The momentum wavefunction fp of a particle in a box (with infinite potential walls) has been described (1) (2) in terms of many plane waves (i.e. many fp’s) even though the energy E=(hbar k)/2/2m expresses only one k. Thus, the energy E appears to be an average. It seems one may have a plane wave exp(ipx) which collides inelastically with the wall and changes to exp(-ip1x) etc.; on average energy is conserved. In (1) and (2), the spatial wavefunction W(x)= Sum over p fp exp(ipx) is obtained by solving the time-independent Schrodinger equation, and fp from taking the Fourier transform of W(x). These are strictly math procedures it seems. In a previous note (3), the time independent Schrodinger equation is written with V(x) and W(x) both expressed in terms of Fourier transforms. This lead to an equation for each p of the form: Sum j+k=p Vk fk = (E-p*p/2m) fp ((1)). In such a case, V(x) exists throughout x. For the case of an infinite wall, there is still a potential, but only at the end walls. In this note, we suggest ((1)) may be used to obtain fp without taking the Fourier transform of W(x) or solving for Vk. This approach also shows cycling is involved, even for a particle in a box. The cycling occurs at the walls.
The momentum wavefunction fp of a particle in a box (with infinite potential walls) has been described (1) (2) in terms of many plane waves (i.e. many fp’s) even though the energy E=(hbar k)/2/2m expresses only one k. Thus, the energy E appears to be an average. It seems one may have a plane wave exp(ipx) which collides inelastically with the wall and changes to exp(-ip1x) etc.; on average energy is conserved. In (1) and (2), the spatial wavefunction W(x)= Sum over p fp exp(ipx) is obtained by solving the time-independent Schrodinger equation, and fp from taking the Fourier transform of W(x). These are strictly math procedures it seems. In a previous note (3), the time independent Schrodinger equation is written with V(x) and W(x) both expressed in terms of Fourier transforms. This lead to an equation for each p of the form: Sum j+k=p Vk fk = (E-p*p/2m) fp ((1)). In such a case, V(x) exists throughout x. For the case of an infinite wall, there is still a potential, but only at the end walls. In this note, we suggest ((1)) may be used to obtain fp without taking the Fourier transform of W(x) or solving for Vk. This approach also shows cycling is involved, even for a particle in a box. The cycling occurs at the walls.
The Schrodinger equation for a bound state includes the classical potential V(r) for which –d/dxj V represents Fj or force in the j direction. It has been argued in a previous note (1), that one may attribute physical meaning to the Fourier series of V(r). In such a case, V(r) and W(r) (the wavefunction) are quantities averaged over momentum (or vectors) as well as being averages over time. One may write the product of V(r)W(r) as one of two Fourier series and then reorganize into a series with basis exp(ikr). In such a case, one examines an interaction with V at each momentum wave level. Then the average is only over time as different k momentum states of in W combine with different p momentum states of the Fourier series of V(r). For a p momentum wave, one may write: p2/2m fp + Sum k Vk f(p-k) = E fp where W= Sum over p fp exp(ipx) in one dimension ((1)) The objective of this note is to observe differences in the interaction of a single plane wave with a potential ((1)) and the interaction of W with V in the Schrodinger equation. In particular, for a particle in a box with infinite walls, V(x)=0 inside the one dimensional box, yet the Fourier series of W includes all p values. From ((1)), even though V(x)=0 as an average, there seems to still be an interaction with various Vk values (Vk= the kth Fourier transform) in the interior of the box as ((1)) holds at all x points.
n an earlier note (1), it was argued the time-independent Schrodinger equation may be written in a momentum form by using W(x)=Sum over p fp exp(ipx)= wavefunction and V(x)=Sum over p Vp exp(ipx). This lead to an equation for each plane wave of the form: p*p/2m fp + Sum over j+k=p Vj fk = E fp ((1)). In this note, we try to argue that ((1)) does not need to follow from the time-independent Schrodinger equation, but rather seems to be a consequence of fp, which is related to conditional probability P(p/x).
In (1), a quantum particle in a box with infinite potential walls is described in terms of a p (momentum) distribution involving all p’s from minus infinite to infinite. The spatial wavefunction which is roughly of the form sin(kx) involves k which is the root mean square momentum. The energy E is an average energy which satisfies Integral dp p*p/2m fp*fp in one dimension. The condition of a zero wavefunction at the walls is also enforced. Here fp is the Fourier transform of W(x), the wavefunction, and is of the form cos(ap-pi/2 n)/ (E-p*p/2m) or sin(ap-pi/2 n)/(E-p*p/2m), where n is the energy level (1). In this note, we point out that fp may be obtained from a variational principle which uses W(a)=0 as a constraint. Next, we argue that there may somehow be a temperature in the wall which forces an average energy of Eave. Normally, one would treat such a system as a Fermi-Dirac gas or Bose-Einstein gas. Recently, however, (2), it has been suggested one may extremize Shannon’s spatial entropy to treat such a system. Here we extremize Shannon’s momentum entropy with the W(a)=0 constraint incorporated and compare with the case of (2).
