Franz Schwabl Quantum Mechanics 4th Edit · Problem 5.1

The Central Potential I

Physics — Quantum Mechanics · angular-momentum-expectation-values

Problem

(a) Show that for the eigenstates of

L_z

the expectation values of

L_+

L_-

L_x

, and

L_y

vanish. (b) Determine the average square deviation

(\Delta L_i)^2

for the components of the angular momentum operator in the states

Y_{ll}

and check the uncertainty relation. (c) Show that for the eigenstates of

\boldsymbol{L}^2

and

L_z

the expression

(\Delta L_x)^2 + (\Delta L_y)^2

is minimal if

m = \pm l