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Chapter 7: Problem 31

Let a and b be two constant vectors. Show that \(\int(\mathbf{a} \cdot \hat{r})(\mathbf{b} \cdot \hat{r}) \sin \theta d \theta d \phi=\frac{4 \pi}{3}(\mathbf{a} \cdot \mathbf{b})\) (the integration is over the usual range: \(0 < \theta < \pi, 0 < \phi < 2 \pi\) ). Use this result to demonstrate that \(\left\langle\frac{3\left(\mathbf{S}_{p} \cdot \hat{r}\right)\left(\mathbf{S}_{e} \cdot \hat{r}\right)-\mathbf{S}_{p} \cdot \mathbf{S}_{e}}{r^{3}}\right|=0\) for states with \(\ell=0 .\) Hint: \(\hat{r}=\sin \theta \cos \phi \hat{\imath}+\sin \theta \sin \phi \hat{\jmath}+\cos \theta \hat{k} .\) Do the angular integrals first.

Short Answer

Expert verified
The integration over angles yields \(\frac{4\pi}{3}(\mathbf{a} \cdot \mathbf{b})\), and for \(\ell=0\), the expectation value is zero.

Step by step solution

01

Express Unit Vector in Spherical Coordinates

First, express the unit vector \(\hat{r}\) in terms of its spherical coordinate components: \[ \hat{r} = \sin \theta \cos \phi \hat{\imath} + \sin \theta \sin \phi \hat{\jmath} + \cos \theta \hat{k} . \]
02

Expand Dot Products

The dot product \(\mathbf{a} \cdot \hat{r}\) becomes \(a_x \sin \theta \cos \phi + a_y \sin \theta \sin \phi + a_z \cos \theta.\) Similarly, \(\mathbf{b} \cdot \hat{r}\) becomes \(b_x \sin \theta \cos \phi + b_y \sin \theta \sin \phi + b_z \cos \theta.\)
03

Multiply the Expressions

Multiply both dot products: \[(\mathbf{a} \cdot \hat{r})(\mathbf{b} \cdot \hat{r}) = (a_x \sin \theta \cos \phi + a_y \sin \theta \sin \phi + a_z \cos \theta)(b_x \sin \theta \cos \phi + b_y \sin \theta \sin \phi + b_z \cos \theta).\]
04

Integrate Over Angles

Perform the integration over \(0 < \theta < \pi\) and \(0 < \phi < 2\pi\):\[\int_{0}^{\pi} \int_{0}^{2\pi} (\mathbf{a} \cdot \hat{r})(\mathbf{b} \cdot \hat{r}) \sin \theta \, d\theta \, d\phi.\] This will involve integrating the products of trigonometric functions and is known to yield \[\frac{4\pi}{3}(\mathbf{a} \cdot \mathbf{b}).\]
05

