In reading Louisa Gilder’s book The Age of Entanglement, I was reminded of Bohr’s correspondence principle1 (originally analogy principle and also referred to as “Bohr’s magic wand”). I hadn’t thought about it much lately. Other than a few times in Lederman’s book Quantum Physics for Poets, the term wasn’t referenced in the other physics books which I read this past year. Is the correspondence principle still relevant in modern physics, in QFT? Or only of historical interest?
Gilder chronicles the debate and distress regarding quantum mechanics (QM) in her book. In the chapter “Uncertainty – Winter 1926-1927,” for example, there’s a strained discussion between Bohr and Heisenberg.
He tried to laugh, apologetically, but Bohr was serious. “Heisenberg, you must understand the torture I am putting myself through in order to get used to the mysticism of nature,” he said. Slowly pacing and slowly explaining, Bohr continued. He wanted to look at both approaches together, and look beyond them to their “epistemological lessons.” His quiet voice pressed on, until Heisenberg felt he could stand it no longer, … — Gilder, Louisa (2008-11-11). The Age of Entanglement (Kindle Locations 1907-1910). Knopf Doubleday Publishing Group. Kindle Edition.
The correspondence principle was centerstage in early quantum theory.
Bohr’s correspondence of large-scale quantum effects with classical physics would guide the young quantum theory for years. But his faithfulness to the spirit of this principle would also prevent him from discovering entanglement. For him, a concept like that—quantum correlation between two separated particles—would belong solely to a shadowy netherworld of subatomic semireality, which could never emerge into the bright Newtonian day of large sizes and distances. — Gilder. Ibid Locations 1141-1144.
The correspondence principle faced some challenges:
- Mapping complimentary (inseparable) properties to classical determinism, e.g., wave-particle, position-momentum.
- Mapping QM “motion” to classical mechanics, e.g., probabilistic particle trajectories, electron jumps/leaps in atoms.
- Mapping discontinuities to classical continuum, e.g., wavefunction collapse.
- Explaining the process of entanglement between quantum doings and “observer,” e.g., the measurement problem (“the distinction between the classical measuring apparatus and the quantum object under observation” — Gilder. Ibid Location 2275).
Bohr’s and Einstein’s positions remain a major divide in the interpretation of QM.
Wiki: Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Bohr … argued that quantum physics does not reduce to classical mechanics similarly as classical mechanics emerges as an approximation of special relativity at small velocities. He argued that classical physics exists independently of quantum theory and cannot be derived from it.
Einstein was searching, as in all his endeavors, for a unified view of the world, not for “complementary” ways of describing it. — Gilder. Ibid Locations 2182-2183.
That historical debate produced the quandary over the “completeness” of QM (particularly the Copenhagen interpretation) and the EPR paradox, which lead eventually to Bell’s Theorem.
Wiki has an interesting example of how large quantum numbers emerge as classical behavior for the quantum oscillator.2 But the contention was over whether in general quantum behavior (as noted above) is epistemologically relevant in a classical framework, even with a statistical interpretation. Local realism, the notion of causality. Hence, the “just do the math” approach to quantum mechanics. Or, as Neil deGrasse Tyson put it, “The universe is under no obligation to make sense to you.”
 As opposed to the complementarity principle, Wiki describes the correspondence principle as:
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.
The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.
The term is also used more generally, to represent the idea that a new theory should reproduce the results of older well-established theories (which become limiting cases) in those domains where the old theories work.
Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle.
For example, Einstein’s special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below). General relativity reduces to Newtonian gravity in the limit of weak gravitational fields. Laplace’s theory of celestial mechanics reduces to Kepler’s when interplanetary interactions are ignored, and Kepler’s reproduces Ptolemy’s equant in a coordinate system where the Earth is stationary.
 Wiki: The quantum harmonic oscillator
Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior.
Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values,
E = (n + 1/2)ℏω, n = 0.1.2.3, … ,
where ω is the angular frequency of the oscillator.
However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of macroscopic systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is
E = (m⋅ω^2⋅A^2)/2 .
Thus, the quantum number has the value
n = E/ℏ⋅ω − 1/2 = (m⋅ω⋅A^2)/2ℏ – 1/2
If we apply typical “human-scale” values m = 1kg, ω = 1 rad/s, and A = 1 m, then n ≈ 4.74×1033. This is a very large number, so the system is indeed in the correspondence limit.
It is simple to see why we perceive a continuum of energy in this limit. With ω = 1 rad/s, the difference between each energy level is ħω ≈ 1.05 × 10−34J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent classical limit.