A particle of mass m and kinetic energy E > 0 approaches an attractive delta-function well located at x = 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. (b) Set [(ma²)Vo/(2h²)] = 1 and plot the reflection coefficient as a function of dimensionless parameter x = E/Vo, (0 < x<00).
A particle of mass m and kinetic energy E > 0 approaches an attractive delta-function well located at x = 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. (b) Set [(ma²)Vo/(2h²)] = 1 and plot the reflection coefficient as a function of dimensionless parameter x = E/Vo, (0 < x<00).
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![Title: Quantum Mechanics - Delta Function Well
**Problem Statement:**
A particle of mass \( m \) and kinetic energy \( E > 0 \) approaches an *attractive* delta-function well located at \( x = 0 \):
\[ V(x) = -V_0 a \, \delta(x) \]
where \( V_0, a \) are positive constants, \( V_0 \) has units of energy, and \( a \) has units of length.
**Tasks:**
(a) Find expressions for the reflection and transmission probabilities.
(b) Set \(\left[(ma^2)V_0/(2\hbar^2)\right] = 1\) and plot the reflection coefficient as a function of the dimensionless parameter \( x = E/V_0 \), \((0 \leq x \leq \infty)\).
**Note:**
For Task (b), the plot should be illustrated with \( x \) representing the ratio \( E/V_0 \), showing how the reflection coefficient varies over this continuous dimensionless parameter.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F751027eb-f090-4412-aa8c-8178130b3fe1%2F5i0gjzg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Quantum Mechanics - Delta Function Well
**Problem Statement:**
A particle of mass \( m \) and kinetic energy \( E > 0 \) approaches an *attractive* delta-function well located at \( x = 0 \):
\[ V(x) = -V_0 a \, \delta(x) \]
where \( V_0, a \) are positive constants, \( V_0 \) has units of energy, and \( a \) has units of length.
**Tasks:**
(a) Find expressions for the reflection and transmission probabilities.
(b) Set \(\left[(ma^2)V_0/(2\hbar^2)\right] = 1\) and plot the reflection coefficient as a function of the dimensionless parameter \( x = E/V_0 \), \((0 \leq x \leq \infty)\).
**Note:**
For Task (b), the plot should be illustrated with \( x \) representing the ratio \( E/V_0 \), showing how the reflection coefficient varies over this continuous dimensionless parameter.
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