Quantum Mechanics

A bird's-eye view

An experiment is a question which science poses to Nature and a measurement is the recording of Nature's answer.¹

-Max Planck

Quantum theory began development during the first half of the twentieth century in order to describe several confounding observations at the smallest scales.

The discovery of the photoelectric effect revealed that the energies of subatomic particles were quantized, meaning that they could only take on discrete values rather than the continuous range of values described by classical physics.

Stranger yet was the double-slit experiment which demonstrated that particles have wave-like characteristics and are thus able to undergo a type of diffraction. Prior to measurement, a particle is not localized and will exhibit an apparent dispersal across space such as in the case of atomic orbitals. Upon measurement, however, a particle will only be found in one location.

These revelations were a shock to many physicists at the time. Quantum mechanical experiments seem to insist that nature is irreducibly probabilistic. Despite being one of the founders of this new domain of physics, Albert Einstein was acutely perturbed by such an outlook—once declaring that:

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'Old One.' I, at any rate, am convinced that He does not throw dice.²

Indeed, the success of Quantum Mechanics in the laboratory cannot be denied. Every quantum system is described by a wave function which can be used to model the state of its various properties. The wave function can be written as a sum of basis vectors representing all potential measurement outcomes:

\( | \Psi \rangle = \sum_{n} a_n | x_n \rangle \)

\( | \Psi \rangle = a_1 | x_1 \rangle + a_2 | x_2 \rangle ... a_n | x_n \rangle\)

It is here that what is commonly called the “measurement problem” comes to the fore. Prior to measurement the basis vectors evolve separately through time via the Schrödinger Equation. But upon measurement, the wave function seems to discontinuously rotate into one of its basis vectors with probability:

\(P(x_n) = |a_n|^2 = |\langle x_n \vert \psi \rangle|^2\)

This is the Born Rule, formulated by physicist Max Born, which states that the square of the wave function amplitude at a position gives the probability of finding the particle at that position.

The startling consequence of these models is that particles do not evolve in a deterministic manner between measurements. The well-known Heisenberg uncertainty principle gives this mathematical concreteness to this fact:

\(\Delta position \Delta momentum \geq h\)

The more precisely one attempts to measure the position of a particle the less precise a measurement of its momentum will be and vice versa. In other words, if you know exactly where a particle is at one moment, you cannot determine with absolute certainty where it will be at the next moment.

Likewise, time and energy are also tied together by the uncertainty principle. If you examine how a quantum system is changing over a shorter and shorter duration, you must accept a large range of uncertainty in its energy.

\(\Delta time \Delta Energy \geq h\)

It is incorrect to assume that the quantum uncertainty here exists solely because one must interfere with a system to measure it. Rather, the uncertainty is built into the very essence of the particles themselves.

There are several schools of thought on what the wave function and the probabilistic nature of quantum mechanics truly mean. There is the standard Copenhagen Interpretation which implies that wave functions do not correspond to any physical object in the real world and are only mathematical tools for describing the outcome of measurements.

Subsequent interpretations include the Pilot Wave Theory of Bohmian Mechanics which conceptualizes the wave function as merely guiding the motion of a particle with inherently classical properties and objective collapse theories which treat the discontinuous change of the wave function as a real physical process.³

The king of modern formulations of quantum mechanics, however, is the Many-Worlds Interpretation which posits the existence of parallel universes that split off from each other upon measurement. The motivations behind this interpretation are not without justification. It allows us to break down the meaning of the wave function into a simple Bayesian dynamic. A higher probability amplitude at a specific position, for instance, would mean that there are more worlds in which a particle will be observed at that location.

Where the Many-worlds interpretation falls short is with respect to a clear ontological stance. What exactly are these “worlds?” No one genuinely believes that a separate cosmological spacetime is created every time we measure a quantum system. Further, even if all measurement outcomes do occur the Many Worlds Interpretation cannot explain why an observer only perceives one of these worlds rather than all simultaneously. It thus remains beholden the notion of wave function collapse which it alleges to repudiate.

The resolution to these conundrums can be found if we examine more closely the implications of quantum probability. If reality is not deterministic, then the essence of reality is not a concrete substance capable of being measured all at once; rather, its essence is possibility itself.

Quantum phenomena are so counter-intuitive because we fail to see possibility as a valid ontological category. The reason for this goes all the way back to prejudices of the Ancient Greeks which have worked their way through the ages into modern scientific thinking.

As we have touched upon in a previous post about pre-socratic philosophy, the greeks saw the process of change as a fundamental enigma. In a fragment from his seminal work entitled On Nature Parmenides writes,

What is cannot have come into being. If it did, it came either from what is or what is not. But it did not come from what is, since if it is existent it did not come to be but already is; nor from what is not, for the nonexistent cannot generate anything.

The Greeks associated the idea of being with the present—that is, with what is actual. Because the past and future are not actual, they are not. Quantum mechanics demands that we at long last unshackle ourselves from this ontological viewpoint.

In an indeterministic reality, the position or path of a particle depends on what is possible in a given measurement setup. A quantum state is thus stretched ahead of itself temporally.

The propagating wave function encodes what might happen to the particle along possible trajectories. Upon observation, one of these possibilities is released from the future as the path the particle has already taken.

If a detector is put at one slit in the double slit experiment, the wave function is altered because the particle can no longer travel through both slits; it has to be observed in either the right or left slit, destroying the initial interference pattern.

The many-worlds interpretation tries to desperately cling onto the notion of determinism by asserting that all potential measurement outcomes get actualized in parallel universes. But, in doing so it undermines the wave function’s relation to possibility and must sweep under the rug why we ourselves don’t become real in all these supposed universes.

1. Planck, Max. Scientific Autobiography and Other Papers. Translated by Frank Gaynor, Philosophical Library, 1949.

2. Einstein, A. (1926). Letter to Max Born, December 4, 1926. In Born, M., The Born-Einstein Letters: Correspondence Between Albert Einstein and Max and Hedwig Born from 1916-1955 (I. Born, Trans., 1971). Walker and Company.

3. Examples include the GRW model, Continuous Spontaneous Localization, and gravitational decoherence