Historical research guide · 1155 · 1989
Precision Tests of Quantum Mechanics
Steven Weinberg
In one minute
Quantum mechanics is linear: add two solutions and you get another. Steven Weinberg asked how well that has actually been tested, and proposed a way to test it far more stringently. He writes down a generalisation in which the wave function evolves under a Hamiltonian function that need not be bilinear, only homogeneous, so that small nonlinear terms can be switched on while the theory keeps its Hamiltonian form. Then he looks for the fingerprint any nonlinear oscillator leaves: detuning. In a nonlinear system the resonant frequency depends on how strongly each mode is excited, so a slow transition driven at a fixed frequency drifts off resonance as it proceeds. The hyperfine transitions of beryllium ions used for frequency standards are driven over seconds, which makes them an extremely sensitive detector. Weinberg shows that existing measurements already bound any nonlinear share of the beryllium nucleus's energy, and names how to tighten the bound by orders of magnitude.
Connection
Chapter 17's section on self-consistent loops needs to say exactly where a door could open. Standard quantum mechanics forbids signalling through entanglement, and those proofs lean on linearity. Weinberg's paper is where a small nonlinearity became a precisely stated, measurable parameter — the thing later work showed would change what entanglement can carry, and the thing precision spectroscopy can bound.
From the source
The generalisation. The wave function obeys Hamilton-form equations with a real Hamiltonian function of the wave function and its conjugate that is homogeneous of first degree in each; if it is bilinear the equation reduces to ordinary time-dependent quantum mechanics, and small non-bilinear terms produce small nonlinearities. Homogeneity guarantees that a solution multiplied by a constant is still a solution.Equations 1 and 2 and the paragraph following
The signature. A nonlinear system's resonant frequency depends on the amplitudes of the modes excited, so a transition driven slowly over a time T can only go to completion if the change in resonant frequency along the way stays within a natural width of order one over T; observing the transition complete therefore bounds the nonlinearity. For a two-level system the resonance condition tuned for the full transition still equals the energy difference, as in ordinary quantum mechanics.Opening paragraphs; Equations 10 and 11
The experiment. The radio-frequency hyperfine transition in beryllium ions used to set frequency standards is driven with transition times of order a second; existing measurements at the National Bureau of Standards already set a limit on the fraction of the beryllium nucleus's energy that could come from nonlinear corrections, and lengthening the transition time, using Ramsey's separated-field method with free-precession times of minutes, or line-splitting methods could improve it by orders of magnitude.Abstract; the beryllium ion analysis and closing paragraph; reference 1, Bollinger, Prestage, Itano and Wineland 1985
The way in
https://doi.org/10.1103/PhysRevLett.62.485How to cite it
Steven Weinberg (1989) Precision Tests of Quantum Mechanics doi:10.1103/PhysRevLett.62.485