In classical physics, there is a potential V(r) for a conservative force and force is linked to changes in momentum through Newton’s second law. For the sake of argument, imagine one may write V(r) as a Fourier series i.e. Sum over k Vk exp(ikx) and attribute physical relevance to this series. In other words, V(r) is a kind of average of microscopic granular system in which each exp(ikx) can impart momentum on a quantum particle. In addition, the Fourier series seems to have the appearance of a vector with exp(ikx) as a basis. It may also be noted that exp(ikx)exp(ipx) = exp(i (k+p)x). Thus, the potential seems to consist of ‘particles’ represented by exp(ikx) which may merge with quantum particles represented by exp(ipx) to form a boosted quantum particle exp(i (p+k)x). At first, it may seem exp(i (p+k)x) represents increased kinetic energy, but actually this merge should create potential energy. This perhaps gives a microscopic picture of potential energy. Thus, a merge of a Fourier series of V(x) with an ensemble of quantum particles Sum over p fp exp(ipx) yields: V(x) Sum over p fp exp(ipx) ((1)) A point which arises, however, is that Sum over p fp exp(ipx) cannot represent physical density because it may be negative at certain x . Thus, ((1)) is a departure from classical physics in which the potential energy is V(x) for one particle or V(x) density(x). ((1)) represents a kind of potential energy interaction, but introduces a new ensemble object Sum over p fp exp(ipx) which does not seem to exist in classical physics. This ensemble also seems to be a vector. The objective of this note is to try to determine properties and conditions which arise from this merge. In particular, we wish to find an equation satisfied by W(x) and also see how the classical picture is linked to this alternative scenario.
Quantum mechanics utilizes a wavefunction W(x)=Sum over p a(p)exp(ipx) (in time independent cases) with exp(ipx) being what we call a free particle complex probability. In such a case, W(x) is an OR scenario of these various probabilities. We have argued in previous notes that exp(ipx) is the probability for a p impulse hit at x, even if the center-of-mass of the particle is not at x. This means that impulse hits may occur probabilistically in space as in the 2-slit example which requires adding two probabilities as an OR case for each slit. The point then seems to be that W(x) (or even its momentum frozen form W*(x)W(x) gives a profile in space of the number of interactions (various p impulse hits) at each x. Classically, the kinetic energy (nonrelativistic) is pp/2m. In a classical bound problem this kinetic energy is considered to exist (in calculations) at the center-of-mass, but forces are also said to be rendered at this point so pp/2m, the center-of-mass and the force hit point all coincide. In quantum mechanics, a p impulse hit and the center-of-mass do not coincide. If p is said to hit at a probabilistic x, then it seems that pp/2m should also be seen to exist at this same hit point. A physical interaction point verifies in a sense the location of pp/2m in a probabilistic manner, i.e. a(p) exp(ipx)/W(x) is the probability used to calculate KEaverage and this is based on impulse hit locations. By normalizing using W(x), one removes the relative number of average hits at each x and obtains a pure energy value at x. (This may then be used in a KEave(x)+V(x)=E equation.) On average W(x) has zero points, which means that if one considers W*(x)W(x) to represent the motion-frozen-out spatial distribution of hits, then no hits are rendered at certain x points (on average). One might argue then that the particle cannot exist at x, but the center-of-mass can exist anywhere. One might next argue that there is no kinetic energy at x, but that cannot be true because 1/2m d/dx d/dx W has the same 0 points as W and the two divide. Thus, there is average kinetic energy at all bound points and this represents an average based on impulse hits (exp(ipx)). Classically, if one has many particles in a potential, one may write density(x)V(x) to find the potential energy of the particles. This implies that the particles interact with V(x), e.g. neutrinos usually do not interact. Density(x) in a classical sense means that the center-of-mass and force hit point are both at x. In quantum mechanics, exp(ipx) represents the probability for a force hit point, W(x) represents an average and W*(x)W(x) represents a motion-frozen average hit point. W*(x)W(x)V(x) seems to represent the average potential energy of the particle at x because W*(x)W(x) is linked with interaction. Even if a particle is not present, however, there is energy V(x) in space. It seems that if W*(x)W(x)=0 at some x, then there is no average particle interaction at this point, but there is still average kinetic energy Sum over p a(p)pp/2m exp(ipx) / W(x) based on the probability for average probability hits. The key idea is that one normalizes probability by dividing by W(x) in the average kinetic energy calculation. W*(x)W(x) is normalized not at x, but integrating over dx and setting this value to 1. One has relative hit at x values represented by W*(x)W(x). For exp(-ipx)exp(ipx)=1, each dx has the same average hit value (with one not being able to distinguish details in hbar/p, i.e. this is a motion frozen result. The question then becomes: Why would one write: W*(x)W(x) KEave(x)? One might argue that this term occurs if one takes the quantum mechanical equation: KEave(x) +V(x) = En and multiplies by W*(x)W(x). One may ask: Does V(x)W*(x)W(x) have physical meaning? V(x) is the potential energy at x. The average interaction probability with V(x) at x should be W(x)V(x) and not W*(x)W(x)V(x). The latter is a classical mechanical idealization. If W*(x)W(x) is 3 at x1 and 2 at x2, then it is as if the particle interacts 1.5 times more at x1 and so if V(x1)=V(x2), then this approach assumes 1.5 the interaction energy at x1. KEave(x), however, is simply free particle energy (average) at x and one does not need to take this average value and multiply it by the relative number of interactions at x. W(x) has zero points meaning that there is no interaction at such a point on average. KEave(x) on the other hand is the average energy at x based on the particle kinetic energy presence pp/2m being at the point of an impulse hit. This does not mean that an interaction is occurring with V(x) at this point in this formalism. Even a free particle has an exp(ipx) probability when it is not interacting. The sum Sum over p a(p)exp(ipx) means that all of the interactions linked with x have occurred and one is making an OR calculation. KEave(x) is free particle kinetic energy at x and V(x) is potential energy (not interaction potential energy with the particle). Multiplying by W*(x)W(x) is a classical minded approach to creating a classical equation: density(x) (KEave(x)+V(x)) = E density(x) ((1)) Here KEave(x) would represent the average kinetic energy of a single particle and density(x) would be the average number of particles at x. density(x)KE(x) has a clear meaning. W*(x)W(x) KEave(x) on the other hand is different. KEave(x) is actually the classical kinetic energy value at x and so it is not clear why one would multiply it by W*(x)W(x). W*(x)W(x) represents the probability for an impulse hit at x in a p-frozen out picture. The notion of exp(ipx) (i.e. probability for a p hit at x) is already buried within the quantum calculation of KEave(x) (which yields the classical value). In order to find the physical meaning of W*(x)W(x) multiplied by KEave(x) or V(x) it seems that one has broken dx into equally sized segments which correspond to different detection hits of the particle. This is then somehow equated with the presence of the particle per unit time in dx which is then somehow rendered as the average energy at x multiplied by the detected presence at x through a steady rate of measurement/time. Thus, multiplying by W*(x)W(x) is not multiplying by a physical density.
In a previous note (1), it was shown that for a ground state oscillator wavefunction, one could analyse the problem in momentum space as a free gas of momentum particles and obtain exp(-p*p/2mT) as fp*fp, fp being the momentum wavefunction. A free energy equal to zero was obtained, with kinetic energy being the energy term in E-TS. This suggests quantum particles scatter (with conservation of momentum) like classical point particles in momentum space, but differ from classical point particles in position space. In particular, one takes a Fourier transform of fp as it seems that each momentum wave carries an exp(ipx) function which is a sort of “baggage” term, possibly due to zitterbewegung. This seems to explain why a free gas in momentum space for the ground state oscillator yields a Gaussian density with W(x)=exp(-ax*x) (W(x)=wavefunction). An oscillator potential is then needed to keep this density with its momentum space pressure in balance. Thus, quantum mechanics and classical statistical mechanics become very similar for this scenario. This begs the question: What about other potentials or oscillator states higher than the ground state? In such a case, it still seems that one may consider “particles colliding in momentum space” with each carrying an exp(ipx) “baggage” term, but one does not have a “classical statistical mechanical” equilibrium scenario any longer. Rather, it seems that one has a net force created by -dV/dx (V(x)=potential) and a difference in pressure d/dx [Average(p*p)d(x)] where d(x) is density. There also seems to be an average velocity (1/W)dW/dx. In momentum space such a velocity would not exist for f(p)=f(-p), but because of the baggage of exp(ipx) it is present i.e. p exp(ipx) - p exp(-ipx). Newton’s law, with force acting on this velocity appears in the equation d/dx [Average(p*p)d(x)]=F(x) d(x). This same equation is equivalent to the gradient of the time-independent Schrodinger equation. For quantum mechanics, one has one particle rather than many as in classical statistical mechanics. Thus, it seems that this “Newton’s law” balance is based on a long term time average. It seems, however, that there should be a period T during which the average can establish itself. After this, one would have periodic time motion where the system keeps scattering back into itself in this characteristic time. This period can be deduced, it seems, from the time-independent Schrodinger equation and is found to be E, the energy.