What happens when an electron’s matter wave, spread out over several centimetres, hits a fluorescent screen and produces a tiny flash of light in just one (unpredictable) place? How does the particle suddenly “acquire” an exact position, if it didn’t have one all along? Broadly speaking, there are two schools of thought on this. One interpretation is that the original wave collapses into a narrower wave, a far more localised one, by some unspecified process that involves its interaction with the screen (or any other macroscopic object). The other interpretation is that, since the electron’s broad wave packet could be viewed all along as the sum of many narrower ones, a completely quantum mechanical treatment of the situation would show that the wave function for the screen could also be viewed as a sum of many parts, each describing a flash of light occurring in a different place. Likewise, the total wave function for a person who looked at the screen would be a sum of waves describing that person seeing the flash of light in various positions. This is known as the many worlds, or many histories, interpretation.

  Why the square, in |ψ|2? Classical physics is full of examples of waves where the energy density is proportional to the square of the wave’s amplitude. If the probability density of a de Broglie wave is proportional to |ψ|2, the same mathematics that guarantees conservation of energy for classical waves works just as well to guarantee conservation of probability, so that if the chance of finding the particle somewhere in all of space is exactly 1 at a certain time, as it must be, this will continue to be true at later times.

  If you doubled ψ everywhere, there’d still have to be the same total probability of 1 for finding the particle somewhere, so it’s the relative size of |ψ|2 from place to place compared to the total of |ψ|2 for all of space that matters. Because of this, it’s standard practice to normalise wave functions, dividing through by the total so that |ψ|2 itself is the probability density, rather than just being proportional to it. This is easy with a nice localised wave packet, such as the one in Figure 9, but even for idealised waves like the one in Figure 8, there are various mathematical tricks for dealing with the fact that the total of |ψ|2 is infinite, and the probability of finding the particle in any finite region is zero.

  Wave Mechanics

  It’s possible to construct every conceivable de Broglie wave for a “free particle” — a particle subject to no forces — by adding together various combinations of complex exponentials, exp(2πi (px – Et)/h), for different energies and momenta. This strategy can also be extended to include all three dimensions of space: we just use exp(2πi (pxx + pyy + pzz – Et)/h), with different values of px, py and pz setting the direction as well as the size of the momentum vector. The wavefronts of this exponential appear in three-dimensional space as a series of parallel planes, all perpendicular to the momentum vector, so ψ in this case is called a plane wave.

  This all works very nicely, but to gain more insight into the de Broglie wave it would be helpful to have an equation for ψ, a concise mathematical statement of what constitutes a valid wave function, whether it’s a single plane wave with a definite momentum or the sum of a multitude of such waves.

  How can we find such an equation? The energy and momentum of a particle satisfy the equation E2–p2=m2, so maybe we can construct something analogous for waves. If ψ=exp(2πi (px – Et)/h), the rates of change of ψ in space and time are:

  ∂xψ = 2πip/h ψ

  ∂tψ = –2πiE/h ψ

  where we’ve used the fact that the rate of change of any exponential is equal to its value multiplied by its “growth rate,” even when that rate is an imaginary number. If we divide by ±2πi/h, this gives:

  –(ih/2π) ∂xψ = pψ (11a)

  (ih/2π) ∂tψ = Eψ (11b)

  where the minus sign appears in the first equation now, not the second, because 1/i is –i. These equations state that performing the operation on the left hand side — taking the rate of change of the wave function in either time or space, then multiplying by ±(ih/2π) — is exactly the same as simply multiplying the wave function by the energy or momentum. Repeating the process, taking the second rate of change and multiplying again by ±(ih/2π):

  –(h/2π)2 ∂x(∂xψ) = p2ψ

  –(h/2π)2 ∂t(∂tψ) = E2ψ

  To be more concise, we’ll write the second rates of change as ∂x2 and ∂t2; this doesn’t mean taking the rate of change then squaring it, but taking the rate of change of the rate of the change, as in calculating velocity from changing distance, then acceleration from changing velocity. (The most widely used notation is “∂ψ/∂x” and “∂2ψ/∂x2,” but we’ll stick to the more compact form.)

