8.5 Failure of the Schrödinger Equation?

Section {8.2} mentioned sending half of the wave function of an electron to Venus, and half to Mars. A scattering setup as described in chapter 7.12 provides a practical means for actually doing this, (at least, for taking the wave function apart in two separate parts.) The obvious question is now: can the Schrödinger equation also describe the physi­cally observed “collapse of the wave function”, where the electron changes from being on both Venus and Mars with a 50/50 proba­bility to, say, being on Mars with absolute certainty?

The answer obtained in this and the next subsection will be most curious: no, the Schrödinger equation flatly contradicts that the wave function collapses, but yes, it requires that measurement leads to the experi­mentally observed collapse. The analysis will take us to a mind-boggling but really unavoidable conclusion about the very nature of our universe.

This subsection will examine the problem the Schrödinger equation has with describing a collapse. First of all, the solutions of the linear Schrödinger equation do not allow a mathemati­cally exact collapse like some non­linear equations do. But that does not necessarily imply that solutions would not be able to collapse physi­cally. It would be conceivable that the solution could evolve to a state where the electron is on Mars with such high proba­bility that it can be taken to be certainty. In fact, a common notion is that, somehow, inter­action with a macroscopic “measurement” apparatus could lead to such an end result.

Of course, the constituent particles that make up such a macroscopic measurement apparatus still need to satisfy the laws of physics. So let’s make up a reasonable model for such a complete macroscopic system, and see what can then be said about the possi­bility for the wave function to evolve towards the electron being on Mars.

The model will ignore the existence of anything beyond the Venus, Earth, Mars system. It will be assumed that the three planets consist of a humongous, but finite, number of conserved classical particles $1,2,3,4,5,\ldots$, with a super­colossal wave function:

$$\Psi({\skew0\vec r}_1, S_{z1}, {\skew0\vec r}_2, S_{z2}, ... ..., {\skew0\vec r}_4, S_{z4}, {\skew0\vec r}_5, S_{z5}, \ldots)$$

Particle 1 will be taken to be the scattered electron. It will be assumed that the wave function satisfies the Schrödinger equation:

$${\rm i}\hbar \frac{\partial\Psi}{\partial t} = - \sum_i ... ...kew0\vec r}_3, S_{z3}, {\skew0\vec r}_4, S_{z4}, \ldots) \Psi$$ (8.1)

Trying to write the solution to this problem would of course be prohibitive, but the evolution of the proba­bility of the electron to be on Venus can still be extracted from it with some fairly standard manipul­ations. First, taking the combin­ation of the Schrödinger equation times $\Psi^*$ minus the complex conjugate of the Schrödinger equation times $\Psi$ produces after some further manipul­ation an equation for the time derivative of the proba­bility:

$${\rm i}\hbar \frac{\partial\Psi^*\Psi}{\partial t} = - \... ...}} - \Psi \frac{\partial\Psi^*}{\partial r_{i,j}} \right)$$ (8.2)

The question is the proba­bility for the electron to be on Venus, and you can get that by integrating the proba­bility equation above over all possible positions and spins of the particles except for particle 1, for which you have to restrict the spatial integr­ation to Venus and its immediate surroundings. If you do that, the left hand side becomes the rate of change of the proba­bility for the electron to be on Venus, regardless of the position and spin of all the other particles.

Inter­estingly, assuming times at which the Venus part of the scattered electron wave is definitely at Venus, the right hand side integrates to zero: the wave function is supposed to disappear at large distances from this isolated system, and whenever particle 1 would be at the border of the surroundings of Venus.

It follows that the proba­bility for the electron to be at Venus cannot change from 50%. A true collapse of the wave function of the electron as postulated in the orthodox inter­pretation, where the proba­bility to find the electron at Venus changes to 100% or 0% cannot occur.

Of course, the model was simple; you might therefore conjecture that a true collapse could occur if additional physics is included, such as non­conserved particles like photons, or other relativistic effects. But that would obviously be a moving target. The analysis made a good-faith effort to examine whether including macroscopic effects may cause the observed collapse of the wave function, and the answer was no. Having a scientifi­cally open mind requires you to at least follow the model to its logical end; nature might be telling you something here.

Is it really true that the results disagree with the observed physics? You need to be careful. There is no reasonable doubt that if a measurement is performed about the presence of the electron on Venus, the wave function will be observed to collapse. But all you established above is that the wave function does not collapse; you did not establish whether or not it will be observed to collapse. To answer the question whether a collapse will be observed, you will need to include the observers in your reasoning.

The problem is with the innocuous looking phrase regardless of the position and spin of all the other particles in the arguments above. Even while the total proba­bility for the electron to be at Venus must stay at 50% in this example system, it is still perfectly possible for the proba­bility to become 100% for one state of the particles that make up the observer and her tools, and to be 0% for another state of the observer and her tools.

It is perfectly possible to have a state of the observer with brain particles, ink-on-paper particles, tape recorder particles, that all say that the electron is on Venus, combined with 100% proba­bility that the electron is on Venus, and a second state of the observer with brain particles, ink-on-paper particles, tape recorder particles, that all say the electron must be on Mars, combined with 0% proba­bility for the electron to be on Venus. Such a scenario is called a “relative state inter­pretation;” the states of the observer and the measured object become entangled with each other.

The state of the electron does not change to a single state of presence or absence; instead two states of the macroscopic universe develop, one with the electron absent, the other with it present. As explained in the next subsection, the Schrödinger equation does not just allow this to occur, it requires this to occur. So, far from being in conflict with the observed collapse, the model above requires it. The model produces the right physics: observed collapse is a consequence of the Schrödinger equation, not of something else.

But all this ends up with the rather disturbing thought that there are now two states of the universe, and the two are different in what they think about the electron. This conclusion was unexpected; it comes as the unavoidable consequence of the mathematical equations that quantum mechanics abstracted for the way nature operates.