Quantum Mechanics 4

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The Uncertainty Principle Constants of the Motion Uncertainty in Energy and Time	Course Home Page Quantum Mechanics 1 Quantum Mechanics 2 Quantum Mechanics 3

The Uncertainty Principle

The wave-particle duality that quantum mechanics entails has been given perhaps its most famous expression in Heisenberg's Uncertainty Principle.

One can discuss this on several levels. Heisenberg proposed some "gedankenexperimente" (thought experiments) to justify his conclusion that simultaneous measurements of certain quantities (notably position and momentum) necessarily involve a level of imprecision.

One of these, the so-called "g-ray microscope", is described in some detail in the text.

But the root of the Uncertainty Principle does not lie in how measurements are made. It is a basic property of nature, as described by quantum mechanics. We will sketch a proof of it, using the rules of quantum mechanics as we have discussed them.

The wavefunction describing a state contains implicity a set of probability distributions, telling the probabilities that a measurement of this or that observable will yield this one or that one of its allowed values.

Call the observable Q. As we have seen, to this observable corresponds an operator, denoted by O_q, whose eigenvalues q_n are the allowed values of the observable.

Now suppose we make a large set of measurements of Q on identical systems, all of which are in the same state described by Y at the time the measurements are made. The average of these measurements is predicted to be

The "spread" of the distribution is given by the variance:

The quantity DQ defines the "uncertainty" in the value of Q. It is the standard deviation of the statistical distribution.

To begin the argument, we quote without proof two facts:

For the kinds of operators that represent observables,

The Schwarz inequality. For any two functions

Now consider two observables, represented by the operators P and Q. We define

Then

This, by Schwarz's inequality, is at least as large as

Now for any complex number

where Imz = (z - z*)/2i is the imaginary part of z.

Thus we have

Writing out all the pieces, we find that

The combination PQ - QP is called the "commutator" of the operators, denoted by [P,Q]. For operators representing observables, this combination results in i times another operator (which might be a constant or zero).

The final general result is this:

The product of uncertainties of two observables is never less that 1/2 the magnitude of the average value of the commutator of the corresponding operators.

Now let us look at the case of position and momentum. The commutator obeys

Thus

In the general rule above this leads to

This is Heisenberg's Uncertaintly principle.

What does it really say? Recall that DQ is the spread in results of measurements of the observable Q, when the system is in the state described by Y(x,t). That means that DxDp is the product of such spreads for those two observables, when each set of measurements is made while the system is in the same state. The measurements must be made simultaneously. so that the state does not change.

The Uncertainty Principle says that the smaller the spread in one of the observables the greater the spread in the other. If by measurement the particle's position is more closely determined, then correspondingly its momentum becomes less definite, and vice versa.

According to the general rule, if the operators of the two observables commute (the commutator is zero) then there is no restriction on the spreads. Both observables can (in principle) be known simultaneously with complete definiteness.

Constants of the Motion

Possibly the most important observable is the energy, represented by the Hamiltonian operator, H. The stationary states are those in which the energy has a definite value, and in a stationary state all probability distributions are constant in time.

Now consider a second observable Q whose operator commutes with H. Then it is possible to have states in which both the energy and Q have definite values.

There is another important property of operators that commute with H. Consider the average value of Q in some arbitrary state. It is given by

Now consider the time derivative of this:

Here we have assumed that the operator O_q does not depnd explicity on time.

Now the S-E for the wavefunction is

and its complex conjugate

Thus we have

This shows that if O_q commutes with H the average of Q is independent of time. One can show similarly that the whole probability distribution for measurements of Q is independent of time.

Such an observable is called a constant of the motion. It is the analog of classical quantities that obey conservation laws.

A trivial example is the case where the external force on the particle is zero. The potential energy function is then a constant, and the momentum operator commutes with H. Consequently the momentum in a constant of the motion.

Less trivial examples occur in three-dimensional situations.

Uncertainty in Energy and Time

Consider an observable that is not a constant of the motion, so that its operator does not commute with H. Then from the above argument we see that

Now according to our general rule from which the Uncertainty Principle followed, the magnitude of the left side above is related to the product of the uncertainties:

Thus we have

Now define a time Dt by

This is the time it takes the average value of Q to change by the amount DQ, i.e., by one standard deviation. It thus represents a characteristic time in the change of the system as represented by the observable Q.

The result is

This is often called the "energy-time uncertainty principle". That is a misnomer. The symbol Dt has nothing to do with the accuracy of time measurements, or of uncertainty over what time it is. In non-relativistic quantum mechanics time is a parameter, assumed to be known and measured with infinite accuracy.

Instead, Dt represents a characteristic evolution time for the system. The relation states that systems that evolve quickly have a correspondingly large uncertainty in their energy, while systems that are nearly stable (large Dt) have nearly definite energy.

As an example, consider an atom in an excited state. It will decay by emitting a photon, so it is not completely stable. (The state is not really a stationary state.) Its average lifetime before decay is t; this plays the role of Dt in the above formulas. The energy of the state is thus uncertain, and the energy of the emitted photon shows a corresponding uncertainty. This shows up as a spread in the distribution of observed photon energies, called the "natural line width".

For atoms the lifetime is relatively long (of order 10^-8 s) so the energy spread is of order 10^-7 eV, which is small enough to be hard to observe. But for highly unstable systems the energy spread can be quite large. The rho particle, for example, has an energy uncertainty (as shown in measurements of its mass) of order 100 MeV. This corresponds to a lifetime of order 10^-23s.