Principles and Concepts

of Quantum Mechanics

of Quantum Mechanics

Implications of

Quantum Mechanics

Quantum Mechanics

8A. Details of Separate, Isolated Universes.

Summary

The different versions of reality in the wave function, such as Schrodinger’s cat alive or Schrodinger’s cat dead, are in separate universes, with no communication possible between the universes.

We will show that the different versions of reality in the wave function are in separate universes which cannot affect each other in any way as time progresses. This result is demonstrated using a particular example, but the same arguments would hold whenever any ‘single-particle’ wave function (except perhaps that for a photon) simultaneously takes at least two different paths. It depends on the linearity of the theory.

Fire a spin silver atom into a Stern-Gerlach device. There will be one ‘trajectory’ traced out by the + wave function and another traced out by the – wave function. Let be the time-dependent three dimensional volume where the silver atom wave function is non-zero on the + trajectory and the corresponding volume on the – trajectory. These regions are well-defined even after the two branches of the silver atom wave function hit the detectors, and they are non-overlapping after the wave function of the silver atom clears the magnet.

Now consider the Schrödinger equation for the full wave function (including both versions of reality) describing the silver atom and the detectors;

(8A-1)

where the

*a*

*’*s give the relative sizes of the two versions, x

_{s}is the coordinate of the silver ‘atom’ and { x

_{d}} represents the coordinates of all the ‘atoms’ in the detectors (plus the observer if one wishes). denotes the branch of the wave function when x

_{s}is in region , and the branch when x

_{s}is in region . Because the two regions do not overlap, we have (assuming no long-range interactions for the silver atom)

when x

when x

_{s}is in region (8A-2)when x

_{s}is in region (8A-3)(Note: TheEqs. (8A-1), (8A-2) and (8A-3) imply the wave functions for the two branches obey their own separate equations of motion,Hin does not (substantially) change the location of the silver atom. Thus in Eq. (8A-2) is a function of x_{s}which has the value 0 when x_{s}is not in .)

(relevant when x

(relevant when x

_{s}is in region )(8A-4)(relevant when x

_{s}is in region )(8A-5)(and the Schrödinger equation is irrelevant (0=0) when x

_{s}is in neither region.) Thus, because they obey their own separate equations, and because H is independent of the wave function in a linear theory, the evolution of is entirely independent of what happens on the branch (and vice versa). That is, the different versions act as if they were in different, non-communicating universes.

Further, we see that,

*once the two parts of the silver atom wave function have separated*the coefficients a

_{+}and a

_{-}can never change (in a unitary theory). See also Mathematical Collapse and Linearity on the need for non-linearity in mathematical collapse theories.