In (1), an outline of the derivation of the Gamow factor for alpha decay from a nucleus is given. In one dimension, the approach considers three regions in space, one with an incident and reflected free particle wavefunction, the second (within a square V barrier) with an exponentially growing and falling solution, and the third, with a single free particle wavefunction exp(-iEt)exp(ipx+phase) representing the escaped alpha particle. Alpha decay is treated as a quantum tunneling problem and the wavefunction and its first derivative d/dx must be continuous. A second step in (1), however, is to assume that one may write a time related wavefunction as exp(- g t), where d/dt partial Integral dx (region) W*W = ½ (-i W* d/dx W + i W d/dx W*). This may appear to be a quantum mechanical equation, but we argue that this is nothing more than the classical continuity equation. In fact, in (2) we argued that quantum mechanics does not exist without classical equations. In particular, the quantum free particle wavefunctions exp(-iEt) and exp(ipx) follow exp(iEt)exp(-iEt) = 1 and exp(-ipx)exp(ipx) = 1 to link them classical densities in time and space for particles moving with a constant speed. If there is a decay in time and space, then the appropriate classical equation is the continuity one. Writing d/dt partial d(t) = constant d(t) yields the familiar exp(-g t) form. We apply the ideas of the Gamow factor to a set of photons bouncing back and forth in an n1 1-dimensional cavity surrounded by n2 (n2>n1) index of refraction material. In this case, if the photons are synchronized then one may follow them classically. The catch is that the probability to reflect or refract is quantum. In general, one would not write exp(-g t) in such a scenario, but would formally write an equation of the form: fraction of remaining photons = R power Integer(t/t1), where R is the probability to reflect and integer(t/t1) is the number of cycles.. If, however, the probability to refract is very small, we show that one may in fact obtain the same result as in the case of exp(- g t) for g <<1. Alternatively, if the cavity is small in length and the speed of the particle (c in the case of light) is large, one could approximate the situation by d/dt partial d(t)= constant d(t) and then obtain exp(-g t), having the photons no longer in deterministic motion.
In (1), an outline of the derivation of the Gamow factor for alpha decay from a nucleus is given. In one dimension, the approach considers three regions in space, one with an incident and reflected free particle wavefunction, the second (within a square V barrier) with an exponentially growing and falling solution, and the third, with a single free particle wavefunction exp(-iEt)exp(ipx+phase) representing the escaped alpha particle. Alpha decay is treated as a quantum tunneling problem and the wavefunction and its first derivative d/dx must be continuous. A second step in (1), however, is to assume that one may write a time related wavefunction as exp(- g t), where d/dt partial Integral dx (region) WW = ½ (-i W d/dx W + i W d/dx W*). This may appear to be a quantum mechanical equation, but we argue that this is nothing more than the classical continuity equation. In fact, in (2) we argued that quantum mechanics does not exist without classical equations. In particular, the quantum free particle wavefunctions exp(-iEt) and exp(ipx) follow exp(iEt)exp(-iEt) = 1 and exp(-ipx)exp(ipx) = 1 to link them classical densities in time and space for particles moving with a constant speed. If there is a decay in time and space, then the appropriate classical equation is the continuity one. Writing d/dt partial d(t) = constant d(t) yields the familiar exp(-g t) form. We apply the ideas of the Gamow factor to a set of photons bouncing back and forth in an n1 1-dimensional cavity surrounded by n2 (n2>n1) index of refraction material. In this case, if the photons are synchronized then one may follow them classically. The catch is that the probability to reflect or refract is quantum. In general, one would not write exp(-g t) in such a scenario, but would formally write an equation of the form: fraction of remaining photons = R power Integer(t/t1), where R is the probability to reflect and integer(t/t1) is the number of cycles.. If, however, the probability to refract is very small, we show that one may in fact obtain the same result as in the case of exp(- g t) for g <<1. Alternatively, if the cavity is small in length and the speed of the particle (c in the case of light) is large, one could approximate the situation by d/dt partial d(t)= constant d(t) and then obtain exp(-g t), having the photons no longer in deterministic motion.
