Classical statistical mechanics (for example a Maxwell-Boltzmann gas) seems to be governed by interactions which affect the momentum of a particle, but do not involve its wavelength hbar/p actively in a calculation i.e. there are no interference effects. On the other hand, a quantum bound state involves both impulse hits which change the momentum (in a one dimensional problem), but also the associated wavelength which restricts the particle’s presence for certain x regions i.e. there is a resolution. In this note, we focus on the latter situation and attempt to formulate an entropy expression which incorporates both the momentum (impulse hits) and wave features in a manner different from: Sp = -Sum over p a*(p)a(p) ln( |a*(p)a(p)|) and Sx = -Integral dx W*(x)W(x) ln(|W*(x)W(x)|) where W(x)=wavefunction = Sum over p a(p)exp(ipx). We use the ground state wavefunction of the quantum oscillator as an example and first present an approximation based on resolution. For a quantum bound state, if the potential delivers an impulse which changes the particle into a p momentum state, then by translational invariance we consider it as having probabilities (resolution) to be at the peaks of cos(px). The overall probability for a given p, using p=hbar/wavelength, is then approximated as: hbar/ (L2p) * |a(p)| / {Sum over |a(p)| } (where hbar/(L2p) is 1/ (the number of half wavelengths which fit into L). This may be used in a Shannon’s entropy scheme to yield: S =- Sum over p hbar/(2Lp) |a(p)|B { -ln(B) + ln(C)-ln(p) + ln(|a(p)| } Where B=1/{Sum over |a(p)| } and C=hbar/2L with L being a cutoff length in the problem (to avoid using infinite). We apply this approach for the ground state of the harmonic oscillator. Next we try to refine the approach by replacing hbar/(2Lp) with an x dependent probability: |cos(px)|/ A(p) where A(p) is a normalization over one crest = A/p We also consider the ground state oscillator result with this modification and then suggest a refinement i.e replace constant1/p with |cos(px)|/p * constant2.
Classical statistical mechanics (for example a Maxwell-Boltzmann gas) seems to be governed by interactions which affect the momentum of a particle, but do not involve its wavelength hbar/p actively in a calculation i.e. there are no interference effects. On the other hand, a quantum bound state involves both impulse hits which change the momentum (in a one dimensional problem), but also the associated wavelength which restricts the particle’s presence for certain x regions i.e. there is a resolution. In this note, we focus on the latter situation and attempt to formulate an entropy expression which incorporates both the momentum (impulse hits) and wave features in a manner different from: Sp = -Sum over p a*(p)a(p) ln( |a*(p)a(p)|) and Sx = -Integral dx W*(x)W(x) ln(|W*(x)W(x)|) where W(x)=wavefunction = Sum over p a(p)exp(ipx). We use the ground state wavefunction of the quantum oscillator as an example and first present an approximation based on resolution. For a quantum bound state, if the potential delivers an impulse which changes the particle into a p momentum state, then by translational invariance we consider it as having probabilities (resolution) to be at the peaks of cos(px). The overall probability for a given p, using p=hbar/wavelength, is then approximated as: hbar/ (L2p) * |a(p)| / {Sum over |a(p)| } (where hbar/(L2p) is 1/ (the number of half wavelengths which fit into L). This may be used in a Shannon’s entropy scheme to yield: S =- Sum over p hbar/(2Lp) |a(p)|B { -ln(B) + ln(C)-ln(p) + ln(|a(p)| } Where B=1/{Sum over |a(p)| } and C=hbar/2L with L being a cutoff length in the problem (to avoid using infinite). We apply this approach for the ground state of the harmonic oscillator. Next we try to refine the approach by replacing hbar/(2Lp) with an x dependent probability: |cos(px)|/ A(p) where A(p) is a normalization over one crest = A/p We also consider the ground state oscillator result with this modification and then suggest a refinement i.e replace constant1/p with |cos(px)|/p * constant2.
