Commuting and non-commuting observables

Calculating commutators

Problem:

Consider the operators whose action is defined by the equations below:
O1ψ(x) = x3ψ(x)
O2ψ(x) = x dψ(x)/dx
Find the commutator [O1, O2].

Solution:

Problem:

Let P and Q be two linear operators and let [P,Q] = -iħ.  Find
(a)  [Q,P],
(b)  [Q,Pn],
(c)  [P,Qn] .

Solution:

Problem:

Consider a one-dimensional system, with momentum operator p and position operator q.
(a) Show that [q,pn] = iħ n pn-1.
(b) Show that [q,F(p)] = iħ ∂F/∂p, if the function F(p) can be defined by a finite polynomial or convergent power series in the operator p.
(c) Show that [q,p2F(q)] = 2iħ pF(q) if F(q) is some function of q only.

Solution:

Problem:

The operators P and Q are represented by matrices in some basis.

image

(a)  Do P and Q commute?
(b)  Find the normalized eigenvectors of P and Q.

Solution:

Problem:

The angular momentum operators {Jx, Jy, Jz} are central to quantum theory.  States are classified according to the eigenvalues of these operators when J is conserved by the respective Hamiltonian H.
(a)  What condition(s) is (are) necessary for all eigenstates of H to be eigenstates of J
An eigenstate of J is usually specified by |j,mz>,
where J2|j,mz>  =  j(j + 1)ħ2|j,mz> and Jz|j,mz> = mzħ|j,mz>.
(b)  We can substitute Jx or Jy for Jz in (a).  However a state cannot be simultaneously an eigenstate of Jz and Jx.  Derive the commutation relation for the angular momentum operators Jx and Jz, (i.e.  [Jx,Jz] = -iħJy) from the definition of the linear momentum operator.
(c)  Prove that it is indeed possible for a state to be simultaneously an eigenstate of J2 = Jx2 + Jy2 +Jz2 and Jz

Solution:

Problem:

Show that if any operator commutes with two of the components of an angular momentum operator, it commutes with the third.

Solution:


Consequences

Problem:

Consider a system with Hamiltonian H.  Two observables A and B commute with H, [H,A] = [H,B] = 0, but do not commute with each other, [A,B] ≠ 0.  Show that the system has degenerate energy level.

Solution:

Problem:

Let A and B be two observables (Hermitian operators).  In any state of the system
ΔAΔB ≥ ½|<i[A,B]>|. 
(a)  Prove this generalized uncertainty principle.
[Hint: Let |ψ> be any state vector and let A1 = A - <A>I  and  B1 = B - <B>I. 
Let |Φ> = A1|ψ> + ixB1|ψ> with x an arbitrary real number.  Use <Φ|Φ>  ≥  0.]

Now consider a single particle in an eigenstate of L2 with wave function Ψ(r,t).
(b)  Calculate the commutators [sinφ, Lz] and [cosφ, Lz], where φ is the azimuthal angle.
(c)  Use these commutation relations and the result from part (a) to obtain uncertainty relations between sinφ, Lz and cosφ, Lz.
Note:  You can complete parts (b) and (c) without completing part (a).

Solution: