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d. Show that the Hamiltonian of the SHO, H = 2m is written as - hwN+ (5) where N- ala is called the mmber operator. е. Show that N is Hermitian. Suggostion: Use the idlentity from Exercise #1, (QR)' = f. A normalized vector |0) (so that (0/0) = 1) is defined to satisfy å|0) = 0. With this the following veetors are constructed: In) = A, (a')" (0) for n = 0, 1,2, . (6) where A, are constant with A, 1. Compute Ñ\n) for n = 0, 1, 2, 3 to show that these are the cigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality.

d. Show that the Hamiltonian of the SHO, H = 2m is written as - hwN+ (5) where N- ala is called the mmber operator. е. Show that N is Hermitian. Suggostion: Use the idlentity from Exercise #1, (QR)' = f. A normalized vector |0) (so that (0/0) = 1) is defined to satisfy å|0) = 0. With this the following veetors are constructed: In) = A, (a')" (0) for n = 0, 1,2, . (6) where A, are constant with A, 1. Compute Ñ\n) for n = 0, 1, 2, 3 to show that these are the cigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality.

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**Exercise #4 (updated)** The Fock operator \(\hat{a}\) is defined by \[ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega} \hat{p} \right) \quad (1) \] where \(\hat{x}\) and \(\hat{p}\) are the position and momentum operators, respectively. a. Write down \(\hat{a}^{\dagger}\) in terms of \(\hat{x}\) and \(\hat{p}\). b. Show that \[ \hat{x} = \sqrt{\frac{\hbar}{2m\omega}} (\hat{a} + \hat{a}^{\dagger}) \quad (2) \] \[ \hat{p} = i \sqrt{\frac{m\omega\hbar}{2}} (\hat{a}^{\dagger} - \hat{a}) \quad (3) \] hold. c. Show that the canonical commutation relation, \([\hat{x}, \hat{p}] = i\hbar\), yields the so-called bosonic commutation relation, \[ [\hat{a}, \hat{a}^{\dagger}] = 1. \quad (4) \] d. Show that the Hamiltonian of the SHO, \(\hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2 \hat{x}^2}{2}\), is written as \[ \hat{H} = \hbar \omega \left( \hat{N} + \frac{1}{2} \right) \quad (5) \] where \(N = \hat{a}^{\dagger} \hat{a}\) is called the number operator. e. Show that \(\hat{N}\) is Hermitian. Suggestion: Use the identity from Exercise #1, \((\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger} \hat{Q}^{\dagger}\). f. A normalized vector \(|0\rangle\) (so that \(\langle 0|0 \rangle = 1\
Transcribed Image Text:**Exercise #4 (updated)** The Fock operator \(\hat{a}\) is defined by \[ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega} \hat{p} \right) \quad (1) \] where \(\hat{x}\) and \(\hat{p}\) are the position and momentum operators, respectively. a. Write down \(\hat{a}^{\dagger}\) in terms of \(\hat{x}\) and \(\hat{p}\). b. Show that \[ \hat{x} = \sqrt{\frac{\hbar}{2m\omega}} (\hat{a} + \hat{a}^{\dagger}) \quad (2) \] \[ \hat{p} = i \sqrt{\frac{m\omega\hbar}{2}} (\hat{a}^{\dagger} - \hat{a}) \quad (3) \] hold. c. Show that the canonical commutation relation, \([\hat{x}, \hat{p}] = i\hbar\), yields the so-called bosonic commutation relation, \[ [\hat{a}, \hat{a}^{\dagger}] = 1. \quad (4) \] d. Show that the Hamiltonian of the SHO, \(\hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2 \hat{x}^2}{2}\), is written as \[ \hat{H} = \hbar \omega \left( \hat{N} + \frac{1}{2} \right) \quad (5) \] where \(N = \hat{a}^{\dagger} \hat{a}\) is called the number operator. e. Show that \(\hat{N}\) is Hermitian. Suggestion: Use the identity from Exercise #1, \((\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger} \hat{Q}^{\dagger}\). f. A normalized vector \(|0\rangle\) (so that \(\langle 0|0 \rangle = 1\
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