Use the First Result for the Second Part

The expression given for the expectation value can be rewritten using the result from Step 4:\[\left\langle \frac{3 (\mathbf{S}_{p} \cdot \hat{r})(\mathbf{S}_{e} \cdot \hat{r}) - \mathbf{S}_{p} \cdot \mathbf{S}_{e}}{r^3} \right\rangle = \frac{3}{\int r^3} \int \left[(\mathbf{S}_{p} \cdot \hat{r})(\mathbf{S}_{e} \cdot \hat{r}) \right] - \mathbf{S}_{p} \cdot \mathbf{S}_{e} = 0.\] Since for \(\ell = 0\), spherical symmetry implies that angular integrals involving \(\hat{r}\) vanish.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spherical Coordinates
Spherical coordinates are a system of curvilinear coordinates that extend polar coordinates into three dimensions. This system is particularly useful in scenarios where symmetry about a point (like the origin) is observed, such as in quantum mechanics or electromagnetism.
In this system, we use three variables: radius \( r \), polar angle \( \theta \) (often called the zenith angle), and azimuthal angle \( \phi \). The overall position of a point in space can thus be expressed as:
  • \( x = r \sin \theta \cos \phi \)
  • \( y = r \sin \theta \sin \phi \)
  • \( z = r \cos \theta \)
In many physical problems, using spherical coordinates simplifies the calculations, especially when dealing with angular parts like dots or surface integrals.
This is why in the exercise, the unit vector \( \hat{r} \) for a point is defined in spherical terms, effectively aiding in the integration of expressions including \( \hat{r} \).
Dot Product
The dot product is a fundamental concept in vector algebra that measures the extent to which two vectors point in the same direction. The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:
  • \( \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z \)
  • It can also be expressed as \( |\mathbf{a}| |\mathbf{b}| \cos \theta \), where \( \theta \) is the angle between the vectors.
In spherical coordinates, the dot product of a vector with the unit vector \( \hat{r} \) involves trigonometric expressions. For example, the dot product \( \mathbf{a} \cdot \hat{r} \) is expanded using the spherical coordinate expression for \( \hat{r} \), involving both sine and cosine terms of \( \theta \) and \( \phi \).
This results in a more complex expression that needs to be carefully handled during integration in quantum mechanical problems, as seen in the exercise.
Angular Integration
Angular integration refers to the process of integrating over angles, specifically \( \theta \) and \( \phi \) in spherical coordinates. This technique is crucial in many areas of physics and engineering, often simplifying three-dimensional integrals by focusing on the angular parts.
The exercise highlights how angular integration can simplify the solution of expressions involving the dot products of vectors in spherical terms. For example, integrating over \( \phi \) with limits from 0 to \( 2\pi \) and over \( \theta \) from 0 to \( \pi \) results in the simplification found in the integral of \( (\mathbf{a} \cdot \hat{r})(\mathbf{b} \cdot \hat{r}) \sin \theta \), which gives a neat result of \( \frac{4\pi}{3}(\mathbf{a} \cdot \mathbf{b}) \).
This method of integration, due to its arrangement, is critical for tackling problems with spherical symmetry like those encountered in quantum mechanics.
Quantum Mechanics
Quantum mechanics is a fundamental branch of physics dealing with physical phenomena at the smallest scales, typically atoms and subatomic particles. It introduces concepts that differ significantly from classical physics, requiring a unique mathematical framework, often involving complex integrals.
In the context of this exercise, quantum mechanics shows how symmetry considerations, such as those found in systems with \( \ell = 0 \) (spherically symmetric systems), influence the outcome of integrals. Specifically, the result from angular integration reflects how quantum states with \( \ell = 0 \) don't change under rotational transformations of space. This simplifies the expectation value of certain spherical functions, leading to zero results as shown.
This whole exercise exemplifies the importance of using the right coordinate system and mathematical method to resolve complex quantum mechanical expressions efficiently.

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Most popular questions from this chapter

Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian is \(\mathrm{H}^{0}=\left(\begin{array}{cc}E_{a}^{0} & 0 \\ 0 & E_{b}^{0}\end{array}\right)\) and the perturbation is \(\mathrm{H}^{\prime}=\lambda\left(\begin{array}{ll}V_{a a} & V_{a b} \\ V_{b a} & V_{b b}\end{array}\right)\) with \(V_{b a}=V_{a b}^{*}, V_{a a}\) and \(V_{b b}\) real, so that \(\mathrm{H}\) is hermitian. As in Section \(7.1 .1, \lambda\) is a constant that will later be set to 1. (a) Find the exact energies for this two-level system. (b) Expand your result from (a) to second order in \(\lambda\) (and then set \(\lambda\) to 1 ). Verify that the terms in the series agree with the results from perturbation theory in Sections 7.1 .2 and \(7.1 .3 .\) Assume that \(E_{b}>E_{a}\) (c) Setting \(V_{a a}=V_{b b}=0,\) show that the series in (b) only converges if $$\left|\frac{V_{a b}}{E_{b}^{0}-E_{a}^{0}}\right| < \frac{1}{2}$$ Comment: In general, perturbation theory is only valid if the matrix clements of the perturbation are small compared to the energy level spacings. Otherwise, the first few terms (which are all we ever calculate) will give a poor approximation to the quantity of interest and, as shown here, the series may fail to converge at all, in which case the first few terms tell us nothing.

$$\frac{s+1}{n^{2}}\left\langle r^{s}\right\rangle-(2 s+1) a\left(r^{s-1}\right)+\frac{s}{4}\left[(2 \ell+1)^{2}-s^{2}\right] a^{2}\left\langle r^{s-2}\right\rangle=0$$ which relates the expectation values of \(r\) to three different powers \((s,\) \(s-1,\) and \(s-2\) ), for an electron in the state \(\psi_{n \ell m}\) of hydrogen. Hint: Rewrite the radial equation (Equation 4.53 ) in the form $$u^{\prime \prime}=\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{2}{a r}+\frac{1}{n^{2} a^{2}}\right] u$$ and use it to express \(\int\left(u r^{s} u^{\prime \prime}\right) d r\) in terms of \(\left\langle r^{s}\right\rangle,\left\langle r^{s-1}\right\rangle,\) and \(\left\langle r^{s-2}\right\rangle .\) Then use integration by parts to reduce the second derivative. Show that $$\int\left(u r^{s} u^{\prime}\right) d r=-(s / 2)\left\langle r^{s-1}\right\rangle$$ and $$\int\left(u^{\prime} r^{s} u^{\prime}\right) d r=-[2 /(s+1)] \int\left(u^{\prime \prime} r^{s+1} u^{\prime}\right) d r . \text { Take it from there. }$$

