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Chapter 7: Problem 26

A box containing two spin-one gyros \(\mathrm{A}\) and \(\mathrm{B}\) is found to have angular-momentum quantum numbers \(j=2, m=1\). Determine the probabilities that when \(J_{z}\) is measured for gyro \(\mathrm{A}\), the values \(m=\pm 1\) and 0 will be obtained. What is the value of the Clebsch-Gordan coefficient \(C(2,1 ; 1,1,1,0)\) ?

Short Answer

Expert verified
Probabilities: \(m=1\): \(\frac{1}{3}\), \(m=0\): \(\frac{2}{3}\). Clebsch-Gordan coefficient \(C(2,1 ; 1,1,1,0): \frac{1}{\sqrt{3}}\).

Step by step solution

01

Understanding the problem

We have a composite system of two spin-one particles labeled as gyros A and B. The total system has angular momentum quantum numbers \(j=2, m=1\). We need to determine the probabilities of measuring \(m=\pm 1\) and \(m=0\) for gyro A when \(J_z\) is observed. Additionally, we are asked to find the Clebsch-Gordan coefficient \(C(2,1 ; 1,1,1,0)\).
02

Identify possible states of the system

The total angular momentum \(j=2, m=1\) can be formed using combinations of individual angular momentum states of each gyro with \(j_1 = 1, m_1\) and \(j_2 = 1, m_2\). These combinations must satisfy \(m_1 + m_2 = 1\). The possible combinations are \((m_1, m_2) = (1, 0), (0, 1), (-1, 2)\).
03

Calculate probabilities for individual states

To find the probabilities for each possible \(m_1\) value, we need to calculate the square of the respective Clebsch-Gordan coefficients. The relevant states here are those for which \((m_1, m_2)\) add up to \(m=1\), in terms of spin particles A and B. Evaluate the Clebsch-Gordan coefficients for \(C(2, 1; 1, 1, 1, 0)\), \(C(2, 1; 1, 0, 0, 1)\), and \(C(2, 1; 1, -1, 1, 2)\). The probability \(P(m_1)\) of measuring \(m_1\) is given by \(P(m_1) = |C(j, m; j_1, m_1, j_2, m_2)|^2\).
04

Determine the Clebsch-Gordan coefficients

Using tables or calculations of Clebsch-Gordan coefficients:- \(C(2, 1; 1, 1, 1, 0) = \frac{1}{\sqrt{3}}\)- \(C(2, 1; 1, 0, 1, 1) = \sqrt{\frac{2}{3}}\)
05

Calculate and verify probabilities

Calculate the probabilities using the coefficients:- For \(m_1 = 1\), \(P(1) = \left|C(2,1;1,1,1,0)\right|^2 = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\)- For \(m_1 = 0\), \(P(0) = \left|C(2,1;1,0,1,1)\right|^2 = \left(\sqrt{\frac{2}{3}}\right)^2 = \frac{2}{3}\)- \(m_1 = -1\) is not applicable since \((m_1, m_2) = (-1, 2)\) is not possible.
06

Compile the result

The probability of measuring \(m = 1\) for gyro A is \(\frac{1}{3}\), the probability of \(m = 0\) is \(\frac{2}{3}\). \(m = -1\) does not occur for this configuration. The Clebsch-Gordan coefficient \(C(2,1 ; 1,1,1,0)\) is \(\frac{1}{\sqrt{3}}\).

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Key Concepts

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

Angular Momentum
Angular momentum is a fundamental concept in physics, especially in quantum mechanics, where it describes the rotation of objects. In the quantum world, it's not just about spinning objects in visible motion, but also about how particles like electrons and nuclei behave. Angular momentum has two crucial components:
  • Magnitude: Represented by the quantum number \( j \), it tells us how much angular momentum is in the system.
  • Direction: Represented by \( m \), it is a component of angular momentum in a specific direction, usually chosen to be the \( z \)-axis.
When dealing with composite systems, such as two particles in a box, the total angular momentum combines the angular momentum of each individual component. The quantum numbers \( j \) and \( m \) specify the possible outcomes of measuring the angular momentum, and these values determine the behavior of the system.
Spin-One Particles
Spin-one particles are particles with a spin quantum number \( s = 1 \). Spin is a type of intrinsic angular momentum carried by quantum particles, similar to how Earth spins on its axis. Unlike integer angular momentum which can take multiple values, the spin for particles like gyros or photons can take values of \(-1, 0, \) and \(+1\).
  • This property makes spin-one particles unique because their total spin can interact in more complex ways, such as adding or canceling out their spins.
  • Spin is a foundational aspect in understanding quantum states, especially when dealing with composite systems where multiple particles are involved.
In the given exercise, particles \( \text{A} \) and \( \text{B} \) are both spin-one, meaning they have three possible orientations when measuring their spin: \( m = -1, 0, \text{ and } +1 \). The challenge is finding how their individual spins combine according to specific quantum mechanical rules.
Quantum Mechanics
Quantum mechanics is an area of physics that explains the nature and behavior of matter and energy on atomic and subatomic levels. Key principles include:
  • Wave-particle duality: Particles exhibit both wave-like and particle-like properties.
  • Uncertainty principle: It is impossible to know both the position and momentum of a particle with absolute precision.
  • Superposition: Particles can exist in multiple states at once until measured.
Applying quantum mechanics to angular momentum, we use mathematical frameworks like quantum numbers and Clebsch-Gordan coefficients to predict probable outcomes of measurements. This is crucial for understanding how individual spins combine in systems such as two spin-one particles, as explored in this problem. It highlights the need to use fundamental quantum principles to account for behaviors that defy classical expectations.
Probability Calculation
Probability calculation in quantum mechanics involves finding the likelihood of a particular measurement outcome. This is based on the quantum state's mathematical description. In this problem, we calculate the probabilities of certain angular momentum values for a particle using Clebsch-Gordan coefficients.The steps include:
  • Identifying the possible individual states \((m_1, m_2)\) of the particles that result in a given total state \((j, m)\).
  • Evaluating the Clebsch-Gordan coefficients, which give the probability amplitude for these combinations.
  • Calculating the square of these amplitudes to get the actual probabilities.
For example, calculating the probability for \( m_1 = 1 \) involves squaring the coefficient \( C(2,1 ; 1,1,1,0) = \frac{1}{\sqrt{3}} \), which yields \( \frac{1}{3} \). This systematic approach uses the coefficients to untangle complex quantum states, providing insights into probable particle behaviors upon measurement.

