There are many examples in classical physics where the behavior of the physical system is practically indeterministic although it is described by deterministic laws.

Lets consider a small sphere placed on a convex surface in Figure 8 in chapter 6.

Lets take a look at the measurement process (figure 19) . We conduct a number of experiments (k times) where we let each time identically prepared wave functions Y_i (i for “initial”) enter the measurement apparatus. We end up each time with a different wave function Y_n after the measurement, the index n = 1,2,3 ... k indicating the number of the particular experiment.

Is it possible that the transition process from the initial state to the final state Y_n is not intrinsically or fundamentally indeterministic as assumed in Copenhagen interpretation but only apparently because the transition is susceptible to slightest differences in initial conditions .

Unfortunately the Schrödinger equation has a property that at first moment disrupts such fantasies. It is linear. The solutions of linear differential equations don't exhibit such a behavior. Slightly differing initial conditions lead to slightly differing final states. ¹

The reason is the following : Lets assume that a particular initial state of the wave function we symbolize by |a > evolves to the state |b > and that another state |c > evolves to the state |d > in a time period of D t. Now lets start with another initial state that is an arbitrary superposition of |a > and |c > where the weight of the state |a > is given by the coefficient a and the weight of the state |c > is given by c so that we have the initial state a|a > + c|c > . If the equation is linear each component in the superposition would evolve independently from each other so that the state after D t would be a|b >+ c|d > . If we change the composition of the initial state slightly by playing with the coefficients a and c the composition of the final state would change also accordingly. There is not a process like the following: If a is slightly greater then c then the contribution of |a > dominates rapidly so that we have a final state with almost no contribution of |d > and exclusively consisting of |b >. ( or if c is slightly greater then a then the contribution of |c > dominates rapidly so that we have a final state with almost no contribution of |b> and exclusively consisting of |d > ).

According to the classical description of motion of electric charges, when an electric charge is in a nonuniform motion namely when it is accelerated or decelerated or changes its direction, electromagnetic radiation is emitted. Because of the law of energy conservation the energy that is carried away by radiation lowers the energy of the particle. It slows down. It is as if there is an extra force, a ghost force acting upon the particle and slowing it down. It is a ghost force in the sense that it doesn’t originate from other electric charges. This force is called radiation reaction force. Newton’s equation F = ma doesn’t describe the motion of an electric charge in full if F is only the sum of the external fields. The validity of the equation F = ma can be maintained only when a term describing the radiation reaction force is added to the left side. The ordinary Schrödinger equation doesn’t take the effects of the dissipation of energy by radiation of an accelerated/decelerated wave packet into the account if only the external fields are considered in the Hamiltonian operator in the Schrödinger equation. To describe the effect of the radiation reaction force we have to include the energy of the electrons own electromagnetic field into the Hamiltonian operator. This has been done by D.H.Sharp ¹⁸ The result is that the we obtain classical expression for the radiation reaction force as a second order term . According to this result the accelerated motion of the average position of the particle leads to dissipation of energy from particle to the electromagnetic field. The calculation predicts also other purely quantummechanical higher order terms. In the following we will present a semiclassical description of radiation reaction force (namely we will threat the particle in usual quantummechanical way while we threat electromagnetic field as a classical field) that is easier to follow. The main result that there is a dissipation of energy from particle to field in form of radiation doesn't differ from the result of purely quantum mechanical description as in ¹⁸.

According to the ordinary Schrödinger equation describing a particle in a static external field the mean value ( the expectation value as it is called in quantum mechanics) of energy remains constant. If we interprete |y|² as the real physical charge density (encouraged by the arguments in the previous chapter) ² and calculate the radiation rate caused by the accelerated motion of this charge distribution in non-stationary states and consider the effects of the resulting energy dissipation from particle to EM field back on the wave function describing the particle, we can well understand how a susceptibility to initial conditions can emerge. It seems very strange to me that in discussions about measurement problem it is always emphasized that unitary evolution cannot describe what happens in measurement ^3,4 but the effect of radiation reaction force is not mentioned at all. Consider for example the position measurement. The electrons are slowed down rapidly and emit therefore radiation (Bremsstrahlung) when they hit the scintillating crystal. It is obvious that we leave in such process the domain of validity of Schrödinger equation if we don't consider the effect of radiation reaction force in the Hamiltonian operator.

