The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [204]
3. Let me clarify one imprecision. Schrödinger’s equation shows that the values attained by a quantum wave (or, in the language of the field, the wavefunction) can be positive or negative; more generally, the values can be complex numbers. Such values cannot be interpreted directly as probabilities—what would a negative or complex probability mean? Instead, probabilities are associated with the squared magnitude of the quantum wave at a given location. Mathematically, this means that to determine the probability that a particle will be found at a given location, we take the product of wave’s value at that point and its complex conjugate. This clarification also addresses an important related issue. Cancellations between overlapping waves are vital to creating an interference pattern. But if the waves themselves were properly described as probability waves, such cancellation couldn’t happen because probabilities are positive numbers. As we now see, however, quantum waves do not only have positive values; this allows cancellations to take place between positive and negative numbers, as well as, more generally, between complex numbers. Because we will only need qualitative features of such waves, for ease of discussion in the main text I will not distinguish between a quantum wave and the associated probability wave (derived from its squared magnitude).
4. For the mathematically inclined reader, note that the quantum wave (wavefunction) for a single particle with large mass would conform to the description I’ve given in the text. However, very massive objects are generally composed of many particles, not one. In such a situation, the quantum mechanical description is more involved. In particular, you might have thought that all of the particles could be described by a quantum wave defined on the same coordinate grid we employ for a single particle—using the same three spatial axes. But that’s not right. The probability wave takes as input the possible position of each particle and produces the probability that the particles occupy those positions. Consequently, the probability wave lives in a space with three axes for each particle—that is, in total three times as many axes as there are particles (or ten times as many, if you embrace string theory’s extra spatial dimensions). This means that the wavefunction for a composite system made of n fundamental particles is a complex-valued function whose domain is not ordinary three-dimensional space but rather 3n-dimensional space; if the number of spatial dimensions is not 3 but rather m, the number 3 in these expressions would be replaced by m. This space is called configuration space. That is, in the general setting, the wavefunction would be a map . When we speak of such a wavefunction as being sharply peaked, we mean that this map would have support in a small mn-dimensional ball within its domain. Note, in particular, that wavefunctions don’t generally reside in the spatial dimensions of common experience. It is only in the idealized case of the wavefunction for a completely isolated single particle that its configuration space coincides with the familiar spatial environment. Note as well that when I say that the quantum laws show that the sharply peaked wavefunction for a massive object traces the same trajectory that Newton’s equations imply for the object itself, you can think of the wavefunction describing the object’s center of mass motion.
5. From this description, you might conclude that there are infinitely many locations that the electron could be found: to properly fill out the gradually varying quantum wave you would need an infinite number of spiked shapes,