3. Let X and P be the position and linear momentum operators of a single particle, respectively. The corresponding representations in one-dimensional position space are X = xb(x) dy(x) Puy = -ih dx where x is position and is a wavefunction. a) Find the commutator X, Consider the case of a particle of mass m in a 1-D box of length L, where the wavefunctions are sin(knx), 0 0 L' b) Show that n is an eigenfunction of the kinetic energy operator corresponding eigenvalue? What is the 2m c) Find < X >.
3. Let X and P be the position and linear momentum operators of a single particle, respectively. The corresponding representations in one-dimensional position space are X = xb(x) dy(x) Puy = -ih dx where x is position and is a wavefunction. a) Find the commutator X, Consider the case of a particle of mass m in a 1-D box of length L, where the wavefunctions are sin(knx), 0 0 L' b) Show that n is an eigenfunction of the kinetic energy operator corresponding eigenvalue? What is the 2m c) Find < X >.
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![**Quantum Mechanics Problem Set**
3. Let \( \hat{X} \) and \( \hat{P} \) be the position and linear momentum operators of a single particle, respectively. The corresponding representations in one-dimensional position space are
\[ \hat{X} \psi = x \psi(x) \]
\[ \hat{P} \psi = -i \hbar \dfrac{d \psi(x)}{dx} \]
where \( x \) is position and \( \psi \) is a wavefunction.
**a) Find the commutator \( [\hat{X}, \hat{P}] \).**
Consider the case of a particle of mass \( m \) in a 1-D box of length \( L \), where the wave functions are
\[
\psi_n(x) =
\begin{cases}
\sqrt{\dfrac{2}{L}} \sin(k_n x), & 0 < x < L \\
0, & \text{otherwise}
\end{cases}
\]
\[ k_n = \dfrac{n \pi}{L}, \, n \in \mathbb{Z}, \, n > 0 \]
**b) Show that \( \psi_n \) is an eigenfunction of the kinetic energy operator \( \dfrac{\hat{P}^2}{2m} \). What is the corresponding eigenvalue?**
**c) Find \( \langle \hat{X} \rangle \).**
**d) Find \( \langle \hat{P} \rangle \).**
**e) Find \( \langle \hat{P}^2 \rangle \).**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6d2f58-ebc5-487b-abb4-91cd16bfc1c1%2Fd7ab1e5c-79a9-4952-ab65-af75e3a1ca41%2Ft57m9u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Quantum Mechanics Problem Set**
3. Let \( \hat{X} \) and \( \hat{P} \) be the position and linear momentum operators of a single particle, respectively. The corresponding representations in one-dimensional position space are
\[ \hat{X} \psi = x \psi(x) \]
\[ \hat{P} \psi = -i \hbar \dfrac{d \psi(x)}{dx} \]
where \( x \) is position and \( \psi \) is a wavefunction.
**a) Find the commutator \( [\hat{X}, \hat{P}] \).**
Consider the case of a particle of mass \( m \) in a 1-D box of length \( L \), where the wave functions are
\[
\psi_n(x) =
\begin{cases}
\sqrt{\dfrac{2}{L}} \sin(k_n x), & 0 < x < L \\
0, & \text{otherwise}
\end{cases}
\]
\[ k_n = \dfrac{n \pi}{L}, \, n \in \mathbb{Z}, \, n > 0 \]
**b) Show that \( \psi_n \) is an eigenfunction of the kinetic energy operator \( \dfrac{\hat{P}^2}{2m} \). What is the corresponding eigenvalue?**
**c) Find \( \langle \hat{X} \rangle \).**
**d) Find \( \langle \hat{P} \rangle \).**
**e) Find \( \langle \hat{P}^2 \rangle \).**
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