well. Thus it seems that a gap must be introduced in order that entropy behave properly as already argued in (1). This gap for the particle in a box, however, violates Newtonian mechanics which allows all momentum p values. As a result given that a gap is needed it may also be the case that e1 does not correspond to p1=0 as in Newtonian mechanics. We used the term box above and argue that the concept of equilibrium and T seem to apply to a confined system i.e one with a length constraint for a 1D example.. For example, the Boltzmann factor exp(-e/T) may be derived considering probability densities n(e(i)) with e1+e2=e3+e4 and n(e1)n(e2) = n(e3)n(e4). This is the reaction balance approach, but it seems that the particle must be confined for such an approach to even be defined. Thus we argue that there is something “special” occurring due to this constraint i.e. confinement in space. If energy has gaps then for a particle in a box: energy = pp/2m so p=momentum itself must have gaps. Furthermore these gaps must be somehow linked to the length of the box. Thus momentum is being linked (and energy too) with length. In fact ,given that one has gaps and uniformity throughout the 1D box, one may consider the length being broken into integer equal sized portions L/n. This is already a crude notion of a wavelength and leads to the idea of allowed p momentum values being associated with L/n in some manner. Finally we note that Fermat’s principle already links an invariant space variable with momentum conservation in the same direction. The classical analogue is Maupertuis’ principle for particles with rest mass. In both cases one may link these approaches with the Lorentz invariant A=-Et+px which happens to be both the relativistic and nonrelativistic classical actions of a free particle. dA/dx partial = p so an eigenvalue equation is: -id/dx exp(ipx) = p exp(ipx) with b/p representing the wavelength. Thus one has a frequency proportional to 1/E and a wavelength proportional to 1/p. Any p, however, is allowed so there is no gap. There is, however, an association of length with p. Thus if a box with a T has a length constraint one may suggest fitting integer numbers of sine wave crests into the box to create the “gap” required by entropy considerations.
Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.
-Et + px is an example of a Lorentz invariant. The two 4-vectors (p,E) and (x,t) transform according to a Lorentz transformation which is linked to viewing a system from different frames moving at constant velocities v. There is, however, a second solution namely t=C/E and p=C/x (C=constant) which also keeps the inner product constant. This is not linked to frame transformations because for a given p or E one has a specific x or t value which has to be thought of in a different manner. In particular, it may represent an interval (“pocket”) in space or time i.e. x as 1/p is a wavelength. This may also be connected with a free particle Action. If x as 1/p represents an interval in space, it seems there should be a probability distribution associated with such an interval. If this is the case, the idea of Shannon’s entropy should hold within that region. Thus increasing energy (and p) leads to smaller wavelength regions i.e. higher resolution. Using ideas from a bound infinite well we argue that entropy for a crest decreases as the resolution increases. Thus high energy of a constant velocity particle is linked to the idea of low entropy if one uses invariance in special relativity and utilizes a second solution which keeps -Et+px constant. This approach is equivalent to exp(-iEt+ipx) the wavefunction of free particle. Thus the idea of entropy and its relationship to energy may be linked with both special relativity and the quantum mechanics of a free particle, something that does not seem to be evident from classical mechanics.