3.3 THE CALCULATIONAL INSTRUCTION BOOK OF THE THEORY

We choose to present that machinery in the Schrödinger rather than in the Heisenberg form because that has long now been the general practice in the field. Schrödinger started his formulation with an analogy between classical mechanics and geometrical optics which William R. Hamilton described in his 1834 paper On the Application to Dynamics of a General Mathematical Method Previously Applied to Optics. Comparing Fermat’s principle which governs the path of a ray of light with his own principle which governs the motion of a mass point, Hamilton showed that the trajectory of a mass-point moving in a conservative field of force described by a potential energy $V (x, y, z)$ is identical with that of a ray of light propagating in a non-homogenous optical medium if the energy of the mass-point is related to the phase velocity of light in that medium as follows:

u = \frac{C}{\sqrt{2 m (E - V)}}

where $u$ is the phase velocity of light, $m$ the mass of the mass-point, $E$ the total energy of the mass-point, and $C$ is a constant independent of the coordinates $(x, y, z)$ but which may depend upon the energy. Schrödinger then proceeds to relate the frequency of the ray of light to the energy of the mass point as follows:

E = h

where $h$ is Planck’s constant. Under the circumstances, the phase velocity of light through the optical medium depends upon its frequency, and that makes the medium dispersive.

But, geometrical optics is the high frequency limit of the wave theory of light originally put forth in 1678 by Christiaan Huygens. Therefore, Schrödinger suggests that classical mechanics, formally equivalent to geometrical optics, should have a wave equivalent. Since a mass-point is clearly localized in space, a monochromatic wave will not do. Schrödinger turns therefore to a superposition of monochromatic waves whose frequencies are closely packed together. Such a superposition, originally introduced into hydrodynamics by Stokes in 1876 and found to be of fundamental importance in optics, is called a group of waves. In a dispersive medium, the velocity $U$ of such a group of waves is different from the phase velocity of the waves involved in it and is given by:

\frac{1}{U} = \frac{d}{d ν} (\frac{ν}{u})

or, in our case:

\frac{1}{U} = \frac{d}{d E} (\frac{E}{u})

By equating the speed of the mass-point with this group velocity, Schrödinger shows that the phase velocity of the equivalent wave is related to the mechanical properties of the mass-point to which it is equivalent as follows:

u = \frac{E}{\sqrt{2 m (E - V)}}

a relationship that de Broglie already found the year before.

Thus possessed by relationships between the mass-point’s energy and momentum, on the one hand, and the frequency and phase speed of the associated wave on the other, Schrödinger transforms the wave equation governing an optical wave—that is:

\frac{\partial^{2} φ (x, y, z)}{\partial x^{2}} + \frac{\partial^{2} φ (x, y, z)}{\partial y^{2}} + \frac{\partial^{2} φ (x, y, z)}{\partial z^{2}} + \frac{4 π^{2} ν^{2}}{u^{2}} φ (x, y, z) = 0

into the time-independent Schrödinger equation:

\frac{\partial^{2} φ (x, y, z)}{\partial x^{2}} + \frac{\partial^{2} φ (x, y, z)}{\partial y^{2}} + \frac{\partial^{2} φ (x, y, z)}{\partial z^{2}} + \frac{8 π^{2} m}{h^{2}} (E - V (x, y, z)) φ (x, y, z) = 0

which embodies the solutions to all those failures of the classical corpuscular theory that we have described above and provides a systematic and universally valid way to answer any additional questions. It is a second-order partial differential equation which can be solved readily once the potential energy function $V (x, y, z)$ for the mechanical system under consideration is given as a function of location. A regular, single-valued, and finite solution for the wave function $φ (x, y, z)$ is known to exist only for certain values of the energy constant $E$ called eigenvalues. These eigenvalues are the only energy levels allowed in quantum mechanics. For instance, if the mechanical system is a harmonic oscillator, the potential energy is proportional to the distance from the oscillation center and the quantized energy levels are, as Planck has already guessed in his famous moment of desperation:

E_{n} = (n + \frac{1}{2}) h ν n = 1,2, 3, \dots .

and, if the mechanical system is a hydrogen atom, the quantized energy levels are, as Bohr found in 1913:

E_{n} = - \frac{2 m e^{2}}{h^{2} n^{2}} n = 1,2, 3, \dots .

