The imaginary number at the heart of quantum mechanics.

[NOTE – this is re-post from the original incarnation of this blog.]

I’ve no idea how many books, articles, podcasts and videos I’ve digested over the years on the subject of quantum mechanics, but it’s a lot.

What they all have in common is that they are aimed at the layperson, and therefore try to describe counterintuitive features of the theory such as superposition, the uncertainty principle, and entanglement using experimental examples and everyday analogies. Almost none of them take even the briefest of toe-dips into the actual mathematics behind the theory.

And that’s not surprising. After all, as Stephen Hawking wrote in A Brief History Of Time, “Someone told me that each equation I included in the book would halve the sales“. No-one outside academia likes to try to get their head around baffling equations, least of all those with no more tools at their disposal than high school maths. Like me.

click image for source

So, some time ago when I made an attempt to dig a little deeper by watching a series of Quantum Mechanics lectures by Leonard Susskind, I wasn’t expecting to get much out of it. I couldn’t have been more wrong.

OK, I couldn’t follow all the intricacies of the maths, and I certainly wouldn’t be able to do any of the calculations myself, but what I did gain was a good overview of the subtler concepts behind the theory, and an understanding of how the maths models and corresponds to those features, including the more counterintuitive ones, like superposition and the uncertainty principle.

The biggest revelation for me was the significance of the constant i, which confusingly is also sometimes known as j, mainly by engineers. i is the symbol that represents an imaginary number, a number that does not exist, specifically the square root of -1.

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This concept is at the mathematical heart of the theory, yet is almost never mentioned in the layperson’s literature, let alone explained. I first encountered i years ago before I started reading about quantum mechanics. A friend who works at a nuclear site who recounted how a scientist there had told him about it. We were both slightly incredulous that the maths behind quantum mechanics, the theory therefore that underpins all our nuclear know-how and much of modern technology is fundamentally based on something we don’t and can never know.

However, it’s not really quite that scary, and quantum mechanics remains by far the most experimentally accurate theory science has ever produced. As with most discoveries, the physical phenomena were uncovered first by experimentation, and only then were mathematical models found to fit what was being seen. These models were then be used to calculate the outcomes of further experiments, proving the theory, and enabling the practical use of the phenomena. The theory isn’t founded on the maths, the maths models the phenomena to give predictive power to the theory. (This is not to say that physical reality is more fundamental than mathematics or vice versa – that’s a philosophical question for another day – but rather it’s just the way scientific discovery usually works.)

So how is i used, and what does it represent? Well, for starters the concept was not created for use in QM, but pre existed and is used in classical physics. The following short clip from icedave33 (who I should also thank for clearing up my many misconceptions whilst writing this piece!) explains very well how i can be used to model 2D rotation on a 1D line:

The imaginary number i and multiples thereof, are used in a series (i, 2i, 3i etc.) to create an extra “imaginary axis” on a graph. The three normal spatial directions are combined on to the “real axis” as shown below:

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Points plotted against the real and imaginary axes then take the form of one real number (e.g. 2 or -2) and one imaginary number (e.g. 3i or -3i), making a composite that is known as a complex number (e.g. 2+3i or 2-3i or -2+3i or -2-3i). This full graph on which both real and complex numbers can be plotted is known as the complex plane.

In quantum mechanics, a system when it is observed can only be found in a few different observable states.  Before observation, the system is in superposition (a combination of these states) and each of the states is assigned a complex number.

A complex number can be represented as an arrow in the complex plane, as shown above. The “size” of the complex number is given by the length of the arrow, but the probability of finding the system in the particular state represented by the complex number of that size is relative to its squared value.

One of the ways to represent a state in quantum mechanics is as what’s known as a quantum state vector. A state vector is just a list of numbers, each corresponding to the value of a parameter of the state – for example quantum spin. States corresponding to the possible observable states are known as eigenstates.

In the maths, the equivalent of observing a parameter is to apply to the state vector a special type of mathematical operator known as a Hermitian operator. This will produce a new state vector.

If the operator is applied to a state vector already in an observable state, then the new state vector is just a multiple of the original. It is still a real number on the real axis. These are known as its eigenvalues.

However, if the operator is applied to a state vector in superposition, then the new state vector will not be like the old one. Instead the system collapses into an observable state and again we have a real eigenvalue.

Intuitively this seems wrong, because in the maths of real numbers it is somewhat like saying that adding one to a number doesn’t always increment it’s value by one, but that instead it depends on which number you begin with. To use an analogy with real whole numbers and fractions, it’s like saying that two plus one equals three, but also any fraction between two and three (2.1, 2.5, 2.999 etc) plus one also equals three.

Another analogy might be a clock’s second-hand that cannot stop between seconds because the mechanism doesn’t allow this to happen. The hand is always at a whole number value. Except that in the case of quantum mechanics, we know that the second-hand can reside between ticks, it’s just then whenever we measure the time by observing it, we always find that the second-hand has ticked into place.

So although superposition is unintuitive, the mathematics of imaginary and complex numbers and their operators models it precisely. Similarly, it’s the use of operators that allows the maths to model other features of QM such as the uncertainty principle. This states that is impossible to measure two parameters of a system’s state to a high degree of accuracy at the same time; for example position and momentum or time and energy. This happens because in quantum mechanical systems, certain mathematical operations do not commute. With real numbers, this would be like saying that two times three does not equal three times two, but that they yield two different answers. This corresponds to the way that measuring position then momentum, or momentum then position, can result in different answers.

So if one can see what the maths is doing, then is it true to say that one has a picture of might be happening physically?

For instance, to me at least, the maths suggests that quantum states in superposition are unobserved in the real world because they are at least partly “off somewhere else” on the imaginary axis, and that the act of measurement seems to “snap” them into existence in the “real” observable world by applying the Hermitian operator.

This is not to suggest that by “off somewhere else” I mean somewhere supernatural. Rather I’m suggesting that the state they are in is not realizable in the external objective word that we experience and measure. Perhaps like virtual particles they reside in some sub-planckian world between the ticks of the clock I used in my earlier analogy.

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However, this is where science meets philosophy, and interpretation is everything. Not always in physics do the apparent properties of the maths correspond to how things are physically. The use of complex numbers in classical wave mechanics described above is a good example, but there, at least as I understand it, the use of the extra dimension introduced by i is more like a shortcut to avoid doing harder, more regular maths. In quantum mechanics I don’t believe that’s the case. The i is a mandatory part of the theory.

Additionally, there’s the possibility (or almost certainly the inevitability) that quantum mechanics will one day be superseded by an even more accurate theory that has different mathematics and thus revises the physical picture again. One thing’s for sure, like i itself, nothing in science is set in stone, but I wish I’d had at least a little introduction to the maths in some of the popular books I’d read previously.


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