THE ELEMENTS OF QUANTUM MECHANICS AND THE PRINCIPLE OF INDETERMINACY

By Linus Pauling

Cornell Philosophy Club, Thursday, January 20, 1938.

Introduction. - In speaking before this organization I feel guilty of unintended misrepresentation. I am not a philosopher nor a theoretical physicist, nor, indeed, am I especially interested in philosophy and physics - instead, I am a chemist who finds quantum mechanics a useful aid in the study of chemical problems. Having, however, made the mistake of coming here tonight, I shall try to present my own simple point of view, trusting that the sophisticated auditor will be kind enough to excuse its naiveté.

Classical Mechanics.- In the treatment of a system by the methods of classical mechanics, as developed 300 years ago by Newton and refined by Lagrange, Hamilton, Jacobi, etc. so far as the mathematical technique is concerned (and by Einstein in a more significant way), an instantaneous configuration is defined by three Cartesian coordinates x_{i}y_{i}z_{i} for each particle. [Drawing of two points with the Cartesian coordinates x_{i}y_{i}z_{i} and x_{2}y_{2}z_{2}, respectively.] The problem is to find how the configuration changes with time. The equations of motion are such that a knowledge of the coordinates x_{i}y_{i}z_{i} and the velocities x´_{i}y´_{i}z´_{ (or the equivalent momenta pxi=mix´i etc.) at the time t = t0 permits the prediction of the state of the isolated system at any later time. It is interesting that xi and pxi is enough - nature was that kind to us, since she might have included [See drawing] also.}

It is the possibility of this prediction in classical mechanics which underlies the doctrine of classical determinism - that, if the present state of an isolated system (eventually the universe) were known, a sufficiently clever and energetic mathematician could foretell exactly the future behavior of the system. Quantum mechanics differs on this point.

History of Quantum Mechanics. - Planck - 1900 - black-body radiation. k = 6.547 · 10^{-27} eng sec. Einstein 1905 - light quantum. Bohr - 1913. Modern physics is a difficult subject partially because it is so extensive - because a multitude of facts is known and must be understood by the student.

The old quantum theory was not a theory - nothing could be said about fundamental questions with use of it, because it was incomplete - a patched-up classical mechanics, representing a first effort to formulate a mechanics of atoms. Then de Broglie, Heisenberg, Schrödinger, Dirac, Born, Jordan etc. set up the quantum mechanics. It is interesting that these men were mostly young - Heisenberg 24 in 1925.

Nature of Quantum Mechanics. - QM permits the formulation of a function Ψ which is compatible with the results of experimental observation of the system at t = t_{o} and permits predictions to be made regarding results of observations at a later time t = t_{1}, the system remaining isolated in between. [Drawing of an observer and system.] In general a statistical prediction, given by a probability distribution function. Let us as an example discuss the hydrogen atom. If we know the atom to have energy -e^{2}/2a_{o} = -13.53 e.v. (normal state) at t = t_{0}, we know that at t = t_{1} it will still have this energy, providing that it remain undisturbed. But for x we can predict only a statistical electron distribution, and for p_{x} another, with no correlation between them. [Drawings of electron distribution graphs.]

It might be asked if more could not be learned about the state of the atom than its energy - could we not know W and x both, and refine Ψ? The answer is no - that the methods of observation are such that W alone is a maximal observation.

The Uncertainty Principle. - Heisenberg in 1926 formulated this; it is ΔxΔp_{x} ≥ h/4π. This holds for q_{i} p_{i} in general.

For mass 1g. this has no importance: Δx = 10^{-13}cm for Δp_{x} = 10^{-13} or Δv' = 10^{-13}cm per sec. But for electron with m = 9.038·10^{-28} or ≈10^{-27}, it gives Δx = 10^{-8} for Δx' = 10^{8}; i.e., compete lack of x, p_{x} correlation in normal state (which has x' ≈ 10^{8}cm·sec^{-1}).

QM thus prevents the deterministic point of view from being adopted - by denying that x_{i} and p_{xi} can ever be known accurately at t = t_{0}
! Heisenberg and Bohr have discussed various experiments to see why this is so, and found that every known method of observation is compatible with this idea. [Drawing of x-ray diffraction.] For example, if x-radiation is used to locate the electron, Compton-effect recoil may change the momentum and energy so that it is no longer known that the atom is in its normal state. Nor can bombardment by particles, such as other electrons, be used, because of their wave character, which leads to diffraction effects.

What About It. - There are several comments that can be made about the uncertainty principle and determinism.

1. Perhaps there really is a deterministic mechanics underlying QM, and perhaps a new physical discovery - such as classical undulatory radiation or real particles - will be made which will permit x and p_{x} to be measured simultaneously. I don't think that this possibility can be ruled out, but until the discovery is made there is no use to talk about it.

2. In a sense we can still say that there is determinism in the universe. If Ψ is known at t = t_{0} for an isolated system, it is known forever after - so long as the system remains isolated. The universe as a whole is an isolated system, and we may thus say that with changing time the Ψ of the universe is changing in the manner prescribed by the Schrödinger wave equation for the universe.

3. These are, however, just words - a possibly more significant question is the following: in a living system is there the possibility that reaction under certain conditions can be varied through the operation of freedom of will, selection being made of one or another of the alternatives provided by the QM wave function? I do not know whether this question has meaning or not - but I think that it hasn't.

The action of will might be of either of the two following kinds:
a. In a large assemblage of identical living systems, individual choices are such that the prob. dist. fn. is retained for the assemblage.
b. The prob. dist. fn. is not retained - instead, every system selects the good alternative (let us say).

Kind a could not be looked for experimentally. Could kind b? I think not - I am afraid that the limits of our experimental technique is such that the uncertainties in determination of the initial state are infinite very much greater than uncertainties due to Heisenberg's principle. These latter are of magnitude ≥h/4π for every degree of freedom. This is comparable to difference between normal and first-excited state for each degree of freedom. But how determine these experimentally?

Actually, for most interatomic interactions of the type of significance in physiology the ΔxΔp_{x} uncertainty in the normal H or other atom has no bearing on the question - the atomic functions Ψ determines the interaction potential. I don't see where free will has a chance to say anything - indeed, it hasn't until an observer comes along to make an objective experiment - if it has anything to say even then.

To support my contention that biological systems are too complex for Heisenberg's h/4π to have experimental significance I might mention ice and water. [Drawing of water molecules.] In ice atom absolute zero there are still (3/2)^{N} configurations. This is 10^{23} for N = 0.606 x 10^{24}! Even for one microgram of ice there are 10^{10,000,000,000,000,000} configurations! This perhaps has something to do with the variety in snow-flake crystals as photographed by Mr. Bentley. We see that in tissues - nerve, for example - the structural possibilities are myriad.