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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Please do D, E, and F
![**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\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa4426dc-92c3-4ac7-bd5a-9ba1dd10b9af%2Fc6a217a2-0a0b-4c4a-b5af-6c060c971b1c%2Fm6klip_processed.png&w=3840&q=75)
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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