Schrödinger's Cormorant

6.11 Total Collapse

Eigenfunctions can be added to form superposition states but these must collapse to a single eigenfunction if a measurement is made.

We used the wave equation, back in Chapter 4, to confirm what experience already told us, that if two waves meet they add up to make a superposition which is also a wave. We’ve already used this idea quite a lot – in constructing stationary waves and in understanding the price in energy that must be paid for localisation. Now we have seen that the Schrödinger equation for our particle in a box generates a whole series of possible wavefunctions for the particle, defining possible standing waves. Based on what we have learned again, back in Chapter 4, we know that superpositions of these waves will also be stationary waves. This leads to one of the key ideas of quantum mechanics: a particle does not necessarily have to be in one of the eigenstates of the Hamiltonian – it can be in a superposition state, where its wavefunction is a linear combination of the eigenfunctions.

This represents a huge increase in the options open to a particle. The question we need to ask, however is are these superpositions themselves eigenstates of the Hamiltonian? If so, this would mean many more possible states and many more possible energies for the particle. To investigate this, lets consider a simple case, using our particle in a box model:

Question 6-13

This exercise has drawn our attention to not just one but two general principles of quantum mechanics.

Orthogonality: this is what we found in step ② of the question. When we took two different eigenfunctions of the same Hamiltonian, ψ1 and ψ2, we found a remarkable definite integral:

When two wave functions have this property, we say they are orthogonal. You can see, in the answer to Question 6-13, how this arises, as a result of cancellation of positive and negative areas in the graph.

In physical terms, orthogonality means that the wavefunctions are completely mutually exclusive. So if a measurement is made, the system can be in one state or the other but it can't be in a state that combines a little bit of both. In fact, this is a general property of all eigenfunctions of a Hamiltonian: if they differ only in the quantum number, they are found to be orthogonal. You'll find this to be a very important principle if you go further into quantum mechanics.

A superposition of eigenfunctions of the Hamiltonian generates another valid wavefunction, BUT this wavefunction is not itself an eigenfunction.

This means that although a particle's wavefunction could exist in the superposition state, it would not have a properly defined energy in this state. If we tried to measure its energy the value we would obtain could only be an eigenvalue of the Hamiltonian. More specifically, the energy measured could be any of the eigenvalues attached to the eigenstates that contribute to the superposition.

Thus in our example, we had a superposition state with wavefunction:

If we made a measurement, we could get either E1 or E2 but not some combination of the two.

Nefertiti now asks the innocent-sounding question “so what happens to the superposition state wave function when this energy measurement is made?” This is when things start to turn ugly. The assembled physicists, ghostly and otherwise, start muttering, then arguing and soon there will be a full-scale brawl. When it comes to contentious issues in physics, this may be the big enchilada.

The mostly widely accepted idea is that the wavefunction “collapses” when – because – the measurement is made. This means that the superposition state ceases to exist at that moment and instead the particle finds itself in one of the eigenstates, where its energy is defined and can be measured. Thus, the particle exists only as a probability density distribution in space until we make an observation that forces it into a specific eigenstate.

This is called the "Copenhagen Interpretation" because it was first developed by Niels Bohr and his mates in that city. Many physicists do not like it, though – not Prince Louis, not Erwin, not Uncle Albert, who famously complained “Am I seriously supposed to believe that the moon is not there in the sky until I see it?”

Erwin didn't like it at all: his attempt to illustrate the interpretation was his infamous cat, shut in a box with the possibility that it might or might not be ruthlessly murdered while we can’t see. Until we open the box, we can’t know whether the cat is dead or alive, so it is – according to the model – in a superposition state in which its state of life/death is not defined. As soon as we open the box and observe, we make the wave function collapse, and the cat’s fate is - one way or the other – sealed. Erwin thought this was patently absurd – the cat was surely already either alive or dead before the box was opened – but, ironically, this gedankenexperiment has become the favourite way of explaining the nature of wavefunction collapse. Erwin is not amused. Nefertiti has an idea. Cormorant is nowhere to be found.

There are alternative interpretations – such as the many worlds model which supposes that each time a choice has to be made – such as where a particle will be detected, all possible outcomes happen, generating alternative universes as a result. This avoids the lack of determinism which is, for some, the unpleasant side of Copenhagen: there is no (apparently) random decision as to which eigenstate the superposition will collapse into. The trouble with many worlds, however, is that it’s an essentially untestable hypothesis. That’s not a good thing in science.

Even amongst those who accept wave function collapse, there is disagreement about how and why it happens. Some think it has to do with interaction of a consciousness with the wavefunction; the more mainstream view is that somehow the interaction process with a measuring device causes the collapse. These are fascinating and, at a deep philosophical level, hugely important questions but we can’t answer them with the level of quantum understanding we have so far accrued. So let’s proceed with the (slightly vague) idea of wavefunction collapse, in the full knowledge that we’ll attract disapproving glances from some quarters.

By extension of Max Born’s principle, the probability that a particular eigenvalue will be measured and, correspondingly, that the wave function will collapse into a specific eigenstate, is given by c2, where c is the coefficient of that eigenstate in the superposition. Thus these coefficients must be normalised: since an energy measurement will certainly yield one of the possible eigenvalues, the sum of the c2 terms must equal 1.

Let’s explore this idea with a little example:

Question 6-14

If we repeatedly make the wavefunction collapse collapse and measure the energy of the eigenstate that results, we can find the mean of the results. We call this the expectation value of the energy measured for a particle in state ψs. The really curious thing is that <E> can never be measured in any individual observation but it emerges as the mean, as a large number of such observations are made.

This concept of wave function collapse can also be applied to the question of finding the position of a particle from its wavefunction. The idea is that in quantum mechanics, any observable quantity – be it energy or momentum or position or whatever – is found by applying the corresponding operator to the wavefunction. However eigenfunctions of the energy operator (the Hamiltonian) are not necessarily eigenfunctions of other operators, such as the position operator. Thus the energy eigenfunctions which we have generated for the particle in a box are also superpositions of the eigenfunctions of the position operator. Consequently, when a position measurement is made, this superposition wavefunction collapses and a single position is measured.

Which brings us back to the question of why there doesn’t seem to be any uncertainty in Cormorant’s position when he has a go at being a particle in a box. “If you know where you are it means you’re observing yourself, so that’s why your wavefunction collapses” suggests Nefertiti. Cormorant doesn’t buy this: ”I might be asleep,” he retorts “but you’re still always going to find me in the same, predictable place, whenever you make your measurement.” The answer, perhaps, is that any interaction with a macroscopic object can – for whatever reason - lead to collapse of the wave function. For something big, like a Cormorant, it’s very difficult to isolate it from the rest of the universe, so there is the potential for many such interactions – with the surface he’s standing on, the air he’s flying through and so on – and so his wavefunction is immediately collapsed before it could even be established in the pipe. Whatever the detail of this, the bottom line is that just as special relativity only really became useful when travelling at speeds close to the speed of light, so quantum theory is only useful when we’re dealing with something really small.

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