Macroscopic quantum phenomena and tunneling

Question

Wikipedia's article on macroscopic quantum tunneling says

Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (typically Avogadro's number) or the quantum states involved are macroscopic in size (up to km size in superconducting wires).

To comply with copyright laws, the following is an edited paraphrase of this reference, pp6-7.

http://assets.cambridge.org/97805218/00020/sample/9780521800020ws.pdf

The term “dynamical degrees of freedom” should be used carefully. Imagine a baseball moving through a wall without being compressed. Certainly, this phenomenon can be called a macroscopic tunneling; since the ball is a collection of atoms, the number of degrees of freedom is comparable to the number of atoms.

Macroscopic tunneling depends on the number of microscopic degrees of freedom like the positions of constituent atoms. Collective degrees of freedom are superior: they are singled out by rearranging the microscopic ones.

Are there any circumstances under which the ball could pass through the wall via macroscopic quantum tunneling, or is this wishful thinking?

Answer 1

Are there any circumstances under which the ball could pass through the wall via macroscopic quantum tunneling...or, is this wishful thinking,

Quantum mechanics and tunneling solutions are dependent on there being a unique wave function describing the system. Wave functions have amplitude and phases, and their complex conjugate squared will give the probability density function for the particular problem.

This simple example illustrates the possibilities:

So your question asks really: is there a single wavefunction describing a ball hitting a wall so that the quantum mechanical calculations would give a probability for the ball to pass the wall.

Evidently the ball is composed of ~$10^{23}$ molecules and the wall too. Each molecule individually would have its quantum mechanical wavefunction , if single. One wave function theoretically describes the whole ball, including all the huge number of variables needed to write it down for the ensemble of molecules.

The density matrix formalism treats the many body quantum mechanical problem. The nutshell I have retained is that for macroscopic objects, as the ball, the quantum mechanical phases are lost because the off diagonal elements become very small and one ends up with a classical macroscopic body, except in cases as superconductivity, where a macroscopic quantum mechanical solution can be defined. In your ball example, the probabilities calculated are essentially zero, and one is back to classical mechanics.

Answer 2

The ball can indeed tunnel through the wall, but a proper description of that astronomically rare event would require a lot of nontrivial analysis. A single wavefunction description is not going to be adequate here, due to decoherence. If you condition on an astronomically low probability event going to happen, then that opens a can of worms containing events that are astronomically more likely but still astronomically rare. A good example is the analysis given in this article about a system spontaneously fluctuating to lower entropy states. While it's clear that an ice cube that has melted in a hot cup of tea can spontaneously unmelt and reappear, what is not clear is how in practice that would happen.

The formalism used in that article may be useful for this problem too, it uses the symmetric two state formulation of quantum mechanics, which allows you to impose the astronomically rare thing having happened as a future boundary condition.