Dear Sir,
We had gone through your excellent analysis that exhibits your intellectual acumen. In addition to the deficiencies of the Schrödinger equation, we will like to add the following.
It is said that quantum mechanical systems are completely described by its wave function? From this it would appear that quantum mechanics is fundamentally about the behavior of wave-functions. But do the scientists really believe that wave-functions describe reality? Even Schrödinger, the founder of the wave-function, found this impossible to believe! He writes (Schrödinger 1935): "That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message". Rather, he was worried about the "blurring" suggested by the spread-out character of the wave-function, which he describes as, "affects macroscopically tangible and visible things, for which the term 'blurring' seems simply wrong".
Schrödinger goes on to note that it may happen in radioactive decay that "the emerging particle is described ... as a spherical wave ... that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however, does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot ..." He observed further that one can easily arrange, for example by including a cat in the system, "quite ridiculous cases" with the ψ-function of the entire system having in it the living and the dead cat mixed or smeared out in equal parts. Resorting to epistemology cannot save such doctrines.
The situation was further made complicated by Bohr with his interpretation of quantum mechanics. But how many scientists truly believe in his interpretation? Apart from the issues relating to the observer and observation, it usually is believed to address the measurement problem. Some say that Quantum mechanics is fundamentally about the micro-particles such as quarks and strings etc, and not the macroscopic regularities associated with measurement of their various properties. But if these entities are somehow not to be identified with the wave-function itself, and if the description is not about measurements, then where is their place in the quantum description? Where is the quantum description of the objects that quantum mechanics should be describing? This question has led to the issues raised in the EPR argument.
The Schrödinger equation and the equations describing the probability waves, which travel, like photons, at the speed of light, actually have two sets of solutions: one equivalent to a positive wave flowing into the future (a "retarded" wave), and the other describing a negative wave flowing into the past (an "advanced" wave). The full version of the wave equation has two sets of solutions (one corresponding to the familiar simple Schrödinger equation, and the other to a kind of mirror image Schrödinger equation describing the flow of negative energy into the past). The proper mathematical description of the wave function actually includes a mixture of both ordinary ("real") numbers and imaginary numbers (those numbers involving i, the square root of -1). Such a mixture is called a complex variable. It is written down as a real part plus (or minus) an imaginary part. The probability calculations needed to work out the chance of finding an electron (say) in a particular place at a particular time actually depend on calculating the square of the complex number corresponding to that particular state of the electron. But calculating the square of a complex variable does not simply mean multiplying it by itself since it is not the same as a real number. Instead, you have to make another variable, a mirror image version called the complex conjugate, by changing the sign in front of the imaginary part (if it was + it becomes - and vice versa). The two complex numbers are then multiplied together to give the probability. This shows that, truly it is not squaring, but a multiplication by manipulation as the negative sign implies non-existence of the second term like that of a mirror image. The mirror image does not make two objects, but only one real object and the other physically non-existent image.
For equations that describe how a system changes as time passes, this process of changing the sign of the imaginary part and finding the complex conjugate is said to be equivalent to reversing the direction of time! The basic probability equation, developed by Max Born back in 1926, itself contains an explicit reference to the nature of time, and to the possibility of two kinds of Schrödinger equations described above. The remarkable implication is that ever since 1926, every time a physicist has taken the complex conjugate of the simple Schrödinger equation and combined it with this equation to calculate a quantum probability, he or she has actually been taking account of the influence of waves that travel backwards in time, without knowing it. There is no problem with the mathematics of the followers of this view point with others. The difference is only in the interpretation that the wave flowing backward in time is real and should be taken seriously, not ignored. A typical quantum "transaction" is in terms of a particle "shaking hands" with another particle somewhere else in space and time. The difficulties with any such description in ordinary language - how to treat interactions that are going both ways in time simultaneously, and are therefore occurring instantaneously as far as clocks in the everyday world are concerned - is waved off as inherent fuzziness of quantum physics.
Some scientists try to solve this problem by effectively standing outside of time, and using the semantic device of a description in terms of some kind of pseudo-time. This is no more than a semantic device. When an electron vibrates, it is assumed that it attempts to radiate by producing a field which is a time-symmetric mixture of a retarded wave propagating into the future and an advanced wave propagating into the past. The retarded wave heads off into the future until it encounters an electron which can absorb the energy being carried by the field. The process of absorption involves making the electron that is doing the absorbing vibrate, and this vibration produces a new retarded field which exactly cancels out the first retarded field. So in the future of the absorber, the net effect is that there is no retarded field. But the absorber also produces a negative energy advanced wave traveling backwards in time to the emitter, down the track of the original retarded wave. At the emitter, this advanced wave is absorbed, making the original electron recoil in such a way that it radiates a second advanced wave back into the past. This "new" advanced wave exactly cancels out the "original" advanced wave, so that there is no effective radiation going back in the past before the moment when the original emission occurred. All that is left is a double wave linking the emitter and the absorber, made up half of a retarded wave carrying positive energy into the future and half of an advanced wave carrying negative energy into the past (in the direction of negative time). Because two negatives make a positive, this advanced wave adds to the original retarded wave as if it too were a retarded wave traveling from the emitter to the absorber.
