The Energy probability distribution of quantum levels of a particle imprisoned in a three dimensional box
Tolga Yarman, B. Akkuş, Md Sadik Hassan Arik, C Marchal, S Cokcoskun, Alexander L. Kholmetskii, Ozan Yarman, Nimet Zaim, Azmi Ali Altıntaṣ, Fatih Özaydin
Abstract This work was trigerred by the earlier achivements of Yarman et al, aiming to bridge themordynamics and quantum mechanics, whence, Planck constant came to replace Boltzmann constant, and “average quantum level number” came to replace “temperature”. This evoked that the classical Maxwell energy probability distribution p(E) with respect to energy E of gas molecules might be taken care of, by the “energy probability distribution of the quantum levels” of a particle imprisoned in a given volume, assuming that in the case we have many particles, following Pauli exclusion principle, no pair of particles can sit at the same level. Thereby, the energy probability distribution of the quantum levels of a particle imprisoned in three dimensions, will be the subject of this essay. Such an outlook becomes interesting from several angles: i) It looks indeed very much like a classical Maxwellian distribution. ii) In the case we have as many free particles in the box as the number of levels depicted by the number of quantum levels in between the predetermined lower bound energy level and the upperbound energy level, all the while assuming that the Pauli principle holds, the distribution we disclose becomes the energy probability distribution of the ensemble of particles imprisoned in the given box. iii) It can even be guessed that, if elastic collisions between the free particles were allowed, and still assuming quantization and the Pauli principle, the outcome we disclose should be about the same as that of the energy probability distribution, molecules in a room would display in equilibrium. iv) The quantized energy being proportional to the sum of three squared integers associated with respectively, each of the spatial dimensions; the property we reveal certainly becomes remarkable from the point of view of mathematics of integer numbers. All the more, we further disclose that, to the probability distribution outlook remains the same, be this qualitatively for higher dimensions than 3.
Abstract This work was trigerred by the earlier achivements of Yarman et al, aiming to bridge themordynamics and quantum mechanics, whence, Planck constant came to replace Boltzmann constant, and “average quantum level number” came to replace “temperature”. This evoked that the classical Maxwell energy probability distribution p(E) with respect to energy E of gas molecules might be taken care of, by the “energy probability distribution of the quantum levels” of a particle imprisoned in a given volume, assuming that in the case we have many particles, following Pauli exclusion principle, no pair of particles can sit at the same level. Thereby, the energy probability distribution of the quantum levels of a particle imprisoned in three dimensions, will be the subject of this essay. Such an outlook becomes interesting from several angles: i) It looks indeed very much like a classical Maxwellian distribution. ii) In the case we have as many free particles in the box as the number of levels depicted by the number of quantum levels in between the predetermined lower bound energy level and the upperbound energy level, all the while assuming that the Pauli principle holds, the distribution we disclose becomes the energy probability distribution of the ensemble of particles imprisoned in the given box. iii) It can even be guessed that, if elastic collisions between the free particles were allowed, and still assuming quantization and the Pauli principle, the outcome we disclose should be about the same as that of the energy probability distribution, molecules in a room would display in equilibrium. iv) The quantized energy being proportional to the sum of three squared integers associated with respectively, each of the spatial dimensions; the property we reveal certainly becomes remarkable from the point of view of mathematics of integer numbers. All the more, we further disclose that, to the probability distribution outlook remains the same, be this qualitatively for higher dimensions than 3.
