/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Consider a spinless particle bou... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 31

Consider a spinless particle bound to a fixed center by a central force potential. (a) Relate, as much as possible, the matrix elements \\[ \left\langle n^{\prime}, l^{\prime}, m^{\prime}\left|\mp \frac{1}{\sqrt{2}}(x \pm i y)\right| n, l, m\right\rangle \quad \text { and } \quad\left\langle n^{\prime}, l^{\prime}, m^{\prime}|z| n, l, m\right\rangle \\] using only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing. (b) Do the same problem using wave functions \(\psi(\mathbf{x})=R_{n l}(r) Y_{l}^{m}(\theta, \phi)\).

Short Answer

Expert verified
Use the Wigner-Eckart theorem; matrix elements are non-zero if angular momentum rules are met.

Step by step solution

01

Understanding the Problem

The task is to relate specific matrix elements involving the position operators in spherical coordinates, using the Wigner-Eckart theorem. We also need to find the conditions under which these matrix elements are non-zero and apply these principles using wave functions given in spherical coordinates.
02

Applying the Wigner-Eckart Theorem

The Wigner-Eckart theorem helps in expressing matrix elements involving vector operators in terms of reduced matrix elements and Clebsch-Gordan coefficients. It simplifies the calculation by breaking down the angular dependence. The theorem implies that \(\langle n', l', m' | T^{(k)}_q | n, l, m \rangle \) is zero unless \(|l' - l| \leq k \leq l' + l\) and \(m' = m + q\).
03

Relate Matrix Elements (a) with the Wigner-Eckart Theorem

To relate \[\langle n', l', m'| \mp \frac{1}{\sqrt{2}}(x \pm iy) | n, l, m \rangle\] and \[\langle n', l', m'| z | n, l, m \rangle,\] we use the correspondence of these operators to spherical tensors of rank 1. Specifically, the combinations \(x \pm i y\) relate to the spherical harmonics and thus can be expressed as raising and lowering operators in the context of angular momentum.
04

Non-vanishing Condition for Matrix Elements

For the matrix elements to be non-zero, the conditions under the Wigner-Eckart theorem must be satisfied: - For \(z\), since it corresponds directly to the \(m = 0\) component of a vector operator, \(m' = m\) and the change in \(l\) can be 0 or 1. - For \(x \pm i y\), these relate to \(m = \pm 1\), so \(m' = m \pm 1\) and similarly, the change in \(l\) can be 0 or 1.
05

Transition to Wave Functions (b)

Switching from the matrix elements to wave functions like \(\psi(\mathbf{x}) = R_{nl}(r) Y_l^m(\theta, \phi)\), we express \(x, y, z\) in spherical coordinates. The wave functions then allow us to directly compute these integrals, using \(Y_l^m\) properties to apply the same conditions and relate the results accordingly.
06

