This Quantum World/Implications and applications/Observables and operators

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Observables and operators[edit]

Remember the mean values

As noted already, if we define the operators

("multiply with ") and

then we can write

By the same token,

Which observable is associated with the differential operator ? If and are constant (as the partial derivative with respect to requires), then is constant, and

Given that and this works out at or

Since, classically, orbital angular momentum is given by so that it seems obvious that we should consider as the operator associated with the  component of the atom's angular momentum.

Yet we need to be wary of basing quantum-mechanical definitions on classical ones. Here are the quantum-mechanical definitions:

Consider the wave function of a closed system with  degrees of freedom. Suppose that the probability distribution (which is short for ) is invariant under translations in time: waiting for any amount of time makes no difference to it:

Then the time dependence of is confined to a phase factor

Further suppose that the time coordinate  and the space coordinates  are homogeneous — equal intervals are physically equivalent. Since is closed, the phase factor cannot then depend on  and its phase can at most linearly depend on  waiting for should have the same effect as twice waiting for  In other words, multiplying the wave function by should have same effect as multiplying it twice by :

Thus

So the existence of a constant ("conserved") quantity or (in conventional units)  is implied for a closed system, and this is what we mean by the energy of the system.

Now suppose that is invariant under translations in the direction of one of the spatial coordinates  say :

Then the dependence of on is confined to a phase factor

And suppose again that the time coordinates  and  are homogeneous. Since is closed, the phase factor cannot then depend on  or  and its phase can at most linearly depend on : translating by should have the same effect as twice translating it by  In other words, multiplying the wave function by should have same effect as multiplying it twice by :

Thus

So the existence of a constant ("conserved") quantity or (in conventional units)  is implied for a closed system, and this is what we mean by the j-component of the system's momentum.

You get the picture. Moreover, the spatial coordiates might as well be the spherical coordinates If is invariant under rotations about the  axis, and if the longitudinal coordinate is homogeneous, then

In this case we call the conserved quantity the  component of the system's angular momentum.




Now suppose that is an observable, that is the corresponding operator, and that satisfies

We say that is an eigenfunction or eigenstate of the operator  and that it has the eigenvalue  Let's calculate the mean and the standard deviation of  for We obviously have that

Hence

since For a system associated with  is dispersion-free. Hence the probability of finding that the value of  lies in an interval containing  is 1. But we have that

So, indeed, is the operator associated with the  component of the atom's angular momentum.

Observe that the eigenfunctions of any of these operators are associated with systems for which the corresponding observable is "sharp": the standard deviation measuring its fuzziness vanishes.

For obvious reasons we also have

If we define the commutator then saying that the operators and commute is the same as saying that their commutator vanishes. Later we will prove that two observables are compatible (can be simultaneously measured) if and only if their operators commute.


Exercise: Show that


One similarly finds that and The upshot: different components of a system's angular momentum are incompatible.


Exercise: Using the above commutators, show that the operator commutes with and