Application of Zentropy to the Quantum Bound State

Application of Zentropy to the Quantum Bound State Francesco R. Ruggeri Hanwell, N.B. Nov. 12, 2025 The notion of zentropy as given in (1) seems to deal with grouping a set of probabilities based on a parameter i, e.g. p(i). In the usual case, Shannon’s entropy is S = -k Sum over i p(i) ln(p(i)) and no grouping appears. There may, however, be physical subgroups that exist within the range of i’s. In such a case, one may consider relative probabilities as these ranges take on a physical identity themselves and so one may wish to have an entropy apply to each of these new physical entities (groupings) as well as overall entropy linked with the uncertainty related to not knowing in which group a particle sits. In this note, we suggest that one may apply this idea to the quantum case of a bound state. In quantum mechanics, one normalizes P(x) = W*(x)W(x), where W(x) is the wavefunction, through Integral dx (all space) W*(x)W(x) = 1. There seem, however, to be two physical states present, namely the classical bound one sitting within the classical turning points and the tunneling portions sitting outside. Thus, instead of using a W*(x)W(x), one may introduce a two value probability q1, q2, where q1= Integral dx (inside turning points) W*(x)W(x) and q2=1-q1. There is then an entropy linked with not knowing whether a particle is in the bound region or the tunneling region, i.e. -k Sum over i qi ln(qi) as well as intrinsic entropies for each region. These entropies, however, should be based on the relative probabilities W*(x)W(x)/q1 in the bound region and W*(x)W(x)/q2 outside. Each “system” is normalized to one in this relative probability picture instead of having the entire system normalized to 1. Thus, we argue that the zentropy formalism of (1) applies to a quantum bound state. Definition of Zentropy We provide a definition of zentropy following (1). Imagine that one has the usual Shannon’s entropy: S = -k Sum over i p(i) ln(p(i)) ((1)) In general, one leaves matters as given by ((1)). It is possible, however, that one may group various i’s into physical subsystems. In such a case, each of these systems might have its own intrinsic entropy and there would exist an entropy linked to the lack of knowledge related to not knowing into which subsystem a particle sits. In the case of separate physical systems, one wishes to have a probability normalized to the specific system, not a probability normalized to the superset of systems which is what p(i) represents. If: qk = Sum over i’s in k p(i) ((2)) one may use the relative probability p(i)/qk for all i’s in the set k. Then: S= -k Sum over k Sum over i’s in a specific k p(i) ln(p(i)) ((3)) One has now broken the overall i system into a set of physical subsystems, each denoted by k. p(i) should be normalized to the subsystem to which it belongs, i.e.: p(i)/qk ((4)) Then ((1)) becomes (setting kb=1): S = - Sum over k Sum over i’s in k p(i) ln( qk p(i)/qk ) S = - Sum over k ln(qk) Sum over i’s in k p(i) - Sum over k qk Sum over i’s in k p(i)/qk ln(p(i)/qk) or S = - Sum over k qk ln(qk) + Sum over k qk Sk ((5)) where Sk = Sum over i’s in k p(i)/qk ln(p(i)/qk) The first term on the RHS is the entropy associated with not knowing in which a subsystem a particle is found, whereas the second term represents a sum over the entropies of each subsystem, each weighted by qk, their representative probability. We now try to give an example for which this definition of zentropy is useful, namely the case of a one-dimensional quantum bound state. Zentropy and a Quantum One-Dimensional Bound State A quantum bound state involves an overall wavefunction which extends to +/- infinite even though a classical bound particle sits within the classical turning points. In quantum mechanics, KEave(x) = { Sum over p a(p) pp/2m exp(ipx) } / { Sum over p a(p) exp(ipx)} =0 at the turning points ((6)) The point is that one cannot simply have a quantum system represent the particle within the turning points. The reason is that even though KEave(x)=0 at the turning point, the p’s which appear in ((6)) are sizable and so there is no stopping the particle from leaving or entering the bound state (within the classical turning points). One cannot have KEave(x)=0 result from one or a few p values close to zero because the wavelengths of these hbar/p would be very large compared to dx in the quantum system. In a classical system, dx is “huge” and one may have a few tiny p values because hbar/p fits within dx. In order to create a localized or bound state, one is forced to impose: W(x)=0 for x= +/- infinite ((7)) Thus: Integral W*(x)W(x) dx (x from - infinite to infinite) = 1 ((8)) One may then introduce a spatial entropy as done in (2) with kb=1: S= - Integral dx (-inifnite to infinite) W*(x)W(x) ln( W*(x)W(x) ) ((9)) One might argue, however, that there are two physical subsystems in the quantum bound state problem, i.e. (A) The classical bound system within the classical turning points defined by KEave(x)=0 (B) The system outside the bound system, i.e. the tunneling system One may introduce two probabilities: q1 = Integral dx (x within the classical turning points) W*(x)W(x) ((10a)) q2=1-q1 = Integral dx (x outside the turning points) W*(x)W(x) ((10b)) There exists an entropy associated with not knowing if the particle is in the bound region or the tunneling region, i.e. S(which subsystem) = - Sum over i qi ln(qi) ((11)) There also exists an intrinsic entropy for each subsystem, but in this case W*(x)W(x) should be normalized for each subsystem, i.e. one should use: W*(x)W(x)/q1 for x within the classical turning points ((12a)) W*(x)W(x)/q2 for x outside ((12b)) Given that Shannon’s entropy is additive, one would expect: S overall = - Integral dx (all x) W*(x)W(x) = S(which subsystem) ((11)) + Sum over k qk S(subsystem k) ((13)) Here, S(subsystem k) = - integral dx (x’s for the subsystem k) W*(x)W(x)/ qk ln{ W*(x)W(x)/qk } ((14)) ((13)), however, is exactly the zentropy definition given in (1). Conclusion In conclusion, given Shannon’s entropy (kb=1), S = - Sum over i p(i) ln(p(i)), it is possible there may exist physical subsystems within the overall system. In other words, a set of i’s may be linked with a physical subsystem k. Then, Sum over (i’s linked to k) p(i) = qk, a probability to have subsystem k within the super-system. In such a case, one may consider two kinds of entropies. The first is the entropy linked with not knowing in which subsystem a particle sits, i.e. - Sum over k qk ln(qk). The second is the set of intrinsic entropies of each subsystem. In such a case, the probabilities of each subsystem k should be normalized to 1 and this done by using p(i)/qk for all i’s in subsystem k. Then: S = - Sum over i p(i) ln(p(i)) = - Sum over k qk ln(qk) - Sum over k qk { Sum over i in the set k p(i)/qk ln(p(i)/qk) }. This results in the definition of zentropy given in (1). We apply this definition to the quantum bound state system for which the two physical subsystems are: the classical bound state within the classical turning points and the tunneling material outside of these bounds. In quantum mechanics, one uses a single W(x) (wavefunction) for the entire supersystem and normalizes W*(x)W(x) over all space to 1. The zentropy approach allows one to define the probabilities q1 and q2 for the two subsystems and create an entropy expression: - Sum over i qi ln(qi) representing a lack of knowledge of which subsystem holds the particle. There is then an intrinsic entropy for each subsystem found by using W*(x)W(x)/q1 for the x region between the classical turning points and W*(x)W(x)/q2 for the x region outside. This leads to the notion of: -Integral dx all space W*(x)W(x) = - Sum over k qk ln(qk) - Sum over k qk { Integral dx (x values linked with subsystem k) W*(x)W(x)/qk ln(W*(x)W(x)/qk). Thus, we argue that the zentropy notion applies to a quantum bound system. References 1. Myers, L., Hew, N, Shang, S. and Liu, Z. Recursive Entropy in Thermodynamics: Establishing the Statistical Physics of Zentropy Approach (2025) https://www.semanticscholar.org/reader/6099766d6f6a998fd60ac414b17be77ac437a0d 4 2. https://en.wikipedia.org/wiki/Particle_in_a_box