Historically statistical mechanics was developed for gasses for which there are very large numbers of molecules (i.e. on the order of 10 to 23rd power). As a result factorial schemes were developed to count the number of arrangements possible, the ln taken of this result and Stirling’s approximation for large N applied. The Shannon’s entropy form S/(-k) = Sum over i P(ei)ln(P(ei)) is consistent with this large N approach. Furthermore, large N statistical mechanics was linked to thermodynamics, in particular the first law dE=TdS - PdV. Surprisingly then, statistical mechanical results were applied to systems with small numbers of particles (e.g. a nucleus with about 100 nucleons etc). In Parts I and II we argued that the first law of thermodynamics describes two separate kinds of changes associated with energy changes. The second, work, is linked to the change of a mechanical parameter such as the volume of a box etc. The energy of a bound quantum particle in a box with an infinite potential is proportional to nn/LL. Thus we argued that a pure quantum bound state may also be associated with the first law of thermodynamics with changes in L with n, the energy level, held constant representing pure work. We further argued that for a bound state two classical like probabilities exists P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx) with P(x) being proportional to 1/L and P(p) to L. For a quantum adiabatic transformation (i.e. changing L with n staying constant) the term dS=0. If one wishes to create S from P(x) and P(p) and desires linearity i.e. S=Sp(n)+Sx, then Sp=Integral dp P(p)ln(P(p)) and Sx= Integral dx P(x) ln(P(x)) is a solution. If one next considers a quantum wavefunction for two levels W(x,t) = a(1)exp(-i E1 t) W1(x) + a(2) exp(-iE2 t) W2(x) with Eave = Sum over n en P(en) where Wn=a(n)a(n), one may again utilize the first law of thermodynamics. In Parts I and II we carried over the entropy form P(ei)ln(P(ei)). In this note we wish to argue that this form follows at the two energy level picture by considerations of a function which consists entirely of probabilities i.e. P(ei) and is linear in the sense that such a function for two identical two state systems should yield S1+S2 as the first law should apply to each system. Furthermore P(ei) should be a function of P(ei/T) where T is a parameter which behaves as 1/LL because ei is proportional to 1/LL. In this way any function of P(ei) remains unchanged under a quantum adiabatic transformation for which i’s stay constant, but L changes. As a result, the entropy at the quantum level seems to follow from such arguments and not from large N approximations of factorial counting approaches. In large N situations it is argued that P(ei)=C(T)exp(-ei/T) follows from maximizing S with respect to the form of P(ei) subject to constant average energy. For small N situations (e.g. the 2 level example) one may solve Sum over i ei d(a(i)a(i)) = T dS directly without maximization. The maximization approach follows directly from dE=TdS (work=0) because dE= Sum ei dP(ei) in such scenarios. This is of the form of a Lagrange constraint This has been pointed out in Part II. In this note, we argue that the maximization of entropy is not needed at all. For dE=Sum over i ei dP(ei) and the ln form of entropy P(ei) is the inverse function of ln. Thus we argue that linearity and not large N arguments are enough to obtain Shannon’s form of entropy and that this ln form suggests P(ei/T) = C(T)exp(-ei/T) if dE=Sum over ei dP(ei)= TdS (for dW=0). Thus maximization of entropy subject to constant average energy is also not needed (although it does follow mathematically from dE=TdS.)
Historically, thermodynamics was formulated for macroscopic systems followed by the development of statistical mechanics which provided microscopic formulations for quantities such as entropy. Entropy is traditionally linked to the ln of the number states available or to the Shannon’s form -k Sum over i P(ei) ln(P(ei)) where P(ei) is the probability for the quantity ei. In addition a variable T (linked to average kinetic energy) appears in both theories. Later the Schrodinger equation was developed to describe bound electron states in an atom etc. These states were characterized by a math object W(x) called the wavefunction and a set of discrete energy levels was calculated which matched data of emitted photons. In a previous note (1) we argued that the average energy En for a quantum bound state plays the same role of characterizing equilibrium as T in classical thermodynamics/statistical mechanics. We further argued that En is linked to parameters in the system. For example, for a particle in a box with infinite potential walls, En is linked to L as well as to n the energy level. One may change En by either changing n or L separately (or both). Holding L constant and changing n leads to a certain type of energy change. We argue this is similar to holding the volume of a gas constant and increasing T. Alternatively, one may leave n unchanged (a quantum adiabatic process) and change L i.e. a mechanical change or work. We argue that one may already consider the concept of the first law of thermodynamics at the quantum bound state level i.e. dE = dQ - dWork where Work is linked to a change in a physical parameter, such as L the length of the box, whereas dQ is linked to energy changes not involving a physical parameter e.g. a change in n the energy level. The Schrodinger equation yields solutions for W(x)=Sum over p a(p)exp(ipx) the wavefunction as well as En’s. We have argued in previous notes that W(x) is part of a conditional probability P(p/x)= a(p)exp(ipx)/W(x) and that En is constant at each point x just as T is constant at each x for classical statistical equilibrium (1). Thus a kind of statistical mechanics exists at the wavefunction level we argue. This statistical mechanics makes use of a free particle wavefunction exp(ipx) and weights a(p), but does not contain the usual P(x), P(p) picture of the classical world. We have shown in (2) that one may obtain P(x)=W*(x)W(x) and P(p)=a*(p)a(p) from the conditional probability. The essential physics we argue, however, is at the level of the wavefunction together with its own unique statistical scenario of exp(ipx)’s interacting with exp(ikx)’s from V(x)=Sum over Vkexp(ikx) (see (1)). Given that P(x) and P(p) exist for a quantum bound state, it may be convenient to use them in formulas because they are linked to a classical statistical picture. We argue that one may use P(x) and P(p) to construct an object S such that T dS does not change when one changes L, the size of box with infinite potential walls. We argue that such an object follows from properties of the Schrodinger equation together with various assumptions. We arrive at S=Sp+Sx where Sp=-k Integral dp a(p)a(p) ln{|a(p)a(p)}| and Sx= -k Integral dx W(x)W(x) ln(W(x)W(x)) for a given energy level. We do not use arguments related to taking ln of the number of states present or Shannon’s entropy. Rather we look for a solution which does not depend on L and is additive. We note, however, that for both W(x)W(x) and a(p)a(p) the solution found is mathematically consistent with the entropy form found by considering independent probabilities. This, however, applies to P(x) and P(p), but we argue that the physical situation is based on W(x) and a(p) as solutions of the Schrodinger equation. Nevertheless P(x) and P(p) are physical observables and not simply math constructs and so may think of them in a somewhat classical manner in terms of probability. Thus we argue that thermodynamics is already present at the level of a quantum bound state and that one may obtain Shannon’s entropy at this level using P(x) and P(p), but without considering the overall probability of independent probabilities as in information theory or numbers of states. Rather obtaining a function which is additive in Sp and Sx and eliminates L is the main consideration. We also argue that S=Sp+Sx carries over into mixed quantum state wavefunctions even though it is usually ignored. Thus we try to argue that it is the bound quantum state which gives rise to the notion of thermodynamics and entropy.