In Cramer's words: The emitter can be considered to produce an "offer" wave which travels to the absorber. The absorber then returns a "confirmation" wave to the emitter, and the transaction is completed with a "handshake" across space-time. But this is only the sequence of events from the point of view of pseudo-time. In reality, this process can be said to be a-temporal; it happens all at once. This is because, according to the special theory of relativity, signals that travel at the speed of light take no time at all to complete any journey. As Einstein puts it:
1
β = ----------------
√ 1 - (υ/V)^2
Since for light β becomes meaningless or infinite, τ also becomes meaningless or infinite. Thus, effectively for light signals every point in the Universe is next door to every other point in the Universe. Whether the signals are traveling backwards or forwards in time doesn't matter, since they take zero time (in their own frame of reference), and +0 is the same as -0 and all the quantum probability waves do travel at the speed of light. The situation is more complicated in three dimensions, but the conclusions are exactly the same. This interpretation makes no predictions that are different from those of conventional quantum mechanics, but it provides a conceptual model which helps many people to think clearly about what is going on in the quantum world. It means that when an electron is faced with a choice of two holes to go through, the offer goes through both but the handshake only comes back through one, so it knows where to go; and in Renninger's experiment, the particle setting out from the radioactive nucleus has already made its handshake and knows which hemisphere it will end up on. There is no more mystery about the quantum mysteries at all; provided you can live with waves that go backwards in time. But as we have shown in our essay, this concept is entirely wrong.
It is true that particles, which are nothing but confined fields, move in waves, which are nothing but the motion of the field that contains the particles. This can be easily derived from fundamental principles. We treat both this wave and its intensity as real. Once we accept this description, the measurement problem vanishes. Wave function describes the movement of the field that contains the particle. Thus, knowing the specific wave function, we can precisely locate a particle in that field, because the particle also has a role in regulating the movement of the wave. Since measurement is taken at "here-now", it is real. We freeze this value and use it at other times when the system is no longer the same as it has evolved temporally. Regarding the superposition of states, we have described in our essay that it is only the combined unknown states of a temporally evolving system at moments other than the moment when measurement is taken. We can know the precise description of the system only at the moment of measurement. At all other times, it could have evolved with time. Knowing the inputs, we can only describe the probability of its state. We cannot precisely describe its state at any other moment. As we have pointed out in our essay, uncertainty in describing the precise state is not due to the laws of Nature. It is a result of natural laws relating to observation that reveal a kind of granularity at certain levels of existence that is related to causality.
Regarding the mysteries of spin behavior, we can explain it easily. As you have pointed out, the electron has a magnetic moment, which is a magnetic field associated with it. If the electron is moving along the z-axis, then the electric and magnetic fields associated with it move along x-axes and y-axes respectively. Since measurement is a process of comparison between similars, any experimental set up to measure the spin properties must use one such field. Thus, while comparing with this field (co-ordinate system), the magnetic moment of the electron will show only two expected values. During other times, it is aligned to the local field. This makes it's spin vector unknown. There is no mystery here.
Regarding the second characteristic of spin - when an electron is rotated through a full 360 degrees, its spin does not return to its original position, but takes an additional 360 degrees to come to its original position, the explanation is simple. If you look at the magnetic field lines of Earth, you will notice that they flow from South Pole sideways in a closed loop towards North Pole, where it closes the loop. When the same field line comes out, due to the movement of Earth, it will come out in the opposite orientation making a figure of 8. After one more rotation, it will regain its original orientation. There is no mystery here also. There is no need to unnecessarily mystify the simple natural phenomena.
We have shown elsewhere that the concept of "light cone" is fallacious as light pulse either propagates in a straight line or in all directions (spheres). There is no reason to assume that it takes a selective direction to validate the imaginative views of some who call themselves scientists. Thus, time evolution of a light pulse will be a set of concentric spheres and not a "light cone". As a consequence, the concept of event horizon is also a hoax.
The basic problem here is not the mysteries of the quantum world, but our way of looking at it and describing it. We know all about the electron and how it behaves except that most do not know what an electron is? This lack of knowledge leads to generation of incomprehensible theories to retain the high position and the perks that come with it at public expenses.
Regards,
basudeba