The notion of entropy originates historically in classical physics and is intertwined with thermodynamics. Shannon’s entropy: - Sum over i P(i) ln(P(i)), where P(i) is the probability for i, however, seems to apply to any probability, but in quantum mechanics appears to be particularly linked to spatial and momentum entropy (1) i.e. - Integral dx W*(x)W(x) ln[ W*(x)W(x)] and - Sum over p a*(p)a(p) ln[ a*(p)a(p) ] where W(x)= wavefunction = Sum over p a(p) exp(ipx). In this note we speculate that one may create an entropy from the real part of exp(ipx) i.e. cos(px) as this term is associated with probability within a wavelength even though the “center of mass” of the particle may be moving uniformly at v=x/t=constant and W*(x)W(x)=1 or a constant. The latter implies an entropy independent of p but there is uncertainty in a free particle due to its wavelength which is experimentally manifested. We argue there is no reason to limit entropy to x or p considerations. For a bound state, a pattern of crests and troughs of different lengths and heights may be produced and so again there is a pattern of unnormalized probability distributions with different weights. The absolute value may be taken so negative wavefunction values are not an issue. For both the free particle and the bound state two features emerge. First the shape of the distributions is linked to the curvature which in turn is associated with the kinetic energy. Thus the probability distributions which describe the quantum system are governed by kinetic energy. Secondly, a pattern of distributions, both negative and positive, ensures that exp(ipx) is perpendicular to other exp(ip1 x) functions and Wn(x) is perpendicular to other Wn1(x)’s. In other words energy identifies the statistical state which exists throughout space and one cannot confuse the identity of one state with another. Thus they are orthogonal throughout space and probability distributions (linked to entropy) seem to enforce this. The inner product of W*(x) and W(x) is like an inner product of a product probability of a state and its time reversed state. We suggest one should be able to define a Shannon’s entropy expression for both |cos(px)|/C1 from exp(ipx) over a half wavelength where C1 is a normalization factor and |W(x)|/C2 roughly over the length over which it has humps i.e. roughly the classical turning points for an oscillator for example. This entropy is not additive and does not map to classical entropy in the large n (energy level) limit. Furthermore it may not be linked to thermodynamics because it is not linked to x or p. Nevertheless, it might be used to describe uncertainty within the quantity system and equilibrium/stable scenarios which are observed in nature. For example, different Wn(x) states are statistically stable states, but one may knock out the particle and it will be in an exp(ipx) state which is also stable. Furthermore, a photon may excite Wn1(x) to Wn2(x), thus this new entropy seems to be linked to stable states allowed in nature. The “price to pay”, however, seems to be uncertainty in wavelength regions for a free particle and the crest/trough regions for a bound state. We also consider some properties of a similar entropy for bound states.
In Part I we suggested that a free quantum particle with wavefunction exp(ipx) should have a spatial Shannon’s entropy associated with the probability cos(px)/C1 over half a wavelength, where C1 is a normalization constant. This new entropy captures the idea of uncertainty due to the wavelength even though the constant velocity suggests uniform motion i.e. W*(x)W(x) = constant. This idea was extended to a bound state and yields different results than the usual P(x)=W*(x)W(x) and P(p) =a*(p)a(p) based Shannon’s entropies where W(x)=wavefunction = Sum over p a(p)exp(ipx). In this note we try to determine some properties of this new entropy. First we consider the high n limit and use a classical approach introduced in (2). Next we expand the bound state entropy to include both spatial and momentum entropies and show that the sum follows from the product probability |W(x)|/C1 |a(p)|/C2 where 1/C1 and 1/C2 are normalization constants. We also examine quantum adiabatic processes. For P(x) = W*(x)W(x) and P(p)=a*(p)a(p), one finds that such a process leads to Sp+Sx remaining unchanged for L changing adiabatically for a box with infinite potential walls or for k the spring constant changing adiabatically in an oscillator. This follows from the fact that W(x/L) or W(x .5sqrt(km) ) (oscillator), from the normalizations Integral P(x)dx= 1 and Integral P(p)dp =1 and from exp(ipx) which shows that p scales in reciprocal manner of x. Thus we argue that these properties still hold for the entropies defined using |W(x)| and |a(p)| as probabilities. In other words Sp+Sx for the new quantum entropy does not change under a quantum adiabatic transformation. Finally we note that the new entropy is not compatible with the first law of thermodynamics.