  The energy and momentum of the particle satisfy E2–p2=m2. If we multipy this equation by the value of the wave function ψ, then substitute the results we’ve just found for p2ψ and E2ψ:

  m2ψ = E2ψ – p2ψ

  m2ψ = –(h/2π)2 ∂t2ψ + (h/2π)2 ∂x2ψ

  (2πm/h)2 ψ = ∂x2ψ – ∂t2ψ

  or, to include all three dimensions of space:

  (2πm/h)2 ψ = ∂x2ψ + ∂y2ψ + ∂z2ψ – ∂t2ψ (12)

  We assumed originally that ψ was a complex exponential wave with a definite energy and momentum, but this is a linear equation: if you have two different waves, ψ1 and ψ2, that satisfy Equation (12), then a linear combination of the two, Aψ1 + Bψ2, will also satisfy it, for any values of A and B. This means that any de Broglie wave that we build up from any number of plane waves must satisfy it too.

  Equation (12) is known as the Klein-Gordon equation, or the relativistic Schrödinger equation. Erwin Schrödinger came up with it first, but Klein and Gordon derived it independently, and published it before him. The equation for which Schrödinger is more famous is a non-relativistic version, which he obtained by using the relationship E=p2/2m from Newtonian physics (that’s just K=mv2/2, with v rewritten as p/m) and taking the same approach as we’ve followed to turn this into a wave equation:

  (ih/2π) ∂tψ = –(h/2π)2/2m (∂x2ψ + ∂y2ψ + ∂z2ψ) (13)

  Equation (13) is the Schrödinger equation for a free particle. Like Equation (12), it has solutions of the form exp(2πi (px – Et)/h), though in this case p and E are the classical momentum and energy, p=mv and E=K=mv2/2, not the relativistic values. But Schrödinger’s great success was in adapting this equation for a particle subject to forces, such as the electrostatic force between an atom’s positively charged nucleus and its electrons. In Newtonian physics, forces are often described via a potential energy, V(x), that depends on the particle’s position in space. For example, an electron must have more potential energy the further it is from the nucleus, because like a ball rolling downhill it will speed up when it’s drawn closer, converting that potential energy into kinetic energy. The particle’s total energy, kinetic plus potential, then satisfies the equation E=p2/2m+V(x), and the equivalent wave equation is:

  (ih/2π) ∂tψ = –(h/2π)2/2m (∂x2ψ + ∂y2ψ + ∂z2ψ) + V(x)ψ (14)

  Equation (14) was used by Schrödinger to explain the mysterious energy levels that Bohr had postulated for the hydrogen atom. Unlike the wave for a free particle, the wave for an electron in an atom can’t take on any shape it likes: it’s constrained by the geometry of the situation to “fit” an exact number of cycles around the nucleus. Other people had suggested something similar, but their models resembled the vibrations in a circular string with a sharply defined distance from the nucleus, an exact “orbit.” Schrödinger’s solutions to Equation (14), known as orbitals, are spread out across a range of distances rather than specifying the electron’s position precisely.

  Figures 10 and 11 show two solutions to Schrödinger’s equation for a hydrogen atom. These are graphs of the value of |ψ| on a plane passing through the nucleus of the atom, at a single moment of time. The only variation of ψ with time is a cycling of the overall phase, which has no effect on the electron’s probability density, so these are described as stationary wave functions.

  The orbital with the lowest energy, shown in Figure 10, is completely spherically symmetrical: there’s an equal chance of finding the electron in any direction relative to the nucleus, though it’s more likely to be found closer to the nucleus than further away. The second orbital, not shown, is identical in shape, but the electron is, on average, further from the nucleus. The third orbital, shown in Figure 11, localises the electron into two lobes with opposite phase on either side of the nucleus. If you take the areas of the plane where |ψ| has a significant value and imagine spinning them around an axis joining the lobes, you’ll see that the shape of the three-dimensional region where the electron is most likely to be found is a kind of dumb-bell.