According to the Copenhagen interpretation, since Y is considered only as a probability wave, and not a real physical field, this “charge density” is not considered to be as a real physical charge density, but only as a statistical average, one would obtain after many position measurements of identically prepared wave functions. If we would strictly hold to Copenhagen interpretation of Y as a probability wave, we would not be allowed to interpret the oscillation of the so obtained “charge density” as the oscillation of a real “charge cloud” that radiates according to Maxwell equations. Arguments listed in chapter 13 show that we can interprete |Y|² as reflecting the actual charge density distribution (directly for a one particle system and indirectly after integrations on particle coordinates for a many particle system).

As mentioned in chapter 6 the wavefunction y is a complex valued . It has a magnitude and a phase. (See appendix) Both magnitude and phase can vary from position to position i.e they are functions of spatial coordinates.

There are special solutions of Schrödinger equation that have such a form that the magnitude of y doesnt change in time and the phase changes in time with the same constant rate everywhere. These special solutions are called stationary solutions. The rate of the change of phase in time is called frequency ( w ).

For a certain given static Potential (given external electric and magnetic fields) There are infinite number of stationary solutions each belonging to different frequencies and having different shapes. For some frequencies there can be more then one solution belonging to the same frequency. The form of the potential determine the shape (namely how the magnitude and the phase vary in space) of stationary solutions . For free particles or for a repulsive potential there is a continuous energy spectrum(I.e. there exist at least one energy eigenfuction for each possible energy value).

 *****skippers mark******

However If the frequencies of two energy eigenstates are different their superposition is not stationary any more. The magnitude of Y changes in time.

In reality however since an accelerated charge radiates according to the maxwell equations, the energy of the charged particle in a nonzero potential dissipates even if the potential is static contrary to the expectations of ordinary Schrödinger equation describing a particle in external field. To further maintain the validity of Schrödinger equation a term describing the energy dissipation by radiation reaction force must be added to the Hamiltonian. It is like adding the radiation reaction force to the external force in Newtons equation F= ma. If we interpret the charge distribution given by |Y|² as the real charge distribution, this additional term must depend on the wave function itself.

This approach was taken in a little known paper of Enrico Fermi ⁵ . The solution of the equation leads to following interesting finding:

If the particle is in a stationary state (lets say state |2 > with energy E2) it remains in this state forever as expected, since there is no motion of the charge cloud. Thus a particle in an excited state remains in this state for ever. But as soon as there is a small contribution from a lower state(lets say state |1 > with the energy E1) , the model predicts a transition towards the lower state namely the contribution of the higher state decrease and the contribution of the lower state increase with elapsing time ⁵ .

Figure 22 - The transition from upper energy state to the lower energy state in Fermi's model 

The weights in superposition come closer to each other. This increases the oscillation of the charge density instead of decreasing it because the more both eigenstates overlap, the greater is the amplitude of the oscillation of the magnitude of the wave function (see Appendix) (This may seem strange from classical point of view, namely that the "motion" increases while the energy decrease. Quantum mechanically the energy is not immediately related to changes in magnitude. There are energy eigenstates with arbitrarily high energy yet they are always stationary states(no time dependence of magnitude). It is the frequency of phase-change-rate that determines the energy of the quantum system. It is somehow an internal motion that doesn’t effect the magnitude.) In short oscillations increase because of radiation, this in turn increases the radiation etc. Thus there is a positive feed back between radiation rate and oscillation amplitude similar to the positive feedback in the case of the sphere rolling down the convex surface in the example given in Figure 8 in chapter 6. . The positive feedback between oscillation of magnitude and the radiation rate continues until the contributions of upper and lower states are equal namely when the overlapping of both states is a maximum. This is the point where the oscillation of the charge density and accordingly the radiation rate is maximum. After this point the oscillation and the radiation begins to slow down. The process proceeds until the lower state is achieved asymptotically. Although it seems at first look as a deficiency of the model that a pure excited state (namely a state without even a slightest contribution from another state) can never radiate, this is not a deficiency for real world. Because of fluctuations in the external field, the Hamiltonian changes slightly all the time so that a state can never remain exactly as an energy eigenstate. ⁶ Since fluctuations are always there, a mixing from other states appear and disappear constantly in the real world. Thus the model provides a very intuitive understanding of spontaneous emission. It suggests also how susceptibility to initial conditions of a certain degree may emerge. It is a pity that it is not taught at the universities in undergraduate courses.

