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Chapter 4: Problem 20

In some cases \(m_{11}\) and \(m_{12}\) may be specified at the same time as \(j\) because although \(\left[f^{2}, j_{1 z}\right]\) is non-zero, the effect of \(\left[j^{2}, j_{12}\right]\) on the state with \(m_{i 1}=j_{1}, m_{i 2}=j_{2}\) is zero. Confirm that \(\left[j^{2}, j_{1 z}\right]\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle=0\) and \(\left[j^{2}, j_{1 z}\right]\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle=0\).

Short Answer

Expert verified
The effect of the commutation of total angular momentum \(j^{2}\) and the \(z\)-component of angular momentum \(j_{1 z}\) on the states \(\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle\) and \(\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle\) is confirmed to be zero.

Step by step solution

01

Understand the components

Begin by understanding the components involved in this exercise. \(j^{2}\) refers to the total angular momentum and \(j_{1 z}\) is the z-component of the angular momentum. Our task is to confirm that the effect of the commutation of these operators on the given states is zero.
02

Solve for the first state

Plug the first state \(\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle\) into the commutation relationship. This gives us \[\left[j^{2}, j_{1 z}\right]\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle\]. By the properties of the commutation relationship, this simplifies to \(j^{2} j_{1 z} \left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle - j_{1 z} j^{2} \left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle\). Given that the action of the \(j_{1 z}\) operator on the state is to multiply it by the eigenvalue, which corresponds to \(j_{1}\) for this particular state, and that the total angular momentum \(j^{2}\) commutes with its z-component \(j_{1 z}\), the result becomes zero.
03

Solve for the second state

Repeat this process for the second state \(\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle\). Here the action of \(j_{1 z}\) is to multiply the state by its eigenvalue, -\(j_{1}\). Despite this change, the same logic applies, and this will also result in zero.

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Most popular questions from this chapter

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 \(l_{2}=l_{x} \pm l_{y} ?\)

Consider a system of two electrons that can have either paired or unpaired spins (e.g. a biradical). The energy of the system depends on the relative orientation of their spins. Show that the operator \(\left(h J / \hbar^{2}\right) s_{1} \cdot s_{2}\) distinguishes between singlet and triplet states. The system is now exposed to a magnetic field in the \(z\) -direction. Because the two electrons are in different environments, they experience different local fields and their interaction energy can be written \(\left(\mu_{\mathrm{B}} / \hbar\right) \times\) \(B\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} ; \mu_{5}\) is the Bohr magneton and \(g\) is the electron \(g\) -value, quantities discussed in Chapter 13 Establish the matrix of the total hamiltonian, and demonstrate that when \(h J \gg \mu_{\mathrm{B}} \mathscr{B},\) the coupled representation is "bctter', but that when \(\mu_{B} D\) : wh \(J\), the uncoupled representation is 'better', Find the eigenvalues of the system in each case. Hint. Use the vector coupling coefficients in Resource section 2 to determine hamiltonian matrix elements.

(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.

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

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.
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