Let's consider a system of atoms which all have two energy levels: a ground state and an excited state. The fracti of atoms that are in the ground state is given by N₁ and the fraction of atoms in the excited state is N₂. The atoms a exposed to an incident radiation field, which can cause (1) atoms in the ground state to become excited or (2) excit atoms to decay via stimulated emission. Atoms in the excited state can additionally undergo spontaneous decay to t ground state. The time rate of change of the fraction of atoms in either state is therefore dependent on the rate of excitation fro he ground to the excited state (pN₁), the rate of decay due to stimulated emission (pN₂), and the rate of spontaneo decay (N₂/t). Note that p is proportional to the power of the ambient radiation field (and that the terms pN₁ and pl must have opposite signs), and is related to the "lifetime" of each excited state (look up "Einstein coefficients" you're interested). We therefore have a set of coupled equations with two variables (N₁ and N₂) and two paramete p and T): dN₁ dt = -pN₁ +pN₂ + N₂ T dN₂ dt To make life easy, let's set t = 1 and assume that all atoms are in the ground state at t = 0. Use the Euler method to solve the above equations from t = 0 to t = 3 assuming a time step of 0.01 for p 0.1, 0.3, 1, 3. Show how the number of atoms in each state evolves with time for each value of p. Note th you are not being asked to solve the equations analytically! -=pN₁-pN₂ N₂ T Does increasing the power of the radiation field ever lead to a population inversion, in which more ator occupy the excited state than the ground state? Present a graph(s) to support your answer, and describe yo findings. Don't just say "The graphs show..." - phrase your answer in terms of the physical system bei evaluated! For example: "As we increase the power of the ambient radiation field we see that 33
Let's consider a system of atoms which all have two energy levels: a ground state and an excited state. The fracti of atoms that are in the ground state is given by N₁ and the fraction of atoms in the excited state is N₂. The atoms a exposed to an incident radiation field, which can cause (1) atoms in the ground state to become excited or (2) excit atoms to decay via stimulated emission. Atoms in the excited state can additionally undergo spontaneous decay to t ground state. The time rate of change of the fraction of atoms in either state is therefore dependent on the rate of excitation fro he ground to the excited state (pN₁), the rate of decay due to stimulated emission (pN₂), and the rate of spontaneo decay (N₂/t). Note that p is proportional to the power of the ambient radiation field (and that the terms pN₁ and pl must have opposite signs), and is related to the "lifetime" of each excited state (look up "Einstein coefficients" you're interested). We therefore have a set of coupled equations with two variables (N₁ and N₂) and two paramete p and T): dN₁ dt = -pN₁ +pN₂ + N₂ T dN₂ dt To make life easy, let's set t = 1 and assume that all atoms are in the ground state at t = 0. Use the Euler method to solve the above equations from t = 0 to t = 3 assuming a time step of 0.01 for p 0.1, 0.3, 1, 3. Show how the number of atoms in each state evolves with time for each value of p. Note th you are not being asked to solve the equations analytically! -=pN₁-pN₂ N₂ T Does increasing the power of the radiation field ever lead to a population inversion, in which more ator occupy the excited state than the ground state? Present a graph(s) to support your answer, and describe yo findings. Don't just say "The graphs show..." - phrase your answer in terms of the physical system bei evaluated! For example: "As we increase the power of the ambient radiation field we see that 33
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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cant seem to code this on python using eulers method help?
![Let’s consider a system of atoms which all have two energy levels: a ground state and an excited state. The fraction of atoms that are in the ground state is given by \( N_1 \) and the fraction of atoms in the excited state is \( N_2 \). The atoms are exposed to an incident radiation field, which can cause (1) atoms in the ground state to become excited or (2) excited atoms to decay via stimulated emission. Atoms in the excited state can additionally undergo spontaneous decay to the ground state.
The time rate of change of the fraction of atoms in either state is therefore dependent on the rate of excitation from the ground to the excited state (\( pN_1 \)), the rate of decay due to stimulated emission (\( pN_2 \)), and the rate of spontaneous decay (\( N_2/\tau \)). Note that \( p \) is proportional to the power of the ambient radiation field (and that the terms \( pN_1 \) and \( pN_2 \) must have opposite signs), and \( \tau \) is related to the “lifetime” of each excited state (look up “Einstein coefficients” if you’re interested). We therefore have a set of coupled equations with two variables (\( N_1 \) and \( N_2 \)) and two parameters (\( p \) and \( \tau \)):
\[
\frac{dN_1}{dt} = -pN_1 + pN_2 + \frac{N_2}{\tau}
\]
\[
\frac{dN_2}{dt} = pN_1 - pN_2 - \frac{N_2}{\tau}
\]
To make life easy, let’s set \( \tau = 1 \) and assume that all atoms are in the ground state at \( t = 0 \).
- Use the Euler method to solve the above equations from \( t = 0 \) to \( t = 3 \) assuming a time step of 0.01 for \( p = 0.1, 0.3, 1, 3 \). Show how the number of atoms in each state evolves with time for each value of \( p \). Note that you are not being asked to solve the equations analytically!
- Does increasing the power of the radiation field ever lead to a population inversion, in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65ba51bd-93fe-4637-8f7c-e52d3fdaa23e%2F047f2a1d-667c-4cee-a156-fa599142a0bd%2Fd37yun9_processed.png&w=3840&q=75)
Transcribed Image Text:Let’s consider a system of atoms which all have two energy levels: a ground state and an excited state. The fraction of atoms that are in the ground state is given by \( N_1 \) and the fraction of atoms in the excited state is \( N_2 \). The atoms are exposed to an incident radiation field, which can cause (1) atoms in the ground state to become excited or (2) excited atoms to decay via stimulated emission. Atoms in the excited state can additionally undergo spontaneous decay to the ground state.
The time rate of change of the fraction of atoms in either state is therefore dependent on the rate of excitation from the ground to the excited state (\( pN_1 \)), the rate of decay due to stimulated emission (\( pN_2 \)), and the rate of spontaneous decay (\( N_2/\tau \)). Note that \( p \) is proportional to the power of the ambient radiation field (and that the terms \( pN_1 \) and \( pN_2 \) must have opposite signs), and \( \tau \) is related to the “lifetime” of each excited state (look up “Einstein coefficients” if you’re interested). We therefore have a set of coupled equations with two variables (\( N_1 \) and \( N_2 \)) and two parameters (\( p \) and \( \tau \)):
\[
\frac{dN_1}{dt} = -pN_1 + pN_2 + \frac{N_2}{\tau}
\]
\[
\frac{dN_2}{dt} = pN_1 - pN_2 - \frac{N_2}{\tau}
\]
To make life easy, let’s set \( \tau = 1 \) and assume that all atoms are in the ground state at \( t = 0 \).
- Use the Euler method to solve the above equations from \( t = 0 \) to \( t = 3 \) assuming a time step of 0.01 for \( p = 0.1, 0.3, 1, 3 \). Show how the number of atoms in each state evolves with time for each value of \( p \). Note that you are not being asked to solve the equations analytically!
- Does increasing the power of the radiation field ever lead to a population inversion, in
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