Absurdities in Modern Physics: A Solution
8-9 Physical Reality of Quantum Tunneling.
Let us look now at another case in which the quantum mechanical interpretation leads to nonsense. It is quantum tunneling. Quantum tunneling is the traversal of a particle to the other side of a potential barrier, even though its energy is less than the height of the barrier. In standard textbooks, it is claimed that tunneling is a part of quantum mechanics that cannot be explained in a rational way. We are told that there is no alternative to believing in imaginary velocities and negative kinetic energy to cross the classically forbidden region of the barrier. In other words, we have to use the absurdities of the Berkeley-Copenhagen interpretation. Since kinetic energy is the product of a mass (positive number) times the velocity squared (also necessary positive), the result cannot give a negative value of kinetic energy. Then, the particle should not be able to cross the potential barrier.
The reason for such an apparently contradictory description of tunneling is due to restrictions unconsciously applied during the description of the phenomenon. For example, let us use for a moment the same terms as in quantum mechanics. We calculate the probability of finding the particle in different regions of space by setting up the Hamiltonian. Then, we use operators to obtain the equation of the particle. Then, the square of the amplitude of the wave function tells the probability of finding the particle.
In this description, perhaps we did not notice that the word "particle" implies a point particle. We all know however, that matter is made out of "waves", as required by the de Broglie wavelength. If particles have a "real" existence and have wave properties compatible with realism, they cannot have the size of a mathematical point.
For over twenty years I have had to teach the basic principles of quantum mechanics and the tunneling of particles through potential barriers. The questions raised by many students about the interpretation of the mathematics of quantum physics were quite interesting. Physics students do not accept absurdities easily. Their fresh and unbiased minds deserve better explanations since the interpretation of tunneling as the real passage of a particle through a tunnel drilled in the potential well is absurd. For many years I have illustrated the problem in a way reproduced by the drawing on Fig. 8-C.
In the upper part of Fig. 8-C I show a rubber band held at about half the height, inside a glass beaker. It is assumed that in this position, the rubber band has no kinetic energy. There is only the gravitational potential between the base of the beaker at this half height. At the bottom of the beaker, the object would then have a positive kinetic energy. We know that the energy is conserved. Therefore, the sum of the potential and kinetic energies of the rubber band is too small, to allow to take the rubber band out of the beaker since no external energy is available. At the top of the beaker the rubber band would have negative kinetic energy.
It is interesting to challenge physicists to take the rubber band out of the beaker without increasing the total energy of the rubber band. They are absolutely convinced that nobody can do it. In fact, no physicist has ever done it in my presence. This is what they have been told when they were students.
rubber band is a deformable body, and it is possible to raise
the rubber band while lowering the other end so that the center
is not raised. Then, one end of the folded rubber band is
above the edge of the beaker without raising the center of
object, as illustrated on the central part of Fig. 8-C. Then the
band can fall freely outside the beaker.
Without external energy, the center of gravity of the rubber band was never risen above the half height line, as in the case of "tunneling". Since we know that a real particle in physics is not a point, as shown by the de Broglie wavelength, it is clear that electrons come out of a potential well the same way the rubber band does.
Even if no material moved through the beaker walls, the center of gravity moved through it. Let me add that I have also used other versions of that rubber band "tunneling" experiment. One is to substitute the rubber band with a paper clip. However, a commercial paper clip cannot be bent easily but we can make our own easily-pliable paper clip, indistinguishable from a standard paper clip, by using a piece of soft solder tin wire from the electronic shop. It is fun to unroll the tin paper clip inside the beaker before taking it out without raising the center of mass. The creeping of the malleable object that allows reaching the outside of the beaker is not described by an appropriate word when using "tunneling" since there cannot be any tunnel. The expression "tunneling" is pedagogically misleading. This crossing of the barrier is achieved successfully by skilled high jumpers. It would be more compatible with reality and rationality to call it the "high jumper effect". This would avoid the misleading expression "tunneling".
the Description of the "High Jumper Effect"
One can find several examples to show that the "high jumper" model is a better description of the phenomenon than the "tunneling" model. There are two main variables in establishing a comparison between the two models. The model of tunneling implies the crossing of the potential barrier by making a tunnel through it. In such a model, the energy required to go through the wall is completely independent of the height of the barrier. It simply depends on its thickness. In the case of classical tunneling however, the difficulty of crossing a barrier depends on the barrier height as well as on the barrier thickness. Furthermore, the existence of a tunnel into a potential barrier does not make sense, since there exists no description of the material from which the potential barrier is made.
In the case of a "high jumper" effect, the difficulty of jumping over an obstacle is a function of both the height and the thickness of the barrier. The problem of the material of the wall is irrelevant since the jumper moves above it, even if its center of gravity moves through it. Consequently, the analogy between quantum tunneling and reality does not hold. We have seen that the "high jumper" model requires the same parameters (thickness and height) as mathematical physics requires.
The mathematical probability of having the high jumper effect (classical tunneling) has been calculated recently by Cohn and Rabinowitz [8.1], [8.2]. These authors calculate the probability of crossing an obstacle with different shapes. Cohn and Rabinowitz show that the probability for ropes of varying lengths to cross the potential barrier of different height and thickness is in excellent agreement with the quantum mechanical calculation. Cohn and Rabinowitz [8.1], [8.2] mention:
Chapter4 Contents Appendix