A particle of mass m and kinetic energy E > 0 approaches an attractive delta-function well located at r = 0 V(x) = -Voa 8(x) where Vo, a are positive contants and Vo has units of energy and a has units of length. (a) Find expressions for the reflection and transmission probabilities.
A particle of mass m and kinetic energy E > 0 approaches an attractive delta-function well located at r = 0 V(x) = -Voa 8(x) where Vo, a are positive contants and Vo has units of energy and a has units of length. (a) Find expressions for the reflection and transmission probabilities.
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![**Quantum Mechanics: Delta-Function Potential Well**
A particle of mass \( m \) and kinetic energy \( E > 0 \) approaches an attractive delta-function well located at \( x = 0 \).
**Potential Function:**
\[ V(x) = -V_0 a \, \delta(x) \]
- \( V_0, a \) are positive constants.
- \( V_0 \) has units of energy.
- \( a \) has units of length.
**Tasks:**
(a) **Objective:** Derive expressions for the reflection and transmission probabilities.
(b) **Objective:**
- Set \( \left[(ma^2)V_0/(2\hbar^2)\right] = 1 \).
- Plot the reflection coefficient as a function of the dimensionless parameter \( x = E/V_0 \), where \( 0 \leq x \leq \infty \).
**Explanation:**
This study involves understanding the interaction of a quantum particle with a delta-function potential well, a common model in quantum mechanics for a localized attractive force. The problem examines the probabilities of reflection and transmission when a particle encounters this potential, crucial for applications like tunneling in quantum systems.
This exercise involves mathematical derivation and visualization through plotting, enhancing comprehension of quantum behavior in simplified systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fd9c89749-7960-4b9f-b608-5a7de360e105%2F9zwfhb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Quantum Mechanics: Delta-Function Potential Well**
A particle of mass \( m \) and kinetic energy \( E > 0 \) approaches an attractive delta-function well located at \( x = 0 \).
**Potential Function:**
\[ V(x) = -V_0 a \, \delta(x) \]
- \( V_0, a \) are positive constants.
- \( V_0 \) has units of energy.
- \( a \) has units of length.
**Tasks:**
(a) **Objective:** Derive expressions for the reflection and transmission probabilities.
(b) **Objective:**
- Set \( \left[(ma^2)V_0/(2\hbar^2)\right] = 1 \).
- Plot the reflection coefficient as a function of the dimensionless parameter \( x = E/V_0 \), where \( 0 \leq x \leq \infty \).
**Explanation:**
This study involves understanding the interaction of a quantum particle with a delta-function potential well, a common model in quantum mechanics for a localized attractive force. The problem examines the probabilities of reflection and transmission when a particle encounters this potential, crucial for applications like tunneling in quantum systems.
This exercise involves mathematical derivation and visualization through plotting, enhancing comprehension of quantum behavior in simplified systems.
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