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Chapter 4: Problem 1

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

Short Answer

Expert verified
The commutator \(\left[l_{x}, l_{y}\right]\) equals to \(-i\hbar l_{z}\) both in position and momentum representations.

Step by step solution

01

Establishing formulas for \(l_{x}\) and \(l_{y}\) operators

Firstly, understand these operators represent the angular momentum in Quantum Mechanics. Their formulas are \(l_{x} = y p_{z} - z p_{y}\) and \(l_{y} = z p_{x} - x p_{z}\).
02

Calculate the commutator in position representation

The commutator of two operators A and B is given by [A, B] = AB - BA. Use the formulas from step 1 to evaluate the commutator. Thus, \( \left[ l_{x}, l_{y} \right] = l_{x} l_{y} - l_{y} l_{x} = (y p_{z} - z p_{y})(z p_{x} - x p_{z}) - (z p_{x} - x p_{z})(y p_{z} - z p_{y}) = -i\hbar l_{z} \). The result is found using the canonical commutation relation between position and momentum, [\(x_{i}, p_{j}\)] = \(i\hbar\) \(\delta_{ij}\).
03

Calculate the commutator in momentum representation

The operators change to position component in the momentum representation by the Fourier transform. The new formulas for \(l_{x}\) and \(l_{y}\) are \(l_{x} = -i\hbar (\theta \partial_{\phi} - \phi cos\theta \partial_{\theta})\) and \(l_{y} = i\hbar \phi sin\theta \partial_{\theta}\). The calculation above is repeated with these formulas. We find that \( \left[ l_{x}, l_{y} \right] = -i\hbar l_{z} \) in momentum space as well, since angular momentum is unchanged by the Fourier transform.

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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 \((1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta ?\)

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) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)

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 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 \(\mathrm{p}_{0}, \mathrm{p}_{+1}, \mathrm{p}_{-1}\).

Construct the vector coupling coefficients for a system with \(j_{1}=1\) and \(j_{2}=1 / 2\) and evaluate the matrix elements \(\left\langle j^{\prime \prime} m_{i}^{\prime}\left|j_{1 z}\right| j m_{i}\right\rangle .\) Hint. Proceed as in Section 4.12 and check the answer against the values in Resource section \(2 .\) For the matrix element, express the coupled states in the uncoupled representation, and then operate with \(j_{1 x}\)
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