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Consider the observable N with eigenvalues wi and corresponding eigenvectors w;). The expectation value of observable N, i.e., (N), for a particle in state |) is O (wi[12/w;)

Consider the observable N with eigenvalues wi and corresponding eigenvectors w;). The expectation value of observable N, i.e., (N), for a particle in state |) is O (wi[12/w;)

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**Quantum Mechanics: Expectation Value of an Observable** In quantum mechanics, consider the observable \( \Omega \) with eigenvalues \( \omega_i \) and corresponding eigenvectors \( |\omega_i \rangle \). The expectation value of this observable, denoted as \( \langle \Omega \rangle \), for a particle in state \( |\psi \rangle \) is calculated using the formula: --- **Options for the Expectation Value:** 1. \( \langle \omega_i|\Omega|\omega_i \rangle \) 2. \( \sum_i \omega_i \langle \psi|\omega_i \rangle \langle \omega_i|\psi \rangle \) 3. \( |\langle \omega_i|\Omega|\omega_i \rangle|^2 \) 4. \( \sum_i \omega_i^2 \langle \omega_i|\psi \rangle \langle \omega_i|\psi \rangle \) --- **Explanation of the Expressions:** - **Option 1:** Incorrect. This expression evaluates to a diagonal matrix element for a specific eigenvalue and is not the expectation value across a superposition state. - **Option 2:** Correct. This represents the expectation value \( \langle \Omega \rangle = \sum_i \omega_i |\langle \omega_i|\psi \rangle|^2 \), which is the weighted sum of the eigenvalues, considering the probability amplitudes. - **Option 3:** Incorrect. This computes the square of a diagonal matrix element and is not relevant for the expectation value. - **Option 4:** Incorrect. This involves the squares of the eigenvalues and does not reflect the standard expectation value calculation. The correct expression for the expectation value of an observable in a given quantum state is expressed by option 2.
Transcribed Image Text:**Quantum Mechanics: Expectation Value of an Observable** In quantum mechanics, consider the observable \( \Omega \) with eigenvalues \( \omega_i \) and corresponding eigenvectors \( |\omega_i \rangle \). The expectation value of this observable, denoted as \( \langle \Omega \rangle \), for a particle in state \( |\psi \rangle \) is calculated using the formula: --- **Options for the Expectation Value:** 1. \( \langle \omega_i|\Omega|\omega_i \rangle \) 2. \( \sum_i \omega_i \langle \psi|\omega_i \rangle \langle \omega_i|\psi \rangle \) 3. \( |\langle \omega_i|\Omega|\omega_i \rangle|^2 \) 4. \( \sum_i \omega_i^2 \langle \omega_i|\psi \rangle \langle \omega_i|\psi \rangle \) --- **Explanation of the Expressions:** - **Option 1:** Incorrect. This expression evaluates to a diagonal matrix element for a specific eigenvalue and is not the expectation value across a superposition state. - **Option 2:** Correct. This represents the expectation value \( \langle \Omega \rangle = \sum_i \omega_i |\langle \omega_i|\psi \rangle|^2 \), which is the weighted sum of the eigenvalues, considering the probability amplitudes. - **Option 3:** Incorrect. This computes the square of a diagonal matrix element and is not relevant for the expectation value. - **Option 4:** Incorrect. This involves the squares of the eigenvalues and does not reflect the standard expectation value calculation. The correct expression for the expectation value of an observable in a given quantum state is expressed by option 2.
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