The problem of an incident beam of light traveling to the right in the x direction and reflecting/ refracting at an interface at x=0 may be solved using Maxwell’s electromagnetic equation for the electric field of a photon. In particular, mathematical continuity of the electric field and its first derivative, where the field is of the form Real( exp(ipx - iEt) ) with E energy being constant, but p allowed to change, may be used to establish relationships between weights of the incident, reflected and refracted electric fields. The modulus of each of these waves is then linked to intensity. The above continuity equation, however, is somewhat unusual if one thinks of a single quantum traveling in time. For instance, before it strikes it is an incident photon, but after (in time) it is either a reflected or refracted one. Thus probability naturally appears in the problem and if one performs the usual grouping by time, then the incident photon is one side and the reflected or refracted on the other. This is the opposite grouping used in the continuity approach so we ask: How may this be understood? On average a photon moves with a certain speed, but it also has a wavelength. We argued in (2) that for a particle with rest mass, there is no time within the wavelength region, but rather a probability to be at an x value is given by exp(ipx). In fact, the two dimensional nature of exp(ipx) accounts for the removal of time as one needs two “variables” or dimensions to describe the direction of motion. On average, however, the particle still moves as x=velocity time. If there is no time in the probabilistic approach within a wavelength, then even treating the particle as a point means that there is no clear separation in time of the incident, reflected and refracted rays. If one uses space as a marker (as in the time-independent Schrodinger approach to a bound quantum particle) then this spatial approach within a wavelength governs the solution to the problem. In such a case, one may write Aexp(ipx) + B exp(-ipx) = Cexp(i p2 x) and a similar conservation of momentum equation A p exp(ipx) -p B exp(-ipx) = Cp2 exp(ip2 x) rather than argue that the electric field and its derivative should be continuous at x=0. We use addition because there exists an OR situation, but we do not employ the usual grouping by time, but rather in space with this probability.On average, however, this approach must be consistent with AA/c1 = BB/c1 + CC/c2 where A*A etc are intensity values much like W*(x)W(x)=Probability(x) for a bound particle where W(x) is the wavefunction. Mathematically the above approach is the same as the continuity of the electric field and its first derivative, but we suggest that this view considers the role of time in the problem. The continuity approach allows for the incident, reflected and refracted photon to exist at the same time which only makes sense if time does not really exist within a wavelength, even though there is a mapping to x=vt. Thus both the loss of time and the idea of probability are needed and they complement each other.
Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.
Note: Dec. 28, 2022 The form exp(-iEt)exp(ipx) for both the photon and a particle with rest mass may be obtained from the Lorentz in variant -Et+px=A by noting that -id/dx A = p and that exp(ipx) is an eigenfunction of exp(ipx). In a series of notes we have considered the problem of light moving in the positive x direction hitting a medium along the y axis at x=0 and reflecting and refracting. We assumed that light is of the form of a photon which has an energy E which is the same in the incident, reflected and refracted beams. We also considered the notion of a steady state stream. Given a single photon in the incident beam, it has a probability a to reflect and 1-a to refract. As discussed in previous notes, one may consider an intensity AA, BB, CC for the incident, reflected and refracted rays (AA>0 etc) and write energy conservation as: AA/c1 = BB/c1 + CC/c2 where c1 is the speed of light in medium 1 and c2 in medium 2. We noted that the energy conservation equation has two unknowns B,C if A is known, but may be broken into two equations if one considers time removed from the problem (i.e steady state) and one groups by space i.e. (AA/c1 -BB/c1) = CC/c2 i.e. A+B=C and A/c1 -B/c1=C/c2. One may note that one does not have a conservation of flux/intensity equation, but that flux/intenisty which physically exists is directly linked to energy conservation. One may also note that one does not have an equation of energy flux conservation. In this note, we consider the dispersion relationship of light E=pc and the notion of a continuity equation which applies to the “square root” probabilities. We consider exp(-iEt+ipx) as the simplest solution, other than a constant and speculate whether exp(ipx) for a steady state picture represents a kind of square root flux. The idea of a square root flux suggests two sets of equations with square root fluxes which may be multiplied to create a flux equation. exp(ipx) has a kind of wavelike structure with two dimensions and may interfere. It also maps to a constant flux through exp(-ipx)exp(ipx). We consider these ideas and conclude that the wavefunction is like a square root flux which helps to associate a picture with the idea of it being a dynamical probability. This is helpful when considering a W(r vector) not equal to W(r) i.e. not spherically symmetric. This suggests unbalanced flux and rotation. In quantum mechanics, one does not have a container and so the fluxes create a static-appearing density in the bound case.
In Parts 1 and 2, we suggested that various physical scenarios such as -EE + p dot p cc = -momocccc and one dimensional reflection refraction probability conservation 1= P(reflect)+P(refract) are not involved with the direction of motion of a particle. If probability is involved (as in a wavefunction), these equations (even though they act on probability) do not show the direction of motion. In any physical test run, however, there is a specific direction of motion. As a result, we argued that there must exist equations linear in probability (exp(ipx) for a free particle) and id/dt partial, -id/dx partial (energy, momentum) which give rise to the physical non-directional cases. Here we retain the general idea of an equilibrium with no direction present, but argue that matters may be more complicated when different time events appear in space in a time-independent approach using exp(ipx)s. In particular, we analyze scattering and place 1-D reflection-refraction in the framework of scattering. We suggest that to create an equilibrium in both directions, one must consider both W=Aexp(ipx) + Bexp(-ipx) (for an incident p and reflected -p) and its flow -i W* d/dx W, together with the opposite direction scenario W*= A*exp(-ipx) +B* exp(ipx) and i(d/dx W*) W. Averaging yields p ( A*A - B*B), i.e. interference terms disappear in this scheme which directly considers motion in both directions i.e. A is linked with motion in both directions which does not match a physical run, but does match classical equilibrium (pressure). This result is not evident if one uses 1-D reflection-refraction at x=0 because one may then use a nd exp(ipx) and d/dx exp(ipx) set of equations to obtain directly 1=P(reflect) +P(refract), but interference terms creep in if one changes the junction to x1 as seen in Part 1. The averaging of both directions removes this issue. We see that exp(ipx) and exp(-ipx) represent equivalent kinetic energies in the time-independent Schrodinger equation. A math solution thus involves any identical energy directional derivative, i.e. exp(ipx) and exp(-ipx). There is no sense of time, so writing Aexp(ipx) + Bexp(-ipx) with A>B introduces a direction which is not in the Schrodinger equation, but is in a physical trial run. Given that the time-independent Schrodinger equation cannot discern time and probabilities add, a full solution should include various same-energy solutions which represent the particle at different times. There should then be a probability conservation linking these different time states, such as 1= P(reflect) + P(refract), but due to interference of W* d/dx W one has interference terms in x, but these may vanish if one considers motion in the opposite direction, i.e. Aexp(-ipx)+ Bexp(ipx), and classically averages expectations of -id/dx with each as one would do classically.