We shall not be particularly interested in the success of the new theory; what we are after instead are the issues which remained unexplored. There are three such issues, all of which are ontological in character: the physical meaning of the wave function thus associated with a material point, the superposition principle and the corresponding measurement-induced collapse of the wave function, and the quantum entanglement. The physical meaning of the wave function engendered considerable debate because the original association of the material point with a wave packet did not stand ontological scrutiny. On the one hand, since the three dimensional medium through which the wave packet propagates is dispersive, each component of the packet propagates with a different phase velocity leading to a quick disintegration of it. On the other hand, the wave associated with a system of two or more material points propagates through a medium with as many sets of three-dimensions as there are individual points, hardly an entity we could associate with physical reality. The preferred solution was offered by Max Born in 1926 and consisted in associating the square of the absolute value of the wave function $φ_{n} (x, y, z)$ to the probability density that, upon measuring the location of a material point of energy $E_{n}$ , one should find it at the point of Cartesian coordinates $(x, y, z)$ :

P (x, y, z) = {{| φ}_{n} (x, y, z) |}^{2}

This association between wave function and probability however left the ontological question open because one never encountered a physical entity whose very Being is a stochastic process. Certainly Einstein felt very uncomfortable with Born’s rule:

I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this theory operates with an incomplete description of physical systems.
P.A. Schilpp, Albert Einstein, Philosopher-Scientist, (1949)

The next ontological issue left unexplored is the collapse of the wave function. Since the Schrödinger equation is linear, the sum of two possible solutions is again a solution. Consequently, the most general solution is:

φ = \sum_{n} a_{n} φ_{n}

where $φ_{n}$ is the eigenfunction corresponding to the eigenvalue $E_{n}$ and $a_{n}$ are arbitrary constants. This is the mathematical representation of the superposition principle which Dirac placed at the foundation of his version of quantum theory and of which we spoke in the first chapter of this essay. The natural question then arises: if we associate the wave function $φ_{n}$ to a material point of given energy, then what are we to associate with the wave function $φ$ ? The quantum mechanical answer is to associate such a wave function to a material point which is simultaneously in all the energy states under the sum. However, any measurement of the energy of that material point will return a specified value which is clearly included in the sum. Therefore, upon measurement, the wave function $φ$ will have to suddenly ‘collapse’ upon the wave function $φ_{n}$ . This process is quite similar to the quantum jumps introduced by Bohr and is thus equally mysterious and ontologically un-intelligible. In fact, Schrödinger, whose foray into the new field was driven by the hope of basing the theory upon an ontologically clearer basis, said:

If we have to go on with these quantum jumps, then I am sorry that I ever got involved.
Erwin Schrödinger, Are there Quantum Jumps?, (1952)

But the most important consequence of the theory is that two states which had once interacted with each other will remain entangled forever:

When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they no longer can be described in the same way as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction, the two representative wave functions have become entangled.
Erwin Schrödinger, Discussion of Probability Relations Between Separated Systems, (1935)

Schrödinger goes on to say:

The best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated and therefore virtually capable of being ‘best possibly known’, i.e., of possessing each of them a representative of their own.

What constitutes entanglement mathematically is that the wave function for the two separated systems is no longer a product of a function of the coordinates $x$ of the first system with a function of the coordinates $y$ of the second system regardless of just how separate they had become after interacting with each other. Instead:

φ (x, y) = \sum_{n} g_{n} (x) f_{n} (y)

where $g_{n} (x)$ are possible wave functions of the first system and $f_{n} (y)$ possible wave functions of the second system. Upon measuring the coordinates of the second system, the total wave function collapses to the wave function associated with the measured $y_{m}$ and the corresponding value $x_{m}$ is thus automatically selected for the first system. One can then say that the entanglement consists in that every observable of one system is determined by the observable of the other one, a result of quantum theory that Schrödinger’s found rather discomforting:

It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter’s mercy in spite of having no access to it.

In quantum mechanics therefore there are no localized system; each system is related to any other system which has ever interacted with it in the past no matter how distantly located from them the system is now. The physical universe implied by quantum mechanics, unlike the divisible universe of our experience, is indivisible and the very premise of the corpuscular conception is thus denied by its offspring.

These three ontological questions had been vigorously debated for a decade after Heisenberg’s introduction of quantum theory. The older generation, Einstein in particular, first tried to show that the theory was internally inconsistent. In a thought experiment presented at the 1927 conference of the Solvay Institute, Einstein tried to prove that both the position and the momentum of a material point could be simultaneously ascertained with any degree of precision in direct contradiction to the uncertainty principle which states otherwise. At the 1930 conference, he presented another thought experiment which he believed would allow exact measurement of both the energy of a photon and the time of its emission despite the fact that the uncertainty principle applies to these physical properties as well. On both these occasions Bohr succeeded in proving to Einstein’s satisfaction that his challenge of the uncertainty principle failed. In 1935, Einstein returned to the debate for the last time with an article written in collaboration with B. Podolski and N. Rosen entitled Can Quantum Mechanical Description of Physical Reality be Considered Complete? In it he employs the entanglement trait of quantum mechanics to set up a dichotomy between the completeness of quantum mechanics, on the one hand, and the physical reality of the system to which it applies, on the other. Bohr answered in an article with the same title and published in the same year but this time he failed to convince Einstein. The latter agreed that one could hold Bohr’s position without contradiction but begged to disagree with the non-locality that it implied.