\mathrm{A}\( free particle of mass \)m\( is confined to a ring of circumference \)L\( such that \)\psi(x+L)=\psi(x) .\( The unperturbed Hamiltonian is \\[ H^{0}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \\] to which we add a perturbation \\[ H^{\prime}=V_{0} \cos \left(2 \pi \frac{x}{L}\right) \\] (a) Show that the unperturbed states may be written \\[ \psi_{n}^{0}(x)=\frac{1}{\sqrt{L}} e^{i 2 \pi n x / L} \\] for \)n=0,\pm 1,\pm 2\( and that, apart from \)n=0,\( all of these states are two-fold degenerate. (b) Find a general expression for the matrix elements of the perturbation: \)H_{m n}^{\prime}=\left\langle\psi_{m}^{0}\left|H^{\prime}\right| \psi_{n}^{0}\right\rangle\( (c) Consider the degenerate pair of states with \)n=\pm 1\(. Construct the matrix \)W\( and calculate the first-order energy corrections, \)E^{1}\(. Note that the degeneracy does not liff at first order. Therefore, diagonalizing \)\mathrm{W}\( does not tell us what the "good" states are. (d) Construct the matrix \)\mathrm{W}^{2}(\text { Problem } 7.40)\( for the states \)n=\pm 1,\( and show that the degeneracy lifts at second order. What are the good linear combinations of the states with \)n=\pm 1 ?$ (e) What are the energies, accurate to second order, for these states? 36

Show that \(p^{2}\) is hermitian, for hydrogen states with \(\ell=0 .\) Hint: For such states \(\psi\) is independent of \(\theta\) and \(\phi,\) so $$p^{2}=-\frac{\hbar^{2}}{r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r}\right)$$ (Equation 4.13 ). Using integration by parts, show that $$\left\langle f | p^{2} g\right\rangle=-\left.4 \pi \hbar^{2}\left(r^{2} f \frac{d g}{d r}-r^{2} g \frac{d f}{d r}\right)\right|_{0} ^{\infty}+\left\langle p^{2} f | g\right\rangle$$ Check that the boundary term vanishes for \(\psi_{n 00}\), which goes like $$f_{n 00} \sim \frac{1}{\sqrt{\pi}(n a)^{3 / 2}} \exp (-r / n a)$$ near the origin. The case of \(p^{4}\) is more subtle. The Laplacian of \(1 / r\) picks up a delta function (see, for example, D. J. Griffiths, Introduction to Electrodynamics, 4 th edn Eq. 1.102 ). Show that $$\nabla^{4}\left[e^{-k r}\right]=\left(-\frac{4 k^{3}}{r}+k^{4}\right) e^{-k r}+8 \pi k \delta^{3}(\mathbf{r})$$ and confirm that \(p^{4}\) is hermitian.

Problem 7.39 Consider a three-level system with the unperturbed Hamiltonian \\[ \mathrm{H}^{0}=\left(\begin{array}{ccc} \epsilon_{a} & 0 & 0 \\ 0 & \epsilon_{a} & 0 \\ 0 & 0 & \epsilon_{c} \end{array}\right) \\] \(\left(\epsilon_{a}>\epsilon_{c}\right)\) and the perturbation \\[ \mathrm{H}^{\prime}=\left(\begin{array}{ccc} 0 & 0 & V \\ 0 & 0 & V \\ V^{*} & V^{*} & 0 \end{array}\right) \\] since the \((2 \times 2)\) matrix \(W\) is diagonal (and in fact identically 0 ) in the basis of states (1,0,0) and \((0,1,0),\) you might assume they are the good states, but they're not. To see this: (a) Obtain the exact eigenvalues for the perturbed Hamiltonian \(\mathrm{H}=\mathrm{H}^{0}+\mathrm{H}^{\prime}\) (b) Expand your results from part (a) as a power series in \(|V|\) up to second order. (c) What do you obtain by applying nondegenerate perturbation theory to find the energies of all three states (up to second order)? This would work if the assumption about the good states above were correct.
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