Let's solve the exercises step by step. --- ## **Exercise 1a** **Given:** - Operator: \( p_x = \frac{\hbar}{i} \frac{d}{dx} \) - Equation: \( p_x \varphi(x) = k \varphi(x) \) - Range: \( 0 < x \leq L \) - Boundary condition: \( \varphi(0) = \varphi(L) \) ### **Step 1: Write the eigenvalue equation** \[ p_x \varphi(x) = k \varphi(x) \implies \frac{\hbar}{i} \frac{d\varphi(x)}{dx} = k \varphi(x) \] \[ \frac{d\varphi(x)}{dx} = \frac{ik}{\hbar} \varphi(x) \] ### **Step 2: Solve the differential equation** This is a standard first-order linear ODE: \[ \frac{d\varphi}{\varphi} = \frac{ik}{\hbar} dx \] Integrate both sides: \[ \ln \varphi(x) = \frac{ik}{\hbar}x + C \] \[ \varphi(x) = A e^{\frac{ikx}{\hbar}} \] ### **Step 3: Apply the boundary condition** \[ \varphi(0) = \varphi(L) \] \[ A e^{0} = A e^{\frac{ikL}{\hbar}} \] \[ e^{\frac{ikL}{\hbar}} = 1 \implies \frac{ikL}{\hbar} = 2\pi i n \implies k = \frac{2\pi \hbar n}{L}, \quad n \in \mathbb{Z} \] ### **Step 4: Eigenfunctions and eigenvalues** - **Eigenvalues:** \( k_n = \frac{2\pi \hbar n}{L} \) for \( n \in \mathbb{Z} \) - **Eigenfunctions:** \( \varphi_n(x) = A e^{i \frac{2\pi n}{L} x} \), typically normalized over \( (0, L) \). --- ## **Exercise 1b** Now for \( p_x^2 \) with boundary conditions \( \varphi(0) = \varphi(L) = 0 \): ### **Step 1: Write the eigenvalue equation** \[ p_x^2 \varphi(x) = \lambda \varphi(x) \] \[ \left( \frac{\hbar}{i} \frac{d}{dx} \right)^2 \varphi(x) = \lambda \varphi(x) \] \[ - \hbar^2 \frac{d^2 \varphi(x)}{dx^2} = \lambda \varphi(x) \] \[ \frac{d^2 \varphi(x)}{dx^2} + k^2 \varphi(x) = 0, \quad \text{where } k^2 = \frac{\lambda}{\hbar^2} \] ### **Step 2: General solution** \[ \varphi(x) = A \sin(kx) + B \cos(kx) \] ### **Step 3: Apply boundary conditions** - \( \varphi(0) = 0 \implies B = 0 \) - \( \varphi(L) = 0 \implies A \sin(kL) = 0 \implies kL = n\pi, \quad n = 1, 2, 3, \ldots \) (since \( A \neq 0 \) for nontrivial solutions) \[ k_n = \frac{n\pi}{L} \] ### **Step 4: Eigenvalues and eigenfunctions** - **Eigenvalues:** \( \lambda_n = \hbar^2 k_n^2 = \hbar^2 \left( \frac{n\pi}{L} \right)^2 \) - **Eigenfunctions:** \( \varphi_n(x) = A \sin \left( \frac{n\pi x}{L} \right) \), \( n = 1, 2, 3, \ldots \) --- ## **Exercise 2** ### **Write the unit operator (projector) in terms of eigenfunctions \(\varphi(x)\):** For a set of orthonormal eigenfunctions \( \{\varphi_n(x)\} \) (discrete) and/or \( \{\varphi_k(x)\} \) (continuum), the unit operator is: \[ \boxed{ \hat{I} = \sum_n |\varphi_n\rangle \langle \varphi_n| + \int dk\, |\varphi_k\rangle \langle \varphi_k| } \] In position representation: \[ \hat{I} = \sum_n \varphi_n(x) \varphi_n^*(x') + \int dk\, \varphi_k(x) \varphi_k^*(x') \] where the summation is over bound states and the integral is over continuum states. --- ## **Summary of Answers** ### **Exercise 1a:** - **Eigenvalues:** \( k_n = \frac{2\pi \hbar n}{L} \), \( n \in \mathbb{Z} \) - **Eigenfunctions:** \( \varphi_n(x) = A e^{i \frac{2\pi n}{L} x} \) ### **Exercise 1b:** - **Eigenvalues:** \( \lambda_n = \hbar^2 \left( \frac{n\pi}{L} \right)^2 \), \( n = 1, 2, 3, \ldots \) - **Eigenfunctions:** \( \varphi_n(x) = A \sin \left( \frac{n\pi x}{L} \right) \) ### **Exercise 2:** - **Unit operator (projector):** \[ \boxed{ \hat{I} = \sum_n |\varphi_n\rangle \langle \varphi_n| + \int dk\, |\varphi_k\rangle \langle \varphi_k| } \] or, in position representation, \[ \boxed{ \hat{I} = \sum_n \varphi_n(x) \varphi_n^*(x') + \int dk\, \varphi_k(x) \varphi_k^*(x') } \]