In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency w of sqrt(w+ w-) (where w+ and w- are defined later). We find this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2. Only one set w,T for a single oscillator should map into the integrated result.
In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency w of sqrt(w+ w-) (where w+ and w- are defined later). We find this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2. Only one set w,T for a single oscillator should map into the integrated result.
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.)
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.
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.
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.
In previous notes (1) we discussed the free particle classical action in the form A= -Et+px in which p,x and E and t are independent for the relative and nonrelativistic cases. This leads to a scenario in which one cannot follow a particle in time i.e. there does not exist an x(t) as in classical mechanics. Instead one may develop eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) for the free particle. The solution exp(-iEt+ipx) is similar to a classical wave (which is a different solution following form A=-Et+px) in that it describes a kind of probability resonance in all of space. If one introduces a potential V(x), this is localized and one needs to think in terms of impulse hit resonances like exp(ikx) creating V(x). In other words, both an ensemble of exp(ipx) representing the particle and V(x)=Sum over k Vk exp(ikx) are described in terms of this resonance type periodic probability which interferes to create local features. One might argue that this feature of V(x) drives the particle. In this note we wish to examine the coupled oscillator H= p1p1/2 + p2p2/2 + w1w1/2(x1x1+x2x2) + .5w2w2(x1-x2)(x1-x2) (with m=1). Transforming variables leads to an equation with a separable solution, but in y=x1+x2 and y1=x1-x2. In this note we try to examine this solution in terms of the above resonance picture and also suggest a possible overall Shannon’s entropy.