Note Nov. 12, 2022 In (1) the discrete energy is e=bn b=constant and n=0,1,2,3... not n=1,2,3 .. as listed below. Ideas of entropy appeared before quantum mechanics was developed. In particular Planck introduced discrete energy levels: en where e= constant and n=1,2,3 to describe allowed energies of an oscillator representing allowed light in a blackbody as shown in (1). In such a case average energy and entropy may be calculated using formulas from Boltzmann. In (1) it is shown that if continuous energy is used entropy is undefined as T=temperature→0 while entropy→0 as T→0 if there are discrete energy levels. As noted this problem applies to light and uses an oscillator model, but leads to the notion of discrete energy levels. In this note we suggest using a particle in a box instead and consider only two levels for simplicity. We still use Boltzmann’s probability factor exp(- energy(i)/T) where i=1,2. In such a case the two probabilities are 1-exp(-e(i)/T) and exp(-e(i)/T). If there is a gap between e(1) and e(2) then as T→0 the particle is in state e(1) i.e. has an entropy of 0. If there is no gap then one has a problem as e(2)-e(1)-->0, but T→0 as well. Thus it seems that a gap must be introduced in order that entropy behave properly as already argued in (1). This gap for the particle in a box, however, violates Newtonian mechanics which allows all momentum p values. As a result given that a gap is needed it may also be the case that e1 does not correspond to p1=0 as in Newtonian mechanics. We used the term box above and argue that the concept of equilibrium and T seem to apply to a confined system i.e one with a length constraint for a 1D example.. For example, the Boltzmann factor exp(-e/T) may be derived considering probability densities n(e(i)) with e1+e2=e3+e4 and n(e1)n(e2) = n(e3)n(e4). This is the reaction balance approach, but it seems that the particle must be confined for such an approach to even be defined. Thus we argue that there is something “special” occurring due to this constraint i.e. confinement in space. If energy has gaps then for a particle in a box: energy = pp/2m so p=momentum itself must have gaps. Furthermore these gaps must be somehow linked to the length of the box. Thus momentum is being linked (and energy too) with length. In fact ,given that one has gaps and uniformity throughout the 1D box, one may consider the length being broken into integer equal sized portions L/n. This is already a crude notion of a wavelength and leads to the idea of allowed p momentum values being associated with L/n in some manner. Finally we note that Fermat’s principle already links an invariant space variable with momentum conservation in the same direction. The classical analogue is Maupertuis’ principle for particles with rest mass. In both cases one may link these approaches with the Lorentz invariant A=-Et+px which happens to be both the relativistic and nonrelativistic classical actions of a free particle. dA/dx partial = p so an eigenvalue equation is: -id/dx exp(ipx) = p exp(ipx) with b/p representing the wavelength. Thus one has a frequency proportional to 1/E and a wavelength proportional to 1/p. Any p, however, is allowed so there is no gap. There is, however, an association of length with p. Thus if a box with a T has a length constraint one may suggest fitting integer numbers of sine wave crests into the box to create the “gap” required by entropy considerations.
