Refuting the laws of physics or humility towards nature

It's incredible how conceited the man is. He attempts to apply his by all means modest experience acquired on the Earth, this little dust in the vast universe, to the whole cosmos, without taking account of the fact that he only knows a very small part of it and, besides, his knowledge is far from complete. This conceit is well seen among physicists, both theorists and experimenters. They tend to identify constructions of the mind, which physical models and theories are, with the reality that the models and theories are intended to describe. I say: are intended to, for I have no doubt that our theories are imperfect and incomplete, and their numerous drawbacks will be presented by heart by a diligent pupil at school within several hundred - or perhaps several dozen - years, if only our conceit does not lead our civilization to a catastrophe. Let me repeat: the laws of Newton, Maxwell and Einstein are not reality, they only represent our feeble attempts to describe it. To persuade you into assuming my point of view and to invite you to learn together humility towards nature, I shall now present an incontestable proof, never published before, of the fact that
No hydrogen atom can exist (in its normal state).

What do we know about reality?

Hydrogen can be rarely observed in its atomic state (it merges easily into particles H_2), but its properties in such a state are so simple and interesting, that they have been investigated in great detail. It has been observed that the electron and the proton, bound by the forces of electrostatic attraction, cannot assume arbitrary energies when the atom as a whole remains in a state of rest. The admissible energy values are given by Rydberg's formula:

The value of R, known as Rydberg number, is 13,6 eV and it corresponds to the binding energy for the electron and the proton in the minimum state called normal, represented by the value of n equal to 1. This property of the hydrogen atom energy has been first described by Niels Bohr in his model. Also modern quantum mechanics assumes quantization of the hydrogen atom energy by introducing minor corrections to the original Rydberg formula, which we shall not consider here, and by pointing out that 2n^2 distinct states with different momentum values correspond to each energy value E(n). For instance, hydrogen atoms usually occur at, say, room temperature in the normal state (n=1), which can be most easily deduced from the fact that hydrogen emits no light, unless it is excited with some additional energy (due to heating, electric discharge, light etc.) So stimulated, it quickly returns to its normal state (in a time of the order of 10^{-8} s, which is sufficient for the light to cover the distance of 3 m), emitting light the wave lengths of which form series of Lyman, Balmer, etc. The radii of electron orbits in states corresponding to the successive values of n are equal to n a(0), where a(0) = 0.53 angstroems is the Bohr radius, i.e. the electron radius in the normal state.

Remark: Everywhere in this text, 10^{n} agrees with TeX notation for n-th power of 10.

Attack: hydrogen cannot exist in the normal state

Before carrying out our proof let's recall the fundamental law of statistical physics: if a physical system can exist in states corresponding to different energies, then under thermodynamic balance at temperature T the probability of the system being in a particular state is proportional to exp(-E/kT), where E is the energy of the state, T represents the temperature (in the absolute scale), while k is the Boltzmann constant equal to 0.000086 eV/K. For instance, at room temperature (T approximately 290 K), kT is around 25 meV. Obviously, if we wish to compute the value of the probability, we must divide this expression by the sum of similar expressions corresponding to all system states, thus ensuring that the sum of all probabilities is 1. (Well, dear reader, in order not to divert your attention, I must explain here that the above law and the probability formula it implies will not be put to doubt.) The subsequent argument is simple. According to what we have just stated,

where the sum is taken over all hydrogen atom states. There are infinitely many such states and each component of the sum in the denominator is greater than 1, each of the E(i)'s being negative. (In fact, we should have also considered states with positive energy, but this would only make the sum bigger.) hence the sum is infinite, which implies that the entire fraction, i.e. the probability of the system being in the normal state, is equal to 0.

Consequence: a new (cosmic?) weapon

Whoever is familiar with the laser and its operation principle will immediately exclaim at this point: "Why, this is an inversion of states!" In the state of thermodynamic balance the hydrogen could be used to construct a laser, in which every atom would emit the energy of 13.6 eV in the form of ultraviolet radiation in a very short time. This means that one kilogram of hydrogen, i.e approximately 1000 gram molecules, would emit the energy of 1600000000 J. If the laser impulse were to have the duration of 10^{-8} s, the corresponding power would be greater than 10000000000000 kW, which is much more then the power of all electric plants in the world, including Chernobyl and Three Mile Island, and more than the joint power of all cars, planes and ships. I wouldn't wish my worst enemy to have such a weapon aimed at him!

Defence: The universe is too small

The sum which appears in the denominator of the probability formula is said to be statistical. It is a very significant magnitude used to characterize a physical system. We have observed that it is infinite, which means: adding successive elements of the sum we can attain arbitrarily large numbers. In particular, the number can be arbitrarily large as compared to the component corresponding to the normal state. As hopeless as the sum of infinitely many numbers may seem, let's try. Let our hydrogen have room temperature, T = 290 K. In fact, two states with opposite spin momentum directions have the lowest energy E(1) = -13.6 eV (n=1). The sum component corresponding to this energy is equal to 2 exp(544). As my readers may have observed, instead of considering components of the statistical sum which correspond to a single state, I prefer to group them into components corresponding to successive values of the energy.

The component which corresponds to the next value of the energy, E(2) = -13.6 eV/4 = -3.4 eV (n = 2), so we have 8 states with different momenta), is equal to 8 exp(136). Its value is obviously 0.25 exp(408) (i.e. 6.3 times 10^{169}) times less than the value of the first component. It is hard to imagine the magnitude of this number. It is usually estimated that the mass of the observable part of the universe is of the order of 10^{51} kilograms, which translated into hydrogen atoms yields a number of the order of 10^{78}. If we take as many universes as there are hydrogen atoms corresponding to the mass of our entire universe and again split these universes into hydrogen atoms, we would still get a number several dozen million millions less than the ratio of the first component of the statistical sum to the second. Now, you may surely ask: If this second component is so unimaginably smaller than the first, then how huge must the number of required components be if their sum were to become greater than the first - or at least be of comparable order!

I shall not bore you with computations. If you wish, you may try to verify that the value of n needed to get a sum of successive components (without the first) greater than the first is of the order of 10^{79}, which is more than the number of hydrogen atoms corresponding to the mass of the universe.

Perhaps another question has occurred to you, too: What would the electron orbit dimension corresponding to such an enormous value of n be? The answer is simple. The radius of the normal state, a(0), must be multiplied by the number n. We get a hydrogen atom of the magnitude of 10^{69} m = 10^{53} light years, a monster with linear dimensions 10^{43} times greater than the radius of the observable universe. No one has ever seen such an atom and certainly no one will in a predictable future! I hope you can see now what I mean by speaking of the conceit of physicists. We extend the scope of applications of our models beyond any reasonable limits, often without even noticing it. I have presented the above paradox to many physicists and in most cases in their minds theory prevailed over practice. (This should not be considered an argument for the low intellectual level of my colleagues. Wasn't it Hegel who, when told that facts speak against his theories, said: "All the worse for the facts"?) Since, as is well known, a change in science is only possible when all adherents of the old ideas pass away, I shall not use my energy to attempt to convince already-formed physicists. I turn to you, future physicist. Just like myself, look at nature with humility and respect and at human knowledge about nature - with due distrust.

Jan Gaj