14. Compute (0|p|0) and (0\f*|0).

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PLEASE ANSWER #14.  The first page is provided for context.

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7. Find \( A_n \) by normalizing \( |n\rangle \). Suggestion: Compute \( \langle n|n\rangle \) for \( n = 0,1,2,3 \) then find the pattern. Use the identity \( \langle \hat{Q} f | g \rangle = \langle f | \hat{Q}^\dagger | g \rangle \) for \( \hat{Q} = \hat{a}^\dagger \).

8. Show that \( |n\rangle \) are the eigenvectors of \( \hat{H} \), i.e.,

   \[
   \hat{H} |n\rangle = E_n |n\rangle \tag{7}
   \]

   is satisfied. Find the energy eigenvalue \( E_n \). Knowing that \( \hat{H} \) is an observable, what can you tell about \( \langle m | n \rangle \) for \( m \neq n \)?

9. Using your results, show that the following formula holds:

   \[
   \hat{a}^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle \tag{8a}
   \]
   \[
   \hat{a} |n\rangle = \sqrt{n} |n-1\rangle \tag{8b}
   \]

   Do this either by giving a general proof or by showing that they hold for \( n = 0,1,2,3 \).

10. Let's go back to Eq.(2). What is the SI unit of the coefficient \( \sqrt{\frac{\hbar}{2m\omega}} \)? Does it make sense to you?

11. Show that the square of position operator is

    \[
    \hat{x}^2 = \frac{\hbar}{2m\omega} (\hat{a}^\dagger \hat{a}^\dagger + \hat{a}^\dagger \hat{a} + \hat{a} \hat{a}^\dagger + \hat{a}\hat{a}) \tag{9}
    \]

12. Compute \( \langle 0 | \hat{x} |
Transcribed Image Text:Certainly! Here is the transcription of the given text for educational purposes: --- 7. Find \( A_n \) by normalizing \( |n\rangle \). Suggestion: Compute \( \langle n|n\rangle \) for \( n = 0,1,2,3 \) then find the pattern. Use the identity \( \langle \hat{Q} f | g \rangle = \langle f | \hat{Q}^\dagger | g \rangle \) for \( \hat{Q} = \hat{a}^\dagger \). 8. Show that \( |n\rangle \) are the eigenvectors of \( \hat{H} \), i.e., \[ \hat{H} |n\rangle = E_n |n\rangle \tag{7} \] is satisfied. Find the energy eigenvalue \( E_n \). Knowing that \( \hat{H} \) is an observable, what can you tell about \( \langle m | n \rangle \) for \( m \neq n \)? 9. Using your results, show that the following formula holds: \[ \hat{a}^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle \tag{8a} \] \[ \hat{a} |n\rangle = \sqrt{n} |n-1\rangle \tag{8b} \] Do this either by giving a general proof or by showing that they hold for \( n = 0,1,2,3 \). 10. Let's go back to Eq.(2). What is the SI unit of the coefficient \( \sqrt{\frac{\hbar}{2m\omega}} \)? Does it make sense to you? 11. Show that the square of position operator is \[ \hat{x}^2 = \frac{\hbar}{2m\omega} (\hat{a}^\dagger \hat{a}^\dagger + \hat{a}^\dagger \hat{a} + \hat{a} \hat{a}^\dagger + \hat{a}\hat{a}) \tag{9} \] 12. Compute \( \langle 0 | \hat{x} |
**The Fock Operator and Harmonic Oscillator**

The Fock operator \( \hat{a} \) is defined by:

\[
\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega} \hat{p} \right) \tag{1}
\]

where \( \hat{x} \) and \( \hat{p} \) are the position and momentum operators, respectively.

**Exercises:**

1. **Determine \( \hat{a}^\dagger \)**:
   - Write down \( \hat{a}^\dagger \) in terms of \( \hat{x} \) and \( \hat{p} \).

2. **Prove the following expressions**:
   - \[
   \hat{x} = \sqrt{\frac{\hbar}{2m\omega}} (\hat{a}^\dagger + \hat{a}) \tag{2}
   \]
   - \[
   \hat{p} = i \sqrt{\frac{\hbar m\omega}{2}} (\hat{a}^\dagger - \hat{a}) \tag{3}
   \]

3. **Canonical Commutation Relation**:
   - Show that the canonical commutation relation, \([\hat{x}, \hat{p}] = i\hbar\), yields the bosonic commutation relation,
   \[
   [\hat{a}, \hat{a}^\dagger] = 1. \tag{4}
   \]

4. **Hamiltonian of the Simple Harmonic Oscillator (SHO)**:
   - Show that the Hamiltonian,
   \[
   \hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2\hat{x}^2}{2},
   \]
   is written as 
   \[
   \hat{H} = \hbar\omega \left( \hat{N} + \frac{1}{2} \right) \tag{5}
   \]
   where \( \hat{N} = \hat{a}^\dagger \hat{a} \) is the number operator.

5. **Hermitian Nature of \( \hat{N} \)**:
   -
Transcribed Image Text:**The Fock Operator and Harmonic Oscillator** The Fock operator \( \hat{a} \) is defined by: \[ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega} \hat{p} \right) \tag{1} \] where \( \hat{x} \) and \( \hat{p} \) are the position and momentum operators, respectively. **Exercises:** 1. **Determine \( \hat{a}^\dagger \)**: - Write down \( \hat{a}^\dagger \) in terms of \( \hat{x} \) and \( \hat{p} \). 2. **Prove the following expressions**: - \[ \hat{x} = \sqrt{\frac{\hbar}{2m\omega}} (\hat{a}^\dagger + \hat{a}) \tag{2} \] - \[ \hat{p} = i \sqrt{\frac{\hbar m\omega}{2}} (\hat{a}^\dagger - \hat{a}) \tag{3} \] 3. **Canonical Commutation Relation**: - Show that the canonical commutation relation, \([\hat{x}, \hat{p}] = i\hbar\), yields the bosonic commutation relation, \[ [\hat{a}, \hat{a}^\dagger] = 1. \tag{4} \] 4. **Hamiltonian of the Simple Harmonic Oscillator (SHO)**: - Show that the Hamiltonian, \[ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2\hat{x}^2}{2}, \] is written as \[ \hat{H} = \hbar\omega \left( \hat{N} + \frac{1}{2} \right) \tag{5} \] where \( \hat{N} = \hat{a}^\dagger \hat{a} \) is the number operator. 5. **Hermitian Nature of \( \hat{N} \)**: -
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The operators can be expressed as,

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