Computing Using Wave Functions

Express \(x, y, z\) in terms of spherical harmonics: - \(x \pm i y = r \sqrt{\frac{4\pi}{3}} Y_1^{\pm 1}(\theta, \phi)\)- \(z = r \sqrt{\frac{4\pi}{3}} Y_1^0(\theta, \phi)\)Integrate these expressions with the given wave functions, ensuring the angular parts are non-zero only when the selection rules from the Wigner-Eckart theorem are satisfied.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Central Force Potential
A central force potential is a type of force field where the force on a particle always points towards or away from a fixed center and its magnitude only depends on the distance from this point. This concept is crucial when dealing with problems in physics where a particle is influenced by a central body like a planet, atom, or nucleus. Central force potentials often take the form of \( V(r) \), where \( r \) is the radial distance from the central point. This symmetry makes these problems ideal for using spherical coordinates, simplifying the mathematical treatment significantly. One of the typical examples of a central force is the gravitational force, which decreases with the square of the distance from the center, represented as \( V(r) = -\frac{Gm_1m_2}{r} \). Here, it's important to emphasize that the force's direction is always along the line connecting the particle and the center, leading to the conservation of angular momentum, which is a key parameter in solving such problems.
Spherical Coordinates
Spherical coordinates are a system used to define the position of a point in space with three numbers: the radial distance \( r \), the polar angle \( \theta \) (measured from the positive z-axis), and the azimuthal angle \( \phi \) (measured in the x-y plane from the positive x-axis). This system is especially convenient for problems with spherical symmetry, such as those involving central force potentials. In spherical coordinates, the position coordinates \( x, y, z \) can be expressed as:
  • \( x = r \sin\theta \cos\phi \)
  • \( y = r \sin\theta \sin\phi \)
  • \( z = r \cos\theta \)
This framework allows physicists to separate variables in differential equations governing the system, particularly when dealing with quantum mechanics and electromagnetism. Understanding spherical coordinates is crucial when calculating matrix elements in quantum systems, as we can convert spatial dependencies into forms that are easier to handle mathematically.
Matrix Elements
Matrix elements in quantum mechanics represent the transition probabilities or amplitudes between different states of a system. Specifically, they quantify how likely it is for a system described by a quantum state \(| n, l, m \rangle\) to transition to a state \(| n', l', m' \rangle\) due to an operator \( \hat{O} \). In our context, this operator relates to the positions \(x, y, \) or \(z\) under spherical coordinates. For example, the matrix element \(\langle n', l', m' | \hat{O} | n, l, m \rangle\) can involve complex calculations but gives essential insights into the probability of different quantum states resulting from specific interactions. Using the Wigner-Eckart theorem simplifies these calculations by reducing the problem to coefficients like Clebsch-Gordan coefficients, significantly easing the computational effort needed. This theorem relates the transformations of matrix elements involving spherical tensors with the inherently coupled angular momentum states.
Angular Momentum
Angular momentum in quantum mechanics plays a central role, much like in classical mechanics, representing the rotational invariance of a system. It is denoted by operators \(L_x, L_y, L_z\), collectively forming the vector \(\mathbf{L}\). In problems involving spherical symmetry, such as those with central force potentials, angular momentum simplifies the analysis. Quantum angular momentum is associated with quantum numbers \( l \) and \( m \), where \( l \) refers to the orbital angular momentum magnitude and \( m \) refers to its projection along a chosen axis (usually the z-axis).
The spherical harmonics \(Y_l^m(\theta, \phi)\) arise naturally when solving the angular part of the Schr枚dinger equation in spherical coordinates, characterizing angular momentum states.
  • \( l = 0, 1, 2, \ldots \)
  • \( m = -l, -(l-1), \ldots, l \)
Within quantum mechanics, the conservation of angular momentum provides powerful symmetry that we utilize, especially when using the Wigner-Eckart theorem, to relate matrix elements of physical observables. This result is essential for understanding the behavior of quantum systems in fields with rotational symmetry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Let \(\mathbf{J}\) be angular momentum. (It may stand for orbital \(\mathbf{L}, \operatorname{spin} \mathbf{S},\) or \(\mathbf{J}_{\text {total. }}\) ) Using the fact that \(J_{x}, J_{y}, J_{z}\left(J_{\pm} \equiv J_{x} \pm i J_{y}\right)\) satisfy the usual angular-momentum commutation relations, prove \\[ \mathbf{J}^{2}=J_{z}^{2}+J_{+} J_{-}-\hbar J_{z} \\] (b) Using (a) (or otherwise), derive the "famous" expression for the coefficient \(c_{-}\) that appears in \\[ \boldsymbol{J}_{-} \psi_{j m}=c_{-} \psi_{j, m-1} \\].

Consider the \(2 \times 2\) matrix defined by \\[ U=\frac{a_{0}+i \sigma \cdot \mathbf{a}}{a_{0}-i \sigma \cdot \mathbf{a}}, \\] where \(a_{0}\) is a real number and a is a three-dimensional vector with real components. (a) Prove that \(U\) is unitary and unimodular. (b) In general, a \(2 \times 2\) unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for \(U\) in terms of \(a_{0}, a_{1}, a_{2},\) and \(a_{3}\).

(a) Evaluate \\[ \sum_{m=-j}^{j}\left|d_{m m^{\prime}}^{(j)}(\beta)\right|^{2} m \\] for any \(j\) (integer or half-integer); then check your answer for \(j=\frac{1}{2}\). (b) Prove, for any \(j\) \\[ \sum_{m=-j}^{j} m^{2}\left|d_{m^{\prime} m}(\beta)\right|^{2}=\frac{1}{2} j(j+1) \sin ^{2} \beta+m^{2} \frac{1}{2}\left(3 \cos ^{2} \beta-1\right) \\] [Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of \(J_{z}^{2}\) using the spherical (irreducible) tensor language.]

A particle in a spherically symmetrical potential is known to be in an eigenstate of \(\mathbf{L}^{2}\) and \(L_{z}\) with eigenvalues \(\hbar^{2} l(l+1)\) and \(m \hbar,\) respectively. Prove that the expectation values between \(|l m\rangle\) states satisfy \\[ \left\langle L_{x}\right\rangle=\left\langle L_{y}\right\rangle=0, \quad\left\langle L_{x}^{2}\right\rangle=\left\langle L_{y}^{2}\right\rangle=\frac{\left[l(l+1) \hbar^{2}-m^{2} \hbar^{2}\right]}{2}, \\] Interpret this result semiclassically.

The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the \(z\) -direction can be written as \\[ H=A \mathbf{S}^{\left(e^{-}\right)} \cdot \mathbf{S}^{\left(e^{+}\right)}+\left(\frac{e B}{m c}\right)\left(S_{z}^{\left(e^{-}\right)}-S_{z}^{\left(e^{+}\right)}\right) \\]. Suppose the spin function of the system is given by \(\chi_{+}^{\left(e^{-}\right)} \chi_{-}^{\left(e^{+}\right)}\) (a) Is this an eigenfunction of \(H\) in the limit \(A \rightarrow 0, e B / m c \neq 0 ?\) If it is, what is the energy eigenvalue? If it is not, what is the expectation value of \(H ?\) (b) Solve the same problem when \(e B / m c \rightarrow 0, A \neq 0\).
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Geometrical and Physical Optics

Read Explanation

Force

Read Explanation

Oscillations

Read Explanation

Circular Motion and Gravitation

Read Explanation

Fields in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.