Classical statistical mechanics is often derived using counting methods applied to all possible configurations a system may take subject to a total volume and a total energy constraint. Given that there are numerous configurations, there should also be an average of these configurations described by f(e) for the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions. In previous notes we have argued that one may apply a dynamical constraint to the average configuration. In particular, we argued that there should be a time reversal balance for two body elastic collisions. This immediately yields the various distributions with only consideration given to the average state and no counting techniques used. The counting of configurations, however, emphasizes that statistical mechanical systems contain “fluctuations” and that a specific dynamic constraint (i.e. time reversal two body scattering) applies only to the average. Quantum bound states, we argue, are very similar. They represent a statistical system which has many fluctuations (i.e. a particle may be knocked out with an momentum p with probability a*(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx)). There is, however, also a dynamical constraint which applies to the average, namely KE(x)+V(x)=E which is a classical mechanical constraint. Here KE(x)= [-1/2m d/dx dW/dx /W] ((1)). If fluctuations disappear in the high energy scenario, one may assume there is an almost fixed momentum p at each x. ((1)) still applies and there is still a wavefunction which is often in the form of crests and troughs. If an almost unique p exists at each x, one expects W(x) to vary in height at peak values. This may be verified from actual W(x) solutions. Given that the constraint on the statistical average is KE(x)+V(x)=E, one would expect that as p becomes an almost unique value at each x, that the quantum expectation Integral dx WW KE(x) with KE(x) given by ((1)) should also approach a classical value in other words Integral dx WW —> integral dt. One does not need to impose the correspondence principle to arrive at this result, one only needs to consider the case in which fluctuations in a quantum system disappear allowing the average p at each x to consist of almost a single p value.
The statistical problem of the distribution of s quanta of equal energy ε0 and total energy E among N distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by p(κ)=(N−1N+s−1)(N−2N+s−κ−2);κ=0,1,2,⋯,s In this context, the special case (N=4, s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of s and N. In the present article we derive alternatively p(κ) by distributing s quanta over N particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; sεo=E∼ fixed, where the energy of the particles becomes continuous, p(κ) transforms to the Boltzmann law P(ε)=⟨ε⟩1e−⟨ε⟩ε;0≤ε<+∞ where ⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
We have argued in previous notes that C(T) exp(-ei/T) (the Maxwell-Boltzmann probability) and exp(-iEt) exp(ipx) (the free particle quantum probability called a wavefunction) have the same informational structure in that product probabilities for a given E1=Ei+Ej for the MB case and Ei,pi and Ej, pj for the quantum case lead to uniform distributions, i.e. ones with minimal information if energy and momentum are conserved. We suggest here that the root for this may be same. We have argued in previous notes that dp/dt = Force cannot have dt be infinitesimally small. This means that there exists a scale at which dt is not 0 or even tiny. We also suggested that the force (even though obeying action-reaction in collisions) may strike at some probabilistic time within dt. We then used special relativity to find that dt =hbar/E and dp=hbar/p. A probability which demonstrates these gradations is exp(-iEt+ipx) (and also Lorentz invariant). This is akin to the notion that p(e1)p(e2)=p(e3)p(e4) and p(p1)p(p2) = p(p3)p(p4) (one dimension for simplicity). In other words, the stochasticity introduced is one of a uniform distribution for product probabilities. One could obtain the same result by starting with this assumption and then suggesting Lorentz invariance. We suggest that this quantum uniform distribution product probability also holds for the Maxwell-Boltzmann case. There, the probability is not complex because p(ei) must be a real number linked to the number of particles with ei. Nevertheless, in an elastic collision of ei and ej, we suggest that the quantum result (and even dp/dt=F) suggest that there is stochasticity in x and t and that one cannot deterministically predict the outcome. Any outcome which conserves energy and momentum has the same likelihood. We suggest that it is the quantum mechanical assumption which is identical to the ideal gas case (which uses quantum mechanical gas molecules albeit at high temperatures). The high temperatures mean that one can only use the conservation laws and the associated stochasticity. We suggest that the stochasticity of interaction inherent in quantum mechanics carries over to collisions in the classical gas. It is not simply that one is approximating the situation in a gas by throwing away information. If one were to try to be as deterministic as possible at some scale (the quantum one) stochasticity arises and there is no full determinism in a collision. The best one can do is use conservation of momentum and sometimes conservation of energy in certain situations like high temperature. If this is the case, one may ask: Why is it possible to perform deterministic classical calculations such as particles bouncing off walls or mirrors or interacting with a potential V(x) in a completely deterministic manner. We suggest that these involve constrained interactions, i.e. the wall or V(x) is associated with a huge mass compared to the particle, unlike in a collision of two identical particles. Action-reaction in the former case is associated with the large mass and so always the same approximately 0 kinetic energy and hence no p(ei)p(ej) =p(ek)p(el), but in the latter, with a mass the same size and so the stochasticity may be seen clearly.