Entropy in quantum mechanics is often based on von Neuman’s prescription (1) which yields zero for a pure eigenstate. Some alternative approaches also yield this result (2). In other words, it seems this type of entropy deals with transitions between different energy eigenstates and excludes the intrinsic entropy of a pure energy eigenstate. There is, however, literature (3) dealing with both spatial and momentum entropy of a single energy eigenstate. In particular, Shannon’s entropy equation yields - W*W ln[W*W] and -a(p)*a(p) ln[a*(p)a(p)] respectively where W, the wavefunction, is Sum over p a(p)exp(ipx). In a previous note (4), we argued that one may use Wn(x) a(p) exp(-ipx) as a probability in Shannon’s entropy equation to obtain an entropy which is the sum of Shannon’s spatial and momentum entropies for a single energy eigenstate. A third term proportional to ipx integrates to zero. In this note, we wish to extend this approach to a time-dependent wavefunction (for example: W(x,t)= Sum over n bn(En) exp(-iEn t) Wn(x).] We suggest W(x,t) bn(En)an(p) exp(iEn t) exp(-ipx) as a candidate probability for Shannon’s entropy and obtain: S= Sum over n bn(En)bn(En) ln(bn(En) + Sum over n bn(En)bn(En) {Sum over p an(p)an(p) ln(an(p))} Sum over x, t W*W ln(W(x,t)) + Sum over x,t,n,p i (Ent-px) W bn(En)an(p) For simplicity, we consider bn(En) and an(p) as being real. For the case of a single n present (i.e. a pure eigenstate) one obtains the entropy of such a state and not zero as in the von Neuman or other approaches. We argue that it might be good to have a single entropy formulation which works for both a single energy eigenstate and multiple states and not the two seemingly independent approaches which seem to be generally used. Furthermore, the approach here includes the intrinsic entropy of pure states together with that of transitions between states in the general W(x,t) (wavefunction) case.
Entropy in quantum mechanics is often based on von Neuman’s prescription (1) which yields zero for a pure eigenstate. Some alternative approaches also yield this result (2). In other words, it seems this type of entropy deals with transitions between different energy eigenstates and excludes the intrinsic entropy of a pure energy eigenstate. There is, however, literature (3) dealing with both spatial and momentum entropy of a single energy eigenstate. In particular, Shannon’s entropy equation yields - WW ln[WW] and -a(p)a(p) ln[a(p)a(p)] respectively where W, the wavefunction, is Sum over p a(p)exp(ipx). In a previous note (4), we argued that one may use Wn(x) a(p) exp(-ipx) as a probability in Shannon’s entropy equation to obtain an entropy which is the sum of Shannon’s spatial and momentum entropies for a single energy eigenstate. A third term proportional to ipx integrates to zero. In this note, we wish to extend this approach to a time-dependent wavefunction (for example: W(x,t)= Sum over n bn(En) exp(-iEn t) Wn(x).] We suggest W(x,t) bn(En)an(p) exp(iEn t) exp(-ipx) as a candidate probability for Shannon’s entropy and obtain: S= Sum over n bn(En)bn(En) ln(bn(En) + Sum over n bn(En)bn(En) {Sum over p an(p)an(p) ln(an(p))} Sum over x, t W*W ln(W(x,t)) + Sum over x,t,n,p i (Ent-px) W bn(En)an(p) For simplicity, we consider bn(En) and an(p) as being real. For the case of a single n present (i.e. a pure eigenstate) one obtains the entropy of such a state and not zero as in the von Neuman or other approaches. We argue that it might be good to have a single entropy formulation which works for both a single energy eigenstate and multiple states and not the two seemingly independent approaches which seem to be generally used. Furthermore, the approach here includes the intrinsic entropy of pure states together with that of transitions between states in the general W(x,t) (wavefunction) case.