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Most popular questions from this chapter

The angular momentum of a hydrogen atom in its ground state is entirely due to the spins of the electron and proton. The atom is in the state \(|1,0\rangle\) in which it has one unit of angular momentum but none of it is parallel to the \(z\)-axis. Express this state as a linear combination of products of the spin states \(|\pm, \mathrm{e}\rangle\) and \(|\pm, \mathrm{p}\rangle\) of the proton and electron. Show that the states \(|x \pm, \mathrm{e}\rangle\) in which the electron has well-defined spin along the \(x\)-axis are $$ |x \pm, \mathrm{e}\rangle=\frac{1}{\sqrt{2}}(|+, \mathrm{e}\rangle \pm|-, \mathrm{e}\rangle) $$ By writing $$ |1,0\rangle=|x+, \mathrm{e}\rangle\langle x+, \mathrm{e} \mid 1,0\rangle+|x-, \mathrm{e}\rangle\langle x-, \mathrm{e} \mid 1,0\rangle $$ express \(|1,0\rangle\) as a linear combination of the products \(|x \pm, \mathrm{e}\rangle|x \pm, \mathrm{p}\rangle .\) Explain the physical significance of your result.

Show that \(\left\langle j, j\left|J_{x}\right| j, j\right\rangle=\left\langle j, j\left|J_{y}\right| j, j\right\rangle=0\) and that \(\langle j, j|\left(J_{x}^{2}+\right.\) \(\left.J_{y}^{2}\right)|j, j\rangle=j\). Discuss the implications of these results for the uncertainty in the orientation of the classical angular momentum vector \(\mathbf{J}\) for both small and large values of \(j\).

The angular part of a system's wavefunction is $$ \langle\theta, \phi \mid \psi\rangle \propto\left(\sqrt{2} \cos \theta+\sin \theta \mathrm{e}^{-\mathrm{i} \phi}-\sin \theta \mathrm{e}^{\mathrm{i} \phi}\right) . $$ What are the possible results of measurement of (a) \(L^{2}\), and (b) \(L_{z}\), and their probabilities? What is the expectation value of \(L_{z} ?\)

Show that \(\left[J_{i}, L_{j}\right]=\mathrm{i} \sum_{k} \epsilon_{i j k} L_{k}\) and \(\left[J_{i}, L^{2}\right]=0\) by eliminating \(L_{i}\) using its definition \(\mathbf{L}=\hbar^{-1} \mathbf{x} \times \mathbf{p}\), and then using the commutators of \(J_{i}\) with \(\mathrm{x}\) and \(\mathbf{p}\).

In the rotation spectrum of \({ }^{12} \mathrm{C}^{16} \mathrm{O}\) the line arising from the transition \(l=4 \rightarrow 3\) is at \(461.04077 \mathrm{GHz}\), while that arising from \(l=36 \rightarrow\) 35 is at \(4115.6055 \mathrm{GHz}\). Show from these data that in a non-rotating CO molecule the intra-nuclear distance is \(s \simeq 0.113 \mathrm{~nm}\), and that the electrons provide a spring between the nuclei that has force constant \(\sim 1904 \mathrm{Nm}^{-1}\). Hence show that the vibrational frequency of CO should lie near \(6.47 \times 10^{13} \mathrm{~Hz}\) (measured value is \(6.43 \times 10^{13} \mathrm{~Hz}\) ). Hint: show from classical mechanics that the distance of \(\mathrm{O}\) from the centre of mass is \(\frac{3}{7} s\) and that the molecule's moment of inertia is \(\frac{48}{7} m_{\mathrm{p}} s^{2}\). Recall also the classical relation \(L=I \omega\).
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