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.
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.
Classical statistical mechanical density maximizes entropy subject to constraints i.e. one has the Maxwell-Boltzmann (MB) distribution P(x)P(p) proportional to exp(- pp/2mT - V(x)/T). Quantum mechanics is also a statistical theory, but it is not based on the maximization of entropy. Rather it is based on a conditional probability P(p/x) = a(p)exp(ipx)/W(x) where W(x) =wavefunction = Sum over p a(p)exp(ipx) such that: Sum over p pp/2m P(p/x) + V(x) = E (bound states). As a result, it has classical densities P(x)=W(x)W(x) and P(p)=a(p)a(p), but these in general do not maximize entropy and so do not match the MB distribution. There is a special case, however, namely the ground state oscillator for which the quantum mechanical results P(x) and P(p) are both Gaussians. It is shown in (1) that for a fixed variance, a Gaussian maximizes entropy. Thus, the MB scenario must be a Gaussian for an oscillator as it is, but the quantum result is also a Gaussian and so matches the classical statistical mechanical result. In this note, we show that the time-independent Schrodinger equation collapses to the equality situation in Heisenberg’s uncertainty principle, suggesting that the solution also represents maximum entropy.
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.
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
Quantum mechanics is usually associated with loss of spatial information. For example, in the case of a superposition Sum over p a(p)exp(ipx), one has the Heisenberg uncertainty relation (standard deviation x) * (standard deviation p) >= hbar/2 and even in the case of a precise p, the probability exp(ipx) suggests an impulse hit can occur within dx =hbar/p. We suggest here, however, that there exists an example in which quantum mechanics retains precise spatial information even in the case in which classical mechanics does not. In particular, we consider the situation of a dx=hbar/p = 1 (i.e. in units of 2*3.14) and a one dimensional set of creation points for a particle: x=0, x=⅓, x=2/3. In such a case, the phase shift of each exp(ipx + phase shift) will identify the precise position (or pretty close if one considers the notion of a wavepacket) of the creation point in a steady stream scenario. In a classical situation, a steady stream scenario will completely wash out any information of the creation points of the particles as there is no distinguishing them in time. If one wishes to consider all different kinds of x creation points, then different p values are required. Thus, discerning space implies a loss of precise information of p.
Note (March 8, 2022) von Neumann's entropy may be written in Shannon's form for the eigenvalues of the density operator. If the density operator is written as Sum over j mj |j><j| with the set |j> not being orthonormal, then von Neumann's entropy would not be Shannon's entropy -Sum over j mj ln(mj) because the set mj are not the eigenvalues of the density matrix. In the literature, Shannon's entropy, however, is sometimes used with the weights of the projection states i.e. mj weights of |j>. For example, see: Kak, S. The Transactional Nature of Quantum Informationhttps://arxiv.org/pdf/0907.2465.pdf Note: (March 10, 2022) As an example of a state linked to the "fictitious" P(x)P(p) consider a single particle in a box with infinite potential walls. Outside the box W(x)=0 so P(x)=0, but P(p)=a(p)a(p) is nonzero for all p. Thus, one needs to consider P(x)P(p) together to make sense of the region outside of the box. Using the argument that one cannot know p and x exactly and so should consider P(x) and P(p) separately seems to lead to considerations of P(p) being nonzero i.e changes of measuring particle momenta outside the box which constradicts P(x)=0 outside. In (1) it is argued that von Neumann’s density operator d = Sum over i bi |i><i| = Sum over j mj |j><j> where in one case |i> represents an orthonormal basis and in the other of |j> a different nonorthogonal basis is used. The problem is that von Neumann entropy -Sum over i bi ln(bi) and -Sum over j mjln(mj) are not the same, they depend on the projection operator used. The author of (1) goes on to argue that historically it was suggested this might mean that a quantum state in quantum statistical mechanics is not defined clearly. We wish to consider the case of a pure bound state for which we have defined S=entropy=Sp+Sx where Sp=-Integral dp P(p) ln(P(p)) and Sx = - Integral dx P(x) ln(P(x)). According to quantum mechanics, one may not know p and x exactly, but S=Sp+Sx is equivalent to using P(p)P(x) in Shannon’s entropy formula. (Alternatively, one may use P(p/x)P(x).) As a result, the so-called state (x,p) associated with P(p)P(x) assumes knowledge of p and x exactly and so is not a real quantum state perhaps in the manner suggested in (1). In order to consider this issue, we argue that a quantum bound problem may be thought of in two ways: the classical mechanics type picture, and the statistical mechanical type one. The first approach is consistent with using the wavefuntion W(x) in the time-independent Schrodinger equation. P(p) and P(x) do not appear, but this is still a statistical picture because KE(x) = {Sum over p pp/2m a(p) exp(ipx)}/W(x). KE(x)+V(x)=E, nevertheless, has the form of a classical mechanical particle. In other words, it has one total E at any x and a potential value V(x) at x. In fact, KE(x) may be considered to represent a vrms(x) which is the velocity of the classical particle, although this is a purely mathematical construct. One may define P(p/x) = a(p)exp(ipx)/W(x) which is complex and allows for interference among exp(ipx) so this picture is very different from that of usual classical probabilities. The second picture of quantum mechanics involves P(x) or P(p) which behave as classical probabilities (i.e. are real, positive and sum to 1). For example: Operator ave = < W | B | W> is associated with this picture (as is von Neumann’s density matrix). One may write energy as: E= Sum over p P(p)pp/2m + Sum over x V(x) P(x), but this equation does not lead to solutions. One must use the Schrodinger to equation to solve for W(x) and find a(p) from W(x)=Sum over p a(p)exp(ipx) and then form P(x)=W(x)W(x) and P(p)=a(p)a(p). Nevertheless quantum statistics, in order to be linked with classical statistical mechanics, seems to be formulated in terms of P(p) and P(x). For example the diagonal elements of von Neumann’s density matrix contain this type of probability. Given that quantum mechanics may be viewed in two different statistical manners, (A) in terms of W(x), the Schrodinger equation and P(p/x) or (B) in terms of P(p) and P(x), a formulation of quantum statistical mechanics based on P(p) and P(x) may lead to unusual results. In particular, P(p) and P(x) represent exact p and x values, but according to quantum theory one cannot know these at the same time. Thus talking of P(p) and P(x) (in a classical way) seems to mean referring to a quantum state that is not really defined, perhaps one of the states noted in (1). This may not be a problem, however, if the underlying statistics follows from W(x) and a(p) and P(p/x) for which p and x are defined, but are linked to complex probabilities.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.