Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.5.6.
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This is an example of how risky it can be to attempt to derive quantum behaviour by using logic based on classical mechanics. In classical mechanics, if we have a system with some total angular momentum with magnitude and rotate this sytem through any angle, the magnitude of the angular momentum remains the same (although, of course, its direction changes). Based on this fact, we might think that if we start with a quantum state such as (where is the total angular momentum number and is the number for ), we should be able to obtain the other states with the same total angular momentum number by rotating this state through various angles about the appropriate rotation axis. To see that this won’t work, suppose we consider a state with and , that is . Classically, such a system has its angular momentum aligned along the axis, so we might think that if we rotate this system by about, say, the axis, we should get a state with , since the angular momentum is now aligned along the axis.
To see if this works, we can use the formula for a finite rotation for a state. Since remains constant, a rotation of a state is given by
For a rotation by an angle about the axis, this formula reduces to
We can copy the matrix from Shankar’s equation 12.5.23:
We therefore have
Plugging these into 3 we have
In the basis, the state is represented by
Thus a rotation about the axis rotates this state into:
If rotation by some angle could change into the state , this result would need to be a multiple of , so we’d need to find such that both of the following equations are true:
This is impossible, so no rotation about the axis can change into the state .
However, there is still a correspondence between classical and quantum mechanics if we compare the average values of the components of . That is, we want to find for the state 10. To do this, we need the other two matrix components and . We can get from Shankar’s equation 12.5.24:
is just the diagonal matrix:
We can now calculate the averages for the state 10:
Thus for the average, we have
In this case, a rotation by does indeed rotate so that it points along the axis, just as it would in classical mechanics.