(a) Work out all of the canonical commutation relations for components of the operators r and p: [x,y], [x,p_y], [x,p_x],[p_y,p_z], and so on.
where the indices stand for x, y, or z, and r_x, = x, r_y, = y, and r_z = z.

(b) Confirm the three-dimensional version of Ehrenfest’s theorem,

(Each of these, of course, stands for three equations—one for each component.) First check that the “generalized” Ehrenfest theorem, Equation 3.73, is valid in three dimensions.
(c) Formulate Heisenberg’s uncertainty principle in three dimensions.

but there is no restriction on, say, σ_xσ_py.

Equation 3.73

Transcribed Image Text:

[ri, Pj] = [P₁,rj] = iħdij, [¹₁,rj] = [P₁, Pj] = 0, - (4.10)