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Chapter 1: Problem 5

Evaluate the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) given that \(\left[l_{x}, l_{y}\right]=i \hbar l,\left[l_{y}, l_{z}\right]=i \hbar l_{x},\) and \(\left[l_{z}, l_{x}\right]=i \hbar l_{y}\)

Short Answer

Expert verified
Evaluating the commutator \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) gives us \(-\hbar^{2}l\).

Step by step solution

01

Distribute the commutator

To begin, we can distribute \(\left[l_{y}\left[l_{y}, l_{z}\right]\right]\) by following the definition of commutator, that is, \(l_{y}\left[l_{y}, l_{z}\right]-\left[l_{y}, l_{z}\right]l_{y} \)
02

Replace the commutators with the given relations

Now, replace the commutators \([l_{y}, l_{z}]\) in these expressions using the given relation \(i \hbar l_{x}\). We get \(l_{y}(i \hbar l_{x})-(i \hbar l_{x})l_{y}\).
03

Distribute \(i \hbar\)

We distribute \(i \hbar\), which can be brought out of commutators without altering the result. We then get \(i \hbar[l_{y}, l_{x}]\).
04

Replace the newly obtained commutator with the corresponding given relation

Replace \([l_{y}, l_{x}]\) using the relation \(i \hbar l\), to get \(i \hbar(i \hbar l) = -\hbar^{2}l\).

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Most popular questions from this chapter

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)

The operator \(e^{A}\) has a meaning if it is expanded as a power scrics: \(\mathrm{e}^{A}=\Sigma_{n}(1 / n !) A^{n} .\) Show that if \(|a\rangle\) is an eigenstate of \(A\) with eigenvalue \(a,\) then it is also an eigenstate of \(\mathrm{e}^{A} .\) Find the latter's eigenvalue.

Construct quantum mechanical operators in the position representation for the following observables: (a) kinetic energy in one and in three dimensions, (b) the inverse separation, \(1 / x,\) (c) electric dipole moment \(\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.\) is the position of a charge \(Q_{i}\) ), (d) \(z\) -component of angular momentum \(\left(x p_{y}-y p_{x}\right),\) (e) the mean square deviations of the position and momentum of a particle from the mean values.

(a) Show that \(\mathrm{e}^{A} \mathrm{e}^{B}=\mathrm{e}^{A+B}\) only if \([A, B]=0 .\) (b) If \([A, B] \neq 0\) but \([A,[A, B]]=[B,[A, B]]=0,\) show that \(\mathrm{e}^{A} \mathrm{e}^{B}=\) \(\mathrm{e}^{A+B} \mathrm{e}^{f},\) where \(f\) is a simple function of \([A, B] .\) Hint. This is another example of the differences between operators \((q-\text { numbers })\) and ordinary numbers (c-numbers). The simplest approach is to expand the exponentials and to collect and compare terms on both sides of the equality. Note that \(\mathrm{e}^{A} \mathrm{e}^{B}\) will give terms like \(2 A B\) while \(\mathrm{e}^{A+B}\) will give \(A B+B A .\) Be careful with order.

Evaluate the commutators (a) \(\left[H, p_{x}\right]\) and (b) \([H, x]\) where \(H=p_{x}^{2} / 2 m+V(x) .\) Choose (i) \(V(x)=V,\) a constant, (ii) \(V(x)=\frac{1}{2} k_{t} x^{2},\) (iii) \(V(x) \rightarrow V(r)=e^{2} / 4 \pi \varepsilon_{0} r .\) Hint. For part (b), case (iii), use \(\left(\partial r^{-1} / \partial x\right)=-x / r^{3}\)
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