In this note we wish to examine three ideas. First we consider the case of W(x,t)=Sum over n an exp(-iEn t) Wn(x) ((1)) (for n=0,1 as an example) and note that the von Neuman entropy is 0 for such a state. This is pointed out in (1) together with the fact that there exists a von Neumann density operator exp(-H/T) (H=Hamiltonian operator) which yields the same result as Shannon’s entropy S using P(ei) if one uses dQ/T=Sum over n en dP(en)/T = dS. It seems then that there is a contradiction if the usual von Neumann entropy is 0 yet using a density operator = exp(-H/T) yields nonzero results. A resolution is suggested in (2) which deals with information theory and we see if it applies to ((1)). Secondly, we consider the Shannon’s form P(e)ln(P(e)) in more detail. In Part I and in previous notes we argued that ln ensures that S=S1+S2 if one has two systems. We also argued that dE= Sum over n en dP(en) = TdS if dWork =0. In this note, we argue that a stronger idea may be at work, namely the idea that Eave= Sum over n en P(en/T) and TS are essentially the same function except for a function of T. This approach may be used to show that ln is the function associated with entropy. Finally, we note as in Part I that when using either the von Neumann approach or Shannon’s entropy one must know which probabilities to use. For a system with energy levels en, P(n)=P(en/T) is an immediate candidate, but there may be more probabilities in the system related to a particle jumping from one ei to another. In such a case, if one uses these probabilities and then sums over some of them leaving only P(en/T), the form of entropy changes from that of Shannon’s form. (Originally, however, Shannon’s form is used before summing.)
Shannon’s entropy: -k Sum over i P(i) ln(P(i)) is expressed in terms of P(i) the probability for state i. If entropy exists as a concept separated from the first law of thermodynamics with dWork=0, then it is not clear what set of states i should be taken in quantum mechanics. In classical statistical mechanics, each particle is automatically described by its position and momentum vectors for no internal motion. For example a pure energy eigenstate wavefunction Wn(x) may be written in terms of many different orthonormal mathematical basis sets e.g. Wn(x)= Sum over i bi phi(i)(x) with probabilities bi*bi. Thus one may create many types of entropy expressions using Shannon’s form without any other physical considerations. For example, W(x)=Sum over p a(p)exp(ipx) and one may create Sp which is usually done in the literature. In previous notes we have argued that Sn = Sp(n)+Sx(n). For a particle in a box with infinite potential walls, Sx(n)=Sx. The sum of the two entropies ensures that the parameter L (box size) vanishes so that a quantum adiabatic transformation is isentropic. Thus a consideration other than Shannon’s form has been used to obtain Sn=Sp(n)+Sx for a particle in a box. Furthermore one may argue Shannon’s form even for a pure state based on S=S1+S2 where 1 and 2 represent two systems. A question still remains: Why are Sx and Sp entropies used and not some other set? Furthermore for a system with several energy levels ei one might ask why -Sum over i P(ei)ln(P(ei)) is used as in (1)? In a previous note (2) we considered that Shannon’s form of entropy follows from dE= TdS (dWork=0) with dE=Sum over i ei dP(ei). Furthermore we argued that P(ei) = C(T)exp(-ei/T) is a solution, but so is C(T) exp(-ei/T) exp(Si) where Si=Sx,i + Sp,i for the pure bound state. In other words, specific states related to the physical situation are chosen and entropy constructed from dE=TdS to reflect this. In this note we argue that entropy may be less related to counting arrangements than to considering the issues involved in energy transfers within a system.
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.
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 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.
In the literature, Shannon’s entropy with spatial density as probability i.e. density(x) ln(density(x)) appears frequently. It seems, however, that this calculation of entropy assumes no knowledge of momentum, yet density=W(x)W(x) (for a bound state) where W(x) = Sum over p a(p) exp(ipx). In other words, the wavefunction W(x) contains information about the momentum distribution associated with P(p/x)= a(p) exp(ipx)/W(x). In classical statistical mechanics, entropy is calculated using exp[ - (p*p/2m + V(x))/T) which includes both momentum and spatial considerations. In a previous note (1), we calculated a possible expression for entropy using P(p/x)P(x). This includes Shannon’s spatial entropy and other terms even if one sums over p after forming Shannon’s entropy with P(p/x)P(x). In this note, we try to consider the interpretation of these two forms of entropy.