Shannon’s entropy is defined for probabilities as - Sum over i P(i) ln(P(i)). For a probability density, such as spatial or momentum density which appear in quantum bound states i.e. P(x) = W*(x)W(x) or P(p) = a(p)* a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx), one defines: Sx = - Integral dx P(x)) ln{P(x)} and Sp = -Integral dp P(p) ln(P(p)) (1) and integrates over al space or momentum. For quantum bound states, however, one may integrate Shannon’s entropy using P(x) and/or P(p) only over the region of a single crest. Such a crest is of the form of an unnormalized probability density which has zero probability at the endpoints. Thus the total P(x) and/or P(p) might consist of a set of crests with possibly different heights. Thus instead of integrating over all x one may consider integrating only over the base of a particular crest to find a local entropy density so to speak. A reason for doing this is that the overall entropy may not satisfy dE = TdS, but an entropy for a crest might. We investigate this for an infinite well and an oscillator. For example, for a infinite potential well Sx over the full range 0 to L is independent of n, but by adding energy one creates pockets (crest entropies) which are very low as one has very high resolution.Thus we argue that a quantum bound system has an overall entropy, but also entropies for each crest (whether in x or p space). The latter may be linked to thermodynamics in a different manner than the former.
In a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
In classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
The von Neumann entropy definition is -Tr ( d ln(d)) and is linked to: = Tr (d A) ((1)). Here d is a matrix as is A i.e Aij = . As a result d is often not diagonal. It seems further that the idea of entropy is linked to an operator (2). In other words, one might have certain probabilities linked to the operator P momentum and a different set to X. For a diagonal d, von Neumann’s form is the same as Shannon’s entropy, but uses the eigenvalues of d. For a pure quantum bound state Wn(x), it seems that one considers the two operators X and P and writes two entropies using Shannon’s entropy form: - Integral dx W(x)W(x) ln(WW) and -Integral dp a(p)a(p) ln(a(p)a(p)) where W(x)=wavefunction= Sum over p a(p)exp(ipx). Here P(x)=W(x)W(x) and P(p)=a(p)a(p). For example in (3), one may see calculations of these “distinct” entropies for a particle in a box with infinite potential walls. X and P are linked to measurables and for a quantum bound system one cannot know exactly x and p at the same time. One may write < KE > for a bound state in two ways: Integral dp a(p)a(p) pp/2m or Integral dx W(x) (-1/2m d/dx dW/dx). Using Shannon’s entropy approach one obtains two different entropy values Sx and Sp. With respect to von Neumann’s entropy, it is zero for a pure state. Although one cannot know x and p at the same time (in terms of a measurement) that does not prevent one from writing x and p together (i.e. both known) in exp(ipx) for the wavefunction of a free particle. In this note we suggest that one might consider product probabilities linked to more than one operator when considering entropy. For example, for <KE> one may write Integral dx P(x) Integral dp pp/2m P(p/x) thus acknowledging that both x and p play a role. In this case, P(x)P(p/x) is the probability to use in Shannon’s entropy as we have shown in (4). In (3) it is suggested that P(x)P(p/x) be used in order to incorporate both variables x and p. Using P(x)P(p) in Shannon’s entropy yields the same overall entropy when integrated over x and p i.e. S=Sx+Sp. One reason that the sum is useful is that it eliminates parameters such as L (box length) for a particle in a box with infinite walls (or k the spring constant for an oscillator. This allows for a quantum adiabatic transformation of a pure state to be isentropic. This follows from a Fourier transformation i.e. W(x)=Sum over p a(p)exp(ipx). The two variables p and x scale with L in the opposite manner i.e. x/L and pL (5). Thus we argue that entropy is not necessarily linked to only one operator, but rather for a pure state is linked to variables which appear in a Fourier transform. Usually a Fourier transform is simply a mathematical procedure, but not in the case of the quantum pure state wavefunction where the variable p is actually momentum. In this note we try to investigate how the quantum relationship between p and x emerges. We argue that it exists at the level of the classical action for a free particle (relativistic or nonrelativistic) A=-Et+px where dA/dx partial =p and dA/dt partial = -E. Thus p and x are variable pairs in classical dynamics and this carries over to classical statistical phase space linked to p and x. We argue that a quantum dynamical link between p and x is also present in A i.e. -id/dx exp(iA) = p exp(iA). In other words p and x are linked dynamically in quantum mechanics as well (for a free particle) with exp(-iEt+ipx) being part of this link. The Fourier series form is already present in this solution as one may create a momentum distribution i.e. Sum over p a(p)exp(ipx). This depends on p and x and both affect any operator based on p and x (or one of these). Thus we expect that probabilities linked to both p and x should appear in an expression of entropy for a quantum bound state.