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 ký = xv(x) dv(x) Ph = -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 { Vž sin(k,7), 0< x < L 0, Un(x) = otherwise NT kin n e Z, n > 0 L' b) Show that pn is an eigenfunction of the kinetic energy operator corresponding eigenvalue? What is the 2m c) Find < X >. d) Find < P >. e) Find < P² >.
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 ký = xv(x) dv(x) Ph = -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 { Vž sin(k,7), 0< x < L 0, Un(x) = otherwise NT kin n e Z, n > 0 L' b) Show that pn is an eigenfunction of the kinetic energy operator corresponding eigenvalue? What is the 2m c) Find < X >. d) Find < P >. e) Find < P² >.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 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:
\[
\hat{X} \psi = x \psi(x)
\]
\[
\hat{P} \psi = -i \hbar \frac{d\psi(x)}{dx}
\]
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 wavefunctions are:
\[
\psi_n(x) =
\begin{cases}
\sqrt{\frac{2}{L}} \sin(k_n x), & 0 < x < L \\
0, & \text{otherwise}
\end{cases}
\]
\[
k_n = \frac{n \pi}{L}, \quad n \in \mathbb{Z}, \quad n > 0
\]
**(b)** Show that \( \psi_n \) is an eigenfunction of the kinetic energy operator \( \frac{\hat{P}^2}{2m} \). 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%2F5b98ebfb-3b99-494e-aebd-be97e1ef777c%2Fyiyd4yl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 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:
\[
\hat{X} \psi = x \psi(x)
\]
\[
\hat{P} \psi = -i \hbar \frac{d\psi(x)}{dx}
\]
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 wavefunctions are:
\[
\psi_n(x) =
\begin{cases}
\sqrt{\frac{2}{L}} \sin(k_n x), & 0 < x < L \\
0, & \text{otherwise}
\end{cases}
\]
\[
k_n = \frac{n \pi}{L}, \quad n \in \mathbb{Z}, \quad n > 0
\]
**(b)** Show that \( \psi_n \) is an eigenfunction of the kinetic energy operator \( \frac{\hat{P}^2}{2m} \). 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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