We saw that bullets do not give rise to an interference pattern but electrons do. The question is why. We can answer this by looking at the difference between a bullet and an electron. A bullet is much more massive than an electron. Its velocity (typically) is also much lower than that of an electron's. So let's first make the velocities equal. Do we now get interference with the bullet? We need to determine the wavelength of a bullet. The wavelength is about 10^-12 smaller than that of an electron even if both are moving at the same speed. This arises because of the mass difference between a bullet and an electron. As the wavelength of a particle gets smaller and smaller, can we resolve its wave-like characteristics? The answer is no. Wavelike nature goes away as the wavelength gets smaller and smaller. Objects with vanishingly small wavelengths appear continuous. Hence, there is no interference. This is the key. Quantum mechanics becomes important when objects have wavelengths that are on the atomic scale. Should the wavelength be less than atomic length scales, then we need not worry about quantum mechanics. Such objects appear continuous and we can think about them in our usual classical way. Another way of thinking about this is to consider what happens when the spacing between the slits goes off to infinity. There is no overlap of the waves and hence no interference. This tells us that if we were to devise an experiment to see the interference of bullets, we would have to have the slits so close together that it would be impossible to determine which slit the bullets went through. The separation between the slits would have to be on the order of half the wavelength of a bullet. Such length scales are not possible. This further shows you that bullets and electrons only differ quantitatively. In quantum mechanics, size matters.

The CI adopts two principles:

The dichotomy between the QM and classical realm is fundamental to the Copenhagen Interpretation. Some other QM interpretations and modifications try to avoid it, and some try to make it a result of a unified theory, not a dualistic assumption.

There are several issues to be studied:

• Is QM (the theory) correct? I.e., does it make correct predictions?
• Is the probabilistic wave function necessary? Are there some "hidden variables" which really determine the outcome?
• Is the CI distinction between macroscopic and microscopic necessary? Does it have to be so qualitative and extreme?
• What other interpretations are there? What do they signify?

We need to make these concepts clear, because they become issues in QM.

Remember that, in principle, with adequate information predictions can be made with certainty. Some difficulties:

• Objectivity ≠ determinism.
• Classical physics is compatible with objectivity. E.g., objects have positions and velocities even when they aren’t observed. (Note: it's not necessary that these particular properties survive as objective, just that which properties apply must not depend entirely on your way of looking at things.)
• We saw with relativity that the "objective", i.e. invariant, quantities don't have to be the same ones that you might guess. We have to be careful in QM interpretation not to describe it as non-objective just because the properties we would like to have be independent of "observation" turn out not to be. The question is, are there some good set of properties which are independent of observation? And what is observation anyway? Is it any interaction with large-scale stuff, or is it something more subjective?

A measurement of some property of a micro-system is an interaction with a macro-system, such that the observed outcomes correspond to eigenvalues of the operator representing that property of the microsystem.

For momentum, there is a continuous range of possibilities, but for spin, the set of results is discrete. The state vector tells us the probability of obtaining each eigenvalue.

Despite QM's probabilistic nature, there remains a lot of determinism:

• The evolution of the wave function is deterministic as long as no "measurement" occurs.
• The statistical distributions are determined by the wave function and are therefore deterministic.
• There is statistical determinism, similar to what we have in thermodynamics.

The frequentist interpretation makes no sense when we are talking about a single atom. The subjectivist interpretation now has to include some irreducible ignorance, which does not contradict the general idea that there is also reducible ignorance, prior knowledge, etc.

Two important questions:

Either answer to 1 gives us problems. If yes, how can it change discontinuously when a measurement is made? This is not only a time discontinuity, but also spatial. If we can avoid the collapse of the wave function, then we might avoid this problem. If no, then how does it give rise to physical effects such as interference? We’ll deal with question 2 later.

If the only problems with QM were that it was not fully deterministic, or that it said that our old ideas about waves and particles didn't describe the world, or that quantities like "momentum" do not generally have precise values, my response would be "Live with it." Relativity has already taught us not to put too much stake in our prior prejudices. We'll see, however, that QM asks us to give up much more, so that it is not quite clear what it is saying about the world, even though we know exactly how to use it to make predictions about almost all experiments. We'll also see that just plain experiment forces us to give up nearly as much of our basic worldview as QM theory does.

So far, we have just described how the wave-function obeys the equation.

Here's the key point: if you have just one particle, going through two slits, the two paths show interference only if they get to the same place. The (x,y,z,t) coordinates must all be the same. The wave function representing MANY particles is a function of ALL their coordinates, so if there are two lumps of this wave function evolving in time, they show interference only if ALL the coordinates of ALL the particles can get to the same places by each path, at the same time. This simply never happens once many particles are involved in a complicated system.

Thus we now have two distinct possibilities, represented by

.

We now have gotten rid of the interference, while postulating nothing different from the linear wave equation, and without worrying about "duality" or any such philosophy.

At this point the solution to the equation gives us: Both distinct possibilities are still there, even though they don't interfere!

Schrödinger's cat:

Say that the micro-variable is a quantum spin, and the measurement apparatus is set up to kill a cat if the spin is up, and give it some food and water if the spin is down. This is not a science-fiction idea, but a relatively trivial thing to set up in an ordinary lab.

The result of the solution of the linear wave equation is that the cat is both alive and dead, in a superposition. This does not mean "in a coma" or "almost dead" but BOTH fully alive and purring or thoroughly dead and decomposing.

Furthermore, once you look, your wave function becomes entangled with those of the cat, etc. The solution of the linear wave equation now describes a superposition of a you who has seen the dead cat and a you who has seen the live cat!

The common folk view among physicists is that the wave function somehow "collapses" to one of the possibilities, following the Born probability rules.

Some physicists hold to a sort of hybrid of these rules.

Many physicists will give this resolution, or something similar.

There are two steps to the argument.

Remember the two slit experiment. The reason we get in trouble when we try to say that the electron really went through one slit or the other is that would destroy the interference between the two paths. However, when we are in a situation where there is no interference pattern to begin with, then what prevents us from making that kind of either/or statement?

The first step of the argument is not very controversial. A macroscopic apparatus is very complex and much larger than any relevant wavelengths, so interference vanishes to truly immeasurable levels. This phenomenon is called decoherence. It appears in the classical physics of waves, and is not a quantum mystery.