As mentioned during the emission process the system is in a superposition of the higher (frequency w₂) and lower (frequency w₁) energy eigenstates. The frequency of the oscillations in charge density distribution (the magnitude of wave function) in the superposed state is w₂ – w₁. (see Appendix)

To avoid a confusion let me repeat . In a stationary solution or energy eigenstate with frequency w it is only the PHASE that oscillates with w . The magnitude doesn’t depend on time . Therefore there is no oscillation of charge density. The magnitude and consequently the charge density oscillates only if there is a superposition of two or more energy eigenstates. If there is a superposition of more then 2 energy eigenstates there is an oscillation mode for any possible pair of energy eigenstates. For a superposition of 4 states there are 6 possible oscillation modes.

According to the Maxwell equations classically a system of charges oscillating with oscillation frequency w emits radiation of the frequency w . This fact allows a completely new interpretation of the emission phenomena without photons.

The electromagnetic radiation of frequency w is made up of portions of energy hw . The reason why a certain frequency w can be absorbed or emitted by an atom is that there are two energy states with of energies E1 and E2 so that E2 – E1 = hw . Using E2 = hw ₂ and E1 = h w₁ this condition can be written as hw ₂ – h w ₁ = hw . Dividing by h gives w = w₂ – w₁ . Thus the conservation of total energy is the reason why the frequency of light must be the difference of the frequencies of the eigenstates .

In the model used in Fermi's paper however the causal relation is different . The reason for why the frequency of the emitted photon w is equal to the difference of frequencies of the eigenstates w ₁ and w ₂ is not the energy conservation but the fact that the charge density oscillates with the frequency w ₂ – w ₁ . The energy conservation demands only that the radiation lasts so long that the total emitted radiation has the energy hw

In Fermis modell the photons with frequency below a certain frequency cannot excite electrons because there is not a suitable oscillation mode namely a suitable pair of Energy eigenstates with appropriate frequencies w₂ and w₁ so that their superposition oscillates with the frequency of the incident radiation . There is no resonance possible because there is no oscillation mode of the charge density with this frequency. Since there is a contiunuous energy/frequency spectrum above a certain energy hw₀ , an oscillation mode exists with any possible frequency w above the certain frequency w₀ . This is why radiation with any frequency above a certain frequency is able to pick up the electrons.

In QED the vacuum fluctuations are considered responsible for the first “kick” in spontaneous emission. But once given the “kick” what sustains the transition? Is it The field of vacuum fluctuations or the radiation reaction force of the oscillating charge density. While majority accepted the vacuum fluctuations as being responsible E.T Jaynes claimed that radiation reaction force is the true reason. This has been a subject of energetic controversy in 1980’s. Careful calculations show that both effects play a role. ^{8, 9, 10}

I have to mention here a misconception regarding the relation of “radiation because of the acceleration” and “spontaneous emission” : It is assumed that spontaneous emission is the quantum mechanical equivalence of “radiation because of the acceleration” ¹¹. in other words the radiation because acceleration of a macroscopic charge is considered as a series of large number of spontaneous emissions . This picture combined with the QED assumption “vacuum fluctuations are the cause for spontaneous emission” leads us to the conclusion that the vacuum fluctuations are responsible for example for the Bremsstrahlung namely the great amount of radiation that is emitted by a rapid deceleration of the electron in a strong electric field. Such a statement would be equivalent to claiming that the Maxwell equations are valid at macroscopical scales because vacuum fluctuations exist. This is a wrong picture. The Maxwell equations are fundamental equations of electromagnetic field. The quantization of the electromagnetic field and the possibility to express the static coulomb force as a phenomena that is a result of the superposition of longitudinal and time like photons ¹² that are solutions of covariant field equations led to the folks theorem “coulomb force is a consequence of photon exchange between particles” This led in turn to the misconception that the electromagnetic field is not a fundamental entity but only a result of statistical average of emission and absorption of large number of quants(photons) as the pressure is a consequence of bouncing of large number of gas molecules on the surface. This is a wrong picture. Field is more fundamental then the quant. Quants are discrete energy levels of the excitation modes of the field. The representation of the coulomb force as a integral over longitudinal and time-like photons means merely that the electric field amplitudes of each photon adds up by a constructive interference so that we obtain the known Coulomb electrostatic field. Therefore radiation reaction force as a consequence of Maxwell equations is a phenomena that exists independent of vacuum fluctuations. In spontaneous emission of an atom, vacuum fluctuations and radiation reaction force both play a role ⁸ but in Bremsstrahlung the radiation reaction force is the far dominating effect. In measurement processes where the wave function interacts with a macroscopic apparatus the second one is the important effect .

