It bears repetition: quantum mechanics assigns probabilities to possible measurement outcomes on the basis of actual measurement outcomes. As we just saw, the probabilities of all possible outcomes are encoded in a density operator, and the trace rule (or, in an important special case, the Born rule) tells us how to extract them. The questions to be addressed next are: how is the density operator determined by actual outcomes, and how does it depend on the time of the measurement to the the possible outcomes of which it serves to assign probabilities.
Suppose that the density operator W1 is appropriate for assigning probabilities to the possible outcomes of any measurement that may be made at the time t1. And suppose that a measurement M is made at t1, and that the projector P represents the outcome. Which density operator W2 is appropriate for assigning probabilities to the possible outcomes of whichever measurement is made next, at a later time t2? (As is customary in discussions of this kind, we focus on repeatable measurements. If a physical system is subjected consecutively to two identical measurements, and if the second measurement invariable yields the same outcome as the first, we call these measurements “repeatable.”)
W2 is uniquely determined by the following requirements:
- It is constructed out of W1 and P.
- It assigns probability 1 to the outcome P of M.
- It assigns probability 0 to every other outcome of M.
If M is a maximal test (that is, if all of its outcomes are represented by 1-dimensional projectors), then W2 equals P. Lo and behold, if we update the density operator to take into account the outcome of a maximal test, it turns into the very projector representing this outcome. It is particularly noteworthy that in this case W2 is independent of W1.
Let us now relax the requirement of repeatability and only demand that measurements be verifiable. A measurement M1, performed at the time t1, is verifiable if it is possible to confirm its outcome by a measurement M2 performed at the later time t2. For this condition to be satisfied, there must be a one-to-one correspondence between the possible outcomes of M1 and those of M2, such that the actual outcome of M1 can be inferred from that of M2.
If the two measurements are maximal tests, so that there are two systems of mutually orthogonal 1-dimensional subspaces, the first representing the possible outcomes of M1, the second representing the possible outcomes of M2, this amounts to requiring the existence of an operator U (of the kind physicists call “unitary”), which transforms any vector in any subspace belonging to the first system into a vector in the corresponding subspace belonging to the second system.
This operator determines how quantum states “evolve.” In other (and less seriously misleading) words, if v1 is the state vector appropriate for calculating the probabilities of the possible outcomes of any measurement that may be made at the time t1, then
v2 = U(t2,t1)v1
is the state vector appropriate for calculating the probabilities of the possible outcomes of any measurement that may be made at t2. If the time between the two measurements is infinitesimal (see below), U can be cast into the form 1 − (i/ℏ) H dt, where dt is the now infinitesimal interval t2 − t1, 1 is the identity operator (it returns unchanged whatever you insert), and H is some self-adjoint operator known as the “Hamilton operator” or, simply, the “Hamiltonian.” The minus sign is a convention, the factor i ensures that U does not change the norm of a vector (plug in a unit vector and out pops a unit vector), and the division by ℏ ensures that the Hamiltonian has the dimension of energy. Thus
v2 = v1 − (i/ℏ) H dt v1.
If dt is infinitesimal, then so is the difference v2 − v1, for which we shall write dv. Dropping the now superfluous index from v1, we can massage this equation to take the form
iℏ dv/dt = Hv.
Whenever we write an equation that contains infinitesimal (“infinitely small”) quantities, what we tacitly intend is to let these quantities go to zero, one (in this case dt) independently, the others dependently. In this limit, ratios of infinitesimal quantities like dv/dt become well-defined ordinary quantities.
If v depends not only on t but also on x, y, and z, it is customary use the symbol ψ(t,x,y,z), and it becomes necessary to replace the derivative dv/dt (which informs us about the rate of change of v) by the partial derivative ∂tψ (which informs us about the rate of change of ψ for any given set of values x,y,z). Thus
iℏ ∂tψ = Hψ.
Comparison with Eq. (2.3.7) shows that this the Schrödinger equation with
H = −(ℏ2/2m) [(∂x)2 + (∂y)2 + (∂z)2] + V.
When we followed the historical route from Planck to Born, the discovery that we were dealing with a probability calculus came last, after the discovery of the Schrödinger equation. When we started afresh, we were looking for a probability calculus capable of handling nontrivial probabilities and thus suitable for dealing with the objective fuzziness in nature, and we found (just now) that the Schrödinger equation is an integral part of this calculus — nothing less, but also nothing more.