A rigid rotor with moment of inertia I is initially in the state: 1 V14 corresponding to the case l = 1. (a) Write this state as a linear combination of the eigenstates of L.. (b) Find the probability that a measurement of Læ yields the value -ħ
A rigid rotor with moment of inertia I is initially in the state: 1 V14 corresponding to the case l = 1. (a) Write this state as a linear combination of the eigenstates of L.. (b) Find the probability that a measurement of Læ yields the value -ħ
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![### Quantum Mechanics: Rigid Rotor
A rigid rotor with a moment of inertia \( I \) is initially in the quantum state:
\[
|\xi\rangle = \frac{1}{\sqrt{14}} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}
\]
This state corresponds to the case where \( \ell = 1 \).
#### (a) Problem
Write this state as a linear combination of the eigenstates of \( L_x \).
#### (b) Problem
Find the probability that a measurement of \( L_x \) yields the value \(-\hbar\).
---
The problem requires understanding the principles of quantum mechanics, particularly the representation of states and operators in terms of eigenstates and eigenvalues. In quantum mechanics, the state of a system can be expressed as a vector in a Hilbert space, and measurements can be predicted through probability amplitudes related to these vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F7e53dc66-1097-4092-b17d-6153877ea3b1%2Fe709ije_processed.png&w=3840&q=75)
Transcribed Image Text:### Quantum Mechanics: Rigid Rotor
A rigid rotor with a moment of inertia \( I \) is initially in the quantum state:
\[
|\xi\rangle = \frac{1}{\sqrt{14}} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}
\]
This state corresponds to the case where \( \ell = 1 \).
#### (a) Problem
Write this state as a linear combination of the eigenstates of \( L_x \).
#### (b) Problem
Find the probability that a measurement of \( L_x \) yields the value \(-\hbar\).
---
The problem requires understanding the principles of quantum mechanics, particularly the representation of states and operators in terms of eigenstates and eigenvalues. In quantum mechanics, the state of a system can be expressed as a vector in a Hilbert space, and measurements can be predicted through probability amplitudes related to these vectors.
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