A one dimensional reflection/refraction problem is by nature probabilistic because an incident photon either reflects or refracts with certain unknown probabilities. The problem is traditionally solved by matching the electric field and its derivative with respect to x at x=0, t=0 where the electric field for a photon follows from Maxwell’s electromagnetic equations and is given by Eo cos(-wt+kx) where p=hbar k. As such, this problem is treated “classically” We argue, however, that this is a quantum mechanical problem (the photon being a quantum particle). Furthermore, no information is needed about the form of the electric field. Reflection/refraction conserves energy. It also occurs at some x point which may be varied i.e. there is translational invariance. A conservation of energy equation may be immediately written. The first medium has an index of refraction n1=1 and the second n2<n1. The conservation of energy equation is: AA/c1 = BB/c1 + CC/c2 where AA, BB and CC are the incident, reflected and refracted intensities = energy per photon * speed * number of photons. AA etc. is written to show that the intensity is explicitly positive. If one considers A known, then B and C are two unknowns, but there is only one equation. Thus energy conservation is not the full reality or information of the problem. This problem may be mathematically solved by using a difference of squares which leads to two equations: (A+B) = C and p1 (A-B) = p2 C. These may be solved for B,C in terms of A. This, however, is not the most general solution in that it does not show invariance of x i.e. the point at which the index of refraction changes. In this note we investigate the consequences of including x in the solution and argue this leads to a kind of square-root probability function which together with its first derivative should be continuous at any x chosen to be the interface of the media. This leads to a solution of Aexp(ipx), instead of A (etc), which is the free particle wavefunction. Thus this wavefunction (or unusual probability) is a direct consequence of the spatial invariance in the problem, but ensures that no x dependence appears if one combines the two square-root probability equations to create an energy conservation equation which has nothing to do with x.
In Parts 1 and 2 (and references therein), we tried to make an argument for the existence of a dynamic instantaneous probability which depends on x. This probability exp(ipx) is essentially the quantum free particle picture. The full probability is exp(-iEt+ipx), but in the 1-D reflection-refraction problem considered, E is the same for the reflected, refracted and incident photons. This means that if one establishes OR (addition) equations for probability, exp(-iEt) factors out, leaving only exp(ipx) pieces. In this note, we reiterate that probability already exists in the 1-D reflection-refraction problem as does a conservation of probability equation, i.e. 1= probability(reflect) + probability(refract) ((1)). Why should there be any other probability and other OR equations? We argue that the above equation ((1)) is a time equation which does not focus on space. One does not need to know that the incident and refracted photon move to the right, while the reflected one moves to the left. One does not need to know that the incident and reflected photons are restricted to x<0 for the n1-n2 index of refraction junction at x=0 and the refracted photon to x>0. In other words, one does not need to know anything about space. There are simply two events which are measured in time called reflection and refraction. If ((1)) is a probability conservation equation in time, and spacetime places x and t on equal footing, one may ask: Where is the spatial probability equation? We suggest that one must be careful. Time is positive, but x may be positive or negative. This, we argue, suggests that there should not only be a spatial probability dependent on x, but also two equations which demonstrate the two one-dimensional directions of space. These two directions are associated with motion or momentum and so one could have a spatial probability and its conservation equation, but also a conservation of momentum equation linked with this spatial motion. It is the spatial picture which shows the two directions of p, not ((1)). As a result, we suggest that one should expect the existence of a spatial probability and that it should be associated with two conservation equations, unlike the time probability with its single equation ((1)). This spatial probability viewpoint leads to free particle quantum mechanics, but these quantum equations which solve the 1-D reflection-refraction problem, also yield ((1)) when one combines the two spatial conservation equations, i.e. removes the spatial direction feature in the second equation and focuses only on the two events, reflection and refraction.