One can easily understand how this mechanism works in a spontaneus emission cascade in a three state system. Lets assume that the system is initially in the highest energy state |3 >. Because of inevitable fluctuations in the Hamiltonian, small contributions from |1 > and |2 > appear as we discussed⁶ . Which transition (|3 > to |1 > or |3 > to |2 >) occurs depend on the answer to the question which one of the lower states |1 > and |2 > has the greater weight in the superposition immediately after the fluctuation happens. This depends completely on the form of fluctuation. Thus in succesive experiments this is random. If the weight of |2 > dominates slightly then the oscillation mode with frequency w₃ – w₂ dominates rapidly over the mode with frequency w₃ – w₁ . Because the maxwell equations work and the radiation proceeds in a self sustaining and self amplificating way as discussed above so that the transition |3> to |2> is completed before the oscillation mode w₃ - w₁ could have time enough to proceed. Thus sensitivity to initial conditions is the reason why the system doesn’t proceed towards a superposition of |1> an |2>. (See figure for a 3 state system) If the perturbation is a vacuum-fluctuation-photon for example of the frequency w₃ – w₁ , the transition from |3> to |1> is certainly supported as time dependent perturbation theory for induced emission shows. However this transition could happen even if there is not a vacuum-fluctuation-photon with exactly required frequency but a fluctuation of any type caused by a distant particle or thermal fluctuations etc as long as it creates a contributions from state |1 >. ¹³

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Notes and references 

Where d = e integral(Y^* r Y)dv

For a two state system where the system is initially almost in the upper state ( |a₂|² = 1 -e with e << 1 at t = 0) the contribution of the upper state decreases in time as |a₂(t)|² = (1 - |a₁(0)|²) /(1 - |a₁(0)|² + exp(At) |a₁(0)|² )

where A = (4e² / 3 c³h )[(E₂ - E₁)³/h³] (integral (Y₁^* r Y₂)dv)²

6 Lets assume that there is a static external field and the system is in an energy eigenstate |n> with the energy En . Lets assume that the external field changes abruptly because of a small fluctuation in a very short time period d t so that we have a slightly different static field after d t. The resulting new Hamiltonian has a different set of eigenfunctions. However if the perturbation is small the new energy eigenstates are very close in their form and in their energy value to the old ones. Therefore the new Hamiltonian has an energy eigenstate |n’> with energy En’ very close to the |n> and En. Our old state is not exactly an energy eigenstate for the acting new Hamiltonian anymore. It can be however expressed as a superposition of the energy eigenstates of the new Hamiltonian. Since the |n’> and |n> very similar, the main contribution to the superposition is given by |n’> and small contributions from other energy eigenstates of the new Hamiltonian occur in the superposition. This leads to oscillations of the magnitude of the wave function and consequently to the oscillations of the charge density.

13 A numeric calculation for a 3 state system was presented in 18^th conference of Turkish Physical association in Adana Turkey .(see appendix ) The three state model shows how a system in the highest energy eigenstate |3> makes a transition either directly to the lowest state |1>or to the intermediate state |2> depending on the initial weights of |1> and |2>. If the initial weights are close to each other superpositions of state |2> and state |1> appear very shortly . In our approach the electromagnetic field was not taken as quantized. We suggest that the quantization of the electromagnetic field would even further increase the susceptibility to initial contributions.