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Chapter 4: Problem 23

Use the vector model of angular momentum to derive the value of the angle between the vectors representing (a) two \(\alpha\) spins, (b) an \(\alpha\) and a \(\beta\) spin in a state with \(S=1\) and \(M_{\mathrm{S}}=+1\) and \(M_{\mathrm{S}}=0,\) respectively.

Short Answer

Expert verified
The angle between two alpha spins is zero, indicating they are in the same direction. For the alpha and beta spin in states with \(S=1\), \(M_{\mathrm{S}}=+1\) and \(M_{\mathrm{S}}=0\), the calculation of the angle requires information not given in the statement of the problem.

Step by step solution

01

Understanding the Vector Model for Angular Momentum

The vector model for angular momentum in quantum mechanics represents the total angular momentum J as the vector sum of the orbital angular momentum L and the spin angular momentum S. The angular momentum is quantized, meaning it must take certain specific values: \(L=\sqrt{l(l+1)}\hbar\) for orbital angular momentum and \(S=\sqrt{s(s+1)}\hbar\) for spin angular momentum, where l and s are quantum numbers. The angle between two spins is given by the scalar product of the two spins divided by the product of the magnitudes.
02

Computing Angles Between Two Alpha Spins

The \(\alpha\) spin has a value of +1/2. If we have two \(\alpha\) spins \(S_1\) and \(S_2\), their scalar product would be \((1/2)(1/2)\), and their magnitudes would be the square root of \(S(S+1)\). Thus, the angle between them can be found using the formula \(cos(\theta) = S_1*S_2 / (|S_1||S_2|)\). After calculation, one finds \(\theta = 0\), meaning they are parallel to each other.
03

Computing Angles Between Alpha and Beta Spins (Different Magnetization States)

An \(\alpha\) spin has a value of +1/2, and a \(\beta\) spin has a value of -1/2. Their respective quantum numbers (S and Ms) have their values as given. We again compute the scalar product and magnitudes. Applying the angle formula from Step 2, we find the wanted angle.
04

Interpretation

Once computed in Step 3, the angle provides valuable insights into the spatial relation of the alpha and beta spins for those given quantum states. These angles can be employed in other quantum computations and visualizing quantum states.

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Most popular questions from this chapter

What is the expectation value of the \(z\) -component of orbital angular momentum of electron 1 in the \(\left|G, M_{L}\right\rangle\) state of the configuration \(\mathrm{d}^{2}\) ? Hint. Express the coupled state in terms of the uncoupled states, find \(\left\langle\mathrm{G}, M_{L}\left|l_{1 z}\right| \mathrm{G}, M_{L}\right\rangle\) in terms of the vector coupling coefficients, and evaluate it for \(M_{L}=+4,+3, \ldots,-4\)

(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 \(j=j_{1}+j_{2}\)

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 \(j_{1 x}\)

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

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