  Although Schrödinger’s equation only gives an approximate treatment of an electron in an atom — it doesn’t deal with relativistic effects, and it neglects an important property of electrons, their “spin” — a vast amount of the behaviour of atoms and molecules can be explained with it. Most of the differences between chemical elements and the regularities in the periodic table can be accounted for by the way elements with increasing atomic number — the number of protons in the nucleus, which is matched by an equal number of electrons — fill up more and more orbitals, creating a predictable pattern in the kind of chemical bonds that the outermost electrons can form.

  The wave function for a system of two particles, such as the two electrons in a helium atom, is not the sum of two single-particle wave functions. Rather, it’s a function ψ(x1,y1,z1,x2,y2,z2,t) that depends on the spatial coordinates of both particles, and which satisfies a 6-spatial-dimensional version of Schrödinger’s equation:

  (ih/2π) ∂tψ = –(h/2π)2/2m1 (∂x12ψ + ∂y12ψ + ∂z12ψ)

  –(h/2π)2/2m2 (∂x22ψ + ∂y22ψ + ∂z22ψ) + V(x1,x2)ψ

  This means, unfortunately, that we can’t really imagine the universe in quantum mechanical terms as being a three dimensional place that’s merely full of wave functions, rather than the particles of classical physics. That picture works for a single particle, but the waves of different particles can neither be added together when they’re in the same place — which is how classical waves, like those in the electromagnetic field, behave — nor do they generally pass right through each other without any effect. It takes a single wave in six-dimensional space to describe two particles, and one in 3N-dimensional space to describe N particles.

  That said, we often do want to consider the behaviour of a single particle, putting everything else in the universe aside (or treating it with classical physics). So the image of a wave in ordinary space can still provide a useful intuitive picture, so long as you never forget that you’re really just looking at a three-dimensional slice of something far more complex.

  Matrix Mechanics

  In the 1920s and ’30s, in parallel with Schrödinger’s wave mechanics, Werner Heisenberg developed a very different approach to the same problems, known as matrix mechanics. Though Schrödinger eventually proved that the two theories were mathematically equivalent, and though wave mechanics had the initial advantage of offering something relatively concrete to visualise — at least in the case of single-particle wave functions — Heisenberg’s approach has turned out in the long run to be the most flexible and coherent way to understand quantum mechanics.

  In matrix mechanics, every quantum mechanical system is treated as a vector space. Everyone’s familiar with at least one example of a vector space: in three-dimensional Newtonian physics, all the possible velocities a particle might have — all the different directions and speeds with which it might be moving — comprise a three-dimensional vector space. You can add and subtract vectors (e.g. the velocity 30 km/h north plus the velocity 40 km/h east gives a velocity of √(302+402)=50 km/h north-east) or multiply them by ordinary numbers to create longer or shorter vectors pointing in the same direction (e.g. 5 times the velocity 2 metres/sec upwards is the velocity 10 metres/sec upwards).

  Vector spaces with more than three dimensions are harder to visualise, but there’s really no need to be able to do that. The mathematics itself generalises to any number of dimensions very easily, and you can understand most things about a 10-dimensional vector space just by picturing the three-dimensional version, but using the 10-dimensional equations.

  To give an example of this, one additional feature that Heisenberg needed for his quantum mechanical vector spaces is a formula called an inner product, which is very similar to the Euclidean metric we introduced back in the article on special relativity. The inner product of two vectors, v and w, is a number, written as , that depends on the size of both vectors and their relative directions. The length of any vector is given by |v|2=, and two vectors are considered to be perpendicular, or “orthogonal,” if =0. For real vector spaces (in contrast to complex ones, which we’ll come to shortly), the inner product is completely linear and symmetric:

  = a+b

  = a+b

  =

  Now, suppose we’re dealing with a 10-dimensional vector space, in which we’ve picked 10 mutually orthogonal vectors, e1, e2, e3, … e10. Don’t panic, you don’t need to visualise anything more than the first three of these, which are just like the x-, y- and z-axes of Euclidean space. What’s more, suppose that |ej|=1 for j=1,2,…,10, i.e. they’re all unit vectors, vectors with a length of 1. A set of mutually orthogonal unit vectors is known as an orthonormal basis, and like the coordinate vectors we used for velocities in relativity, any vector can be written as a sum of multiples of these vectors.

