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Chapter 4: Problem 13

Calculate the values of the following matrix elements between p-orbitals: (a) \(\left\langle\mathrm{p}_{x}\left|l_{z}\right| \mathrm{p}_{y}\right\rangle,\) (b) \(\left\langle\mathrm{p}_{x}\left|l_{+}\right| \mathrm{p}_{y}\right\rangle\) \((\mathrm{c})\left\langle\mathrm{p}_{z}\left|l_{y}\right| \mathrm{p}_{x}\right\rangle\) \((d)\left\langle p_{z}\left|l_{x}\right| p_{y}\right\rangle,\) and (e) \(\left\langle\mathrm{p}_{z}\left|l_{x}\right| \mathrm{p}_{x}\right\rangle .\) Hint. Use the relations between \(\mathrm{p}_{x}, \mathrm{p}_{y}, \mathrm{p}_{z}\) and \(\mathrm{p}_{0}, \mathrm{p}_{+1}, \mathrm{p}_{-1}\).

Short Answer

Expert verified
The matrix elements are: (a) The element \( \langle p_x | l_z | p_y \rangle \) is equal to iħ. (b) The element \( \langle p_x | l_+ | p_y \rangle \) is zero. (c) The element \( \langle p_z | l_y | p_x \rangle \) is zero. (d) The element \( \langle p_z | l_x | p_y \rangle \) is zero. (e) The element \( \langle p_z | l_x | p_x \rangle \) is also zero.

Step by step solution

01

Conversion Between Representations

Convert the x, y, and z representations of the p-orbitals into 0, +1, and -1 representations respectively. This can be done using the relations:\(p_{0}=\frac{1}{\sqrt{2}}(p_{x}-i p_{y})\), \(p_{+1}=p_{z}\), \(p_{-1}=\frac{1}{\sqrt{2}}(p_{x}+i p_{y})\)
02

Substituting the Converted Representations

Substitute the relations from Step 1 into the matrix elements to convert them into a form which can be solved using the commutation relations.
03

Applying Commutation Relations

Use the commutation properties of the angular momentum operators to calculate the matrix elements. The commutation relations are \( [l_{x},l_{y}]=i \hbar l_{z} \), \( [l_{y},l_{z}]=i \hbar l_{x} \), and \( [l_{z},l_{x}]=i \hbar l_{y} \). Also, use the relation \( l_{\pm} = l_x \pm il_y \). Calculate the different elements (a-e) separatedly.
04

Simplify Results

Simplify the results. Any orbital elements in which the initial and final state are not the same (for example \( |\langle p_x | l_z | p_y \rangle |\)) will be zero, due to the orthogonality of the p-orbitals.

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Most popular questions from this chapter

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 \(\left[j^{2}, j_{1 z}\right]\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle=0\).

(a) Demonstrate that if \(\left[j_{1 q}, j_{2 q^{\prime}}\right]=0\) for all \(q, q^{\prime},\) then (b) Go on to show that if \(j_{1} \times j_{1}=i \hbar j_{1}\) and \(j_{1} \times j_{2}=-j_{2} \times j_{1}\) \(j_{2} \times j_{2}=i \hbar j_{2},\) then \(j \times j=\) ihj where \(j=j_{1}+j_{2}\)

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

Evaluate the commutator \(\left[l_{x}, l_{y}\right]\) in (a) the position representation, (b) the momentum representation.

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).
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