The End of Time: The Next Revolution in Physics

One of the equations that Schrödinger found governs this process. If ψ is known everywhere in Q at a certain time, you know what ψ will be slightly later. From this new value, you can go on another small step in time, and another, and so on arbitrarily far into the future. The role played here by the red and green mists, the two primary components of ψ, is quite interesting: the way the red mist varies in space determines the rate of change of the green mist in time, and vice versa. The two components play a kind of tennis. This equation is sometimes called the time-dependent Schrödinger equation because time features in it. This is not in fact the first equation that Schrödinger discovered.

The first one he found is now usually called the stationary or time-independent Schrödinger equation. This determines what happens in certain special cases in which the two components of ψ, the red and green ‘mists’, oscillate regularly, the increase of one matching the decrease of the other. This has the consequence, as we have already seen for a momentum eigenstate, that the blue mist (the probability density) has a frozen value – it is independent of time (though its value generally changes over Q). Such a state is called a stationary state. This explains the name given to the second equation – its solutions are stationary states. The standard view is that the time-dependent equation is the fundamental equation of quantum mechanics; the stationary equation is seen as a special case derived from it. This corresponds to an overall scheme in which some state of ψ is created at some time and then evolves until a measurement is made.

There are intriguing hints that in the quantum mechanics of the universe the roles of these two equations are reversed. The stationary equation (or something like it) may be the fundamental equation, from which the time-dependent equation is derived only as an approximation. We think it is fundamental because we have been fooled by circumstances that make it valid for the description of the phenomena we find around us. However, these phenomena deceive us greatly when it comes to the overall story of the universe. In particular, they lead us to believe time exists when it does not.

That this is likely to be so follows from an important property of the two Schrödinger equations. For any quantum system, we can use the time-independent equation to find all the stationary states it can have. Each of these states corresponds to a definite energy, and in each of them the red and green mists oscillate with the same fixed frequency while the blue mist remains constant. These solutions are also solutions of the time-dependent equation, though they are special, being stationary. I have mentioned linearity in quantum mechanics. Here, linearity means that two or more solutions of the time-dependent Schrödinger equation can be simply added together to give another solution. If the special stationary solutions are added, something significant results. In each solution, considered separately, the red and green mists oscillate at a fixed frequency while the blue mist remains constant. However, when we add two such solutions with different frequencies, they interfere: the added intensities of the red and green mists no longer oscillate regularly. More significantly, the blue mist varies in time.

Now this is very characteristic – indeed, it is the essence of quantum evolution. All solutions of the time-dependent equation can be found by adding stationary solutions with different frequencies. Each stationary solution on its own has regular oscillations of its red and green mists, but a constant – in fact static – distribution of its blue mist. But as soon as stationary states with different energies, and hence frequencies, are added together, irregular oscillations commence – in particular in the blue mist, the touchstone of true change. All true change in quantum mechanics comes from interference between stationary states with different energies. In a system described by a stationary state, no change takes place.