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.
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.
In Newtonian mechanics there exists a continuous x i.e. for a conservative force: .5mv(x)v(x) + V(x) = E with x taking any value. This is associated with the idea of dx→0. As a result one may choose an x and x+dx with dx→0 and v(x) and v(x+dx) become arbitrarily close in value. It is almost as if their identities merge. In (1) entropy and average energy are calculated using the Maxwell-Boltzmann MB factor exp(-e/T) and a continuous energy. The results E= NT (k=1) and S = -ln(T/To) with To being an integration constant show that one must have T>To because T→0 does not lead to S→0. Thus there is a problem if one uses continuous energy. In other words continuous energy is rejected.. (1) then considers an equally spaced energy i.e. e=bn, n=0,1,2,3 … and calculates energy and entropy again. In this case S→0 as T→0. Thus (1) concludes that it is entropy which leads to the discretization of energy In this note we try to argue that discretization of energy is already implied by the MB factor. The MB factor exp(-energy/T) where T is a math parameter called temperature represents a probability. In other words, different energies have different probabilities i.e. one must distinguish between them. Having two energy values merge is a problem now because one has a second parameter T which may also go to zero i.e. e2-e1→0 and T→0 leads to trouble because exp(-e2/T) does not tend to zero as it should if e1=0. We argue that the underlying reason for e1 and e2 being discretely different is that one cannot use the probability/T formalism i.e. MB factor unless interactions occur. We argue that such two body interactions cannot ultimately transfer or remove a de→0 and keep e1 and e2 distinct. In other words the interaction/collision is ultimately linked to a finite energy transfer even though T→0. As a result energy should be discretized in an equilibrium scenario. This argument does not make use of entropy explicitly although implicitly having exp(-e2/T)-->0 as T→0 (say for a 2 level system) means that the particle is in state e1 which may be associated with an entropy of 0 if one ignores internal features of this state. We also note that the continuous energy result E=NT, and not just the entropy, leads to trouble because as T becomes extremely tiny E which is an average of say e1 and e2 with e1=0 as an example approaches T in magnitude. The smaller T becomes the smaller e2 becomes which e2 and e1 no longer have separate identities, but in such a case one does not have an equilibrium scenario either. In other words exp(-e/T) implies interactions between distinguishable states with respect to energy.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
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.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
Early experiments linked to bound state quantum mechanics noticed discrete energy E levels. (Ignoring the issue of radiation from accelerating charges), classical physics predicts continuous E levels. If one tries a statistical approach i.e. thinks in terms of a “perceived” density dt=C dx/v(x) where v(x) is velocity, one again has continuous energy levels. Thus, experimental results may suggest a non-classical condition which affects both acceptable densities and energy levels. Bypassing Newtonian mechanics which traces a particle through x(t), we note that even classically a potential interacts with p and not v i.e. pp/2m - popo/2m = -V(x)+V(xo). This equation holds for p= mo v = (.5mo) 2v etc. Thus, energy flow i.e. p is of interest and if one does not trace the particle as x(t) or consider acceleration (which is not considered statistically), one may view energy flux p as being “spread” over the x axis, it seems. The length 1/p then corresponds to one unit of energy flux for the case of p. In other words, each p is associated with a length 1/p. If this length may be physically resolved, one immediate trial result suggests a spatial probability which varies over 1/p. In other words one may have a periodic function such as cos(px). This physical scenario seems to arise only when the quantum particle interacts with a potential. Calling 2*3.14/p = wavelength, this length is not associated with the velocity of the particle, but with energy flow. Each particle has an identical type of cycling within its own wavelength i.e. cos(px) behaviour as each wavelength represents one unit of energy flux. The wavelength limits resolution and leads to a probability scheme. In a bound quantum system dx associated with V(x) is usually much smaller than the larger wavelengths and so classical ideas such as p and pp/2m still apply, but are modified i.e. multiplied by probabilities which may be positive or negative. Having a spatial probability linked to density which also creates a probability scheme used to calculate average kinetic energy at each x based on 1/p type wavelengths is independent of classical physics. Trying to force average kinetic energy at x in the wavelength scheme to match KE(x) classical leads to discrete E values.
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.