This Quantum World/Implications and applications/Observables and operators

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Observables and operatorsEdit

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