Comment on: “Critical assessment of the Schrödinger picture of quantum mechanics”

Abstract

Recently, Faria et al [Phys. Lett. A 305 (2002) 322] discussed an example in which the Heisenberg and the Schrödinger pictures of quantum mechanics gave different results. We identify the mistake in their reasoning and conclude that the example they discussed does not support the inequivalence of these two pictures.PACS: 03.65.Ta Keywords: Foundations of quantum mechanics A long time ago, Dirac argued [1] that, in systems with an infinite number of the degrees of freedom, the Schrödinger picture of quantum mechanics may not be equivalent to the Heisenberg picture. The standard general proof of the equivalence of these two pictures may fail to be valid because, when an infinite number of the degrees of freedom is present, the formal unitary transformation between these two pictures may not exist in a rigorous sense. In [1], Dirac argued that it was the Heisenberg picture that was correct in the case of an infinite number of the degrees of freedom, while the Schrödinger picture was wrong. The two pictures are really equivalent only when the number of the degrees of freedom is finite.Recently, Faria et al [2] considered an example in which they explicitly obtained a discrepancy between the results obtained in the two pictures. Specifically, they considered a charged harmonic oscillator in 3 dimensions (having 3, i.e. a finite number of the degrees of freedom!) interacting with the electromagnetic field (having an infinite number of the degrees of freedom). For simplicity, they studied only the x-direction of the harmonic oscillator. In the Heisenberg picture, they quantized both the harmonic oscillator and the electromagnetic field and obtained that the average square of the position operator of the harmonic oscillator in the ground state is equal to x 2 =h/2mω 0 (where m is the mass of the harmonic oscillator and ω 0 is its frequency), the same result as if the interaction with the electromagnetic field was absent. In the Schrödinger picture, they used a semiclassical 1