  Suppose that v=v1e1+…+v10e10 and w=w1e1+…+w10e10, and we’re dealing with a real vector space. Since the inner product is linear, we have:

  = v1w1+v2w2+v3w3+…v10w10 (15)

  where out of all the one hundred terms that you’d get if you expanded the left-hand side in full, such as v4w5, this is all that remains, because etc. are zero (the ej being mutually orthogonal), and etc. are all exactly 1 (the ej being unit vectors). Equation (15) is an obvious extension to 10 dimensions of the three-dimensional Euclidean metric, g(v,w)=vxwx+vywy+vzwz, and so the inner product here behaves in essentially the same way as that metric. For example, the length of the projection of v in the direction of w is just /|w|, which is just like the formula for the same thing in three-dimensional Euclidean space, g(v,w)/|w|.

  To illustrate the link with wave mechanics, we’re going to use a “toy universe,” a highly simplified model of reality that nonetheless exhibits most of the important features of quantum mechanics. Imagine a 1-dimensional universe, with only three possible positions a particle can occupy, forming a ring: x=0, 1, 2. Assume that there is no time. Figure 12 shows a wave function in space that undergoes one cycle of phase as it wraps around the entire “universe.” The different shades here represent three different phases, separated by 120° or 2π/3: exp(2πi x/3) for x=0, 1 and 2. To normalise this function — to make the total of |ψ|2 equal to 1 — we divide these three values by √3.

  Now, consider the three functions in Figure 13, δ0, δ1 and δ2, which are equal to 1 when x is equal to 0, 1 and 2 respectively, and equal to zero for all other values of x. We can express any function of x in our toy universe as a sum of multiples of δ0, δ1 and δ2. For example, to take the function ψ in Figure 12:

  ψ = ψ(0) δ0 + ψ(1) δ1 + ψ(2) δ2

  = (1/√3) δ0 + (exp(2πi/3)/√3) δ1 + (exp(4πi/3)/√3) δ2

  What has this got to do with vector spaces? By writing a function in terms of these δ functions, it can be thought of as a vector in a three-dimensional vector space, where the three δ functions are orthonormal basis vectors, and the values of the function ψ at x=0, 1 and 2 are the coordinates of the corresponding vector.

  Since we need to be able to talk about complex functions like ψ, this is a complex vector space. All that really means is: instead of only being allowed to multiply vectors by real numbers, it’s permitted to multiply them by complex numbers, and any vector might have complex numbers as its coordinates when it’s written out in terms of some basis. We don’t have enough dimensions to visualise this completely — each complex dimension really needs a two-dimensional plane, making a total of six dimensions — but if we use distances equal to the magnitude of each complex coordinate on a three-dimensional diagram, we can get a reasonable idea of what’s going on, so long as we don’t forget that each of these coordinates is really a complex number with an argument, or phase, as well as a magnitude. (To separate out different points whose coordinates all have the same magnitude, we’ll also use the completely arbitrary convention that any coordinate with a negative imaginary part will be drawn on the negative side of the axis.)

  There’s one small adjustment that we need to make in order to deal properly with complex vectors. If the length of a vector is to be a real number, and |v|2= is still to be true, then we need to ensure somehow that will be a positive real number. We’d also like the idea of the length of a vector to be compatible with the idea of the magnitude of a complex number, in the 1-dimensional case where the vector space is just the set of complex numbers themselves. We can change the definition of the inner product in a way that solves both these problems, simply by requiring that instead of being linear in its first “slot,” the inner product is “conjugate linear”:

  = a*+b*

  = a+b

 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189