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Chapter 4: Problem 10

Suppose that in place of the actual angular momentum commutation rules, the operators obeyed \(\left[l_{x}, l_{y}\right]=-\mathrm{i} \hbar l_{z^{*}}\) What would be the roles of \(l_{2}=l_{x} \pm l_{y} ?\)

Short Answer

Expert verified
The new roles of \(l_2 = l_x \pm l_y\) will be uncovered by looking at the new commutation relation, which could hint at how the new system would behave or could suggest a new physical interpretation for the system with the modified commutation relations.

Step by step solution

01

Express \(l_2\) in Terms of \(l_x\) and \(l_y\)

Define two operators \(l_{2+}=l_x+l_y\) and \(l_{2-}=l_x-l_y\). Each of these operators can be expressed in terms of \(l_x\) and \(l_y\) and their corresponding commutation relations can be derived from the given commutation relation.
02

Calculate the Commutators

Now, compute the new commutators between \(l_{2+}\) or \(l_{2-}\) and \(l_{z^*}\). This will involve the calculation of \([l_{2+}, l_{z^*}]\) and \([l_{2-}, l_{z^*}]\), using the relation \([A,B+C]=[A,B]+[A,C]\).
03

Evaluation of the New Roles

From the evaluated commutators it can be seen how \(l_{2+}\) and \(l_{2-}\) are affected by this new commutation relationship. The kind of commutation relations arising will give a hint to the roles of \(\l_{2+}\) and \(l_{2-}\) under such a system.

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Most popular questions from this chapter

What are the possible outcomes of a single measurement of the \(z\) -component of spin angular momentum of an electron in the spin state \((1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta ?\)

(a) Confirm that the Pauli matrices \\[ \sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \sigma_{y}=\left(\begin{array}{rr} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array}\right) \quad \boldsymbol{\sigma}_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \\] satisfy the angular momentum commutation relations when we write \(s_{q}=\frac{1}{2} \hbar \sigma_{q},\) and hence provide a matrix representation of angular momentum. (b) Why does the representation correspond to \(s=1 / 2\) ? Hint. For the second part, form the matrix representing \(s^{2}\) and establish its eigenvalues.

Using the Pauli matrix representation, reduce each of the operators (a) \(s_{x} s_{y}\) (b) \(s_{x} s_{y}^{2} s_{z}^{2},\) and (c) \(s_{x}^{2} s_{y}^{2} s_{z}^{2},\) to a single spin operator.

Determine what total angular momenta may arise in the following composite systems: (a) \(j_{1}=3, j_{2}=4 ;\) (b) the orbital momenta of two electrons (i) both in p-orbitals, (ii) both in d-orbitals, (iii) the configuration \(\mathrm{p}^{1} \mathrm{d}^{1} ;\) (c) the spin angular momenta of four electrons. Hint. Use the ClebschGordan series, eqn \(4.42 ;\) apply it successively in (c).

Calculate the matrix elements (a) \(\left\langle 0,0\left|l_{z}\right| 0,0\right\rangle\) (b) \(\left\langle 2,1\left|l_{+}\right| 2,0\right\rangle\) (c) \(\left\langle 2,2\left|l_{+}^{2}\right| 2,0\right\rangle,\) (d) \(\left\langle 2,0\left|l_{+} l_{-}\right| 2,0\right\rangle\) (e) \(\left\langle 2,0\left|l \downarrow_{+}\right| 2,0\right\rangle,\) and (f) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)
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