(2) An electron of mass 9.11 x 10-31 kg in the ground state is trapped inside a one-dimensional box of width L=1x 10-10 meters (about the size of an atom). In other words, the potential energy U(x) =0 inside the box (between 0 < x < L), but U = 00 outside the box (for r <0 and x > L). If the width of the box is reduced by 1/2, how many eV (where 1 eV =1 electron-volt = 1 electron volt) will be the energy of the electron inside the box? (This shows that the particle-in-a-box model works approximately for electrons in single-electron atom, assuming no ionization.) %3D %3D
(2) An electron of mass 9.11 x 10-31 kg in the ground state is trapped inside a one-dimensional box of width L=1x 10-10 meters (about the size of an atom). In other words, the potential energy U(x) =0 inside the box (between 0 < x < L), but U = 00 outside the box (for r <0 and x > L). If the width of the box is reduced by 1/2, how many eV (where 1 eV =1 electron-volt = 1 electron volt) will be the energy of the electron inside the box? (This shows that the particle-in-a-box model works approximately for electrons in single-electron atom, assuming no ionization.) %3D %3D
Related questions
Question
100%
Show all steps to solve
![## Quantum Mechanics: Particle in a Box
**Problem Statement:**
An electron of mass \(9.11 \times 10^{-31}\) kg in the ground state is trapped inside a one-dimensional box of width \(L = 1 \times 10^{-10}\) meters (about the size of an atom). In other words, the potential energy \(U(x) = 0\) inside the box (between \(0 \leq x \leq L\)), but \(U = \infty\) outside the box (for \(x < 0\) and \(x > L\)).
If the width of the box is reduced by half, how many electron volts (eV) will be the energy of the electron inside the box?
This demonstrates that the particle-in-a-box model works approximately for electrons in a single-electron atom, assuming no ionization.
### Explanation
In quantum mechanics, the particle-in-a-box model (also known as the infinite potential well) provides insights into the behavior of particles, like electrons, in a confined space. The model illustrates that the energy levels are quantized, meaning electrons can only occupy specific energy levels.
For an electron in a one-dimensional box of width \(L\):
- The energy levels are defined as:
\[
E_n = \frac{n^2 h^2}{8mL^2}
\]
where:
- \(E_n\) is the energy of the electron at level \(n\),
- \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \, \text{m}^2 \, \text{kg/s}\)),
- \(m\) is the mass of the electron,
- \(L\) is the width of the box,
- \(n\) is a quantum number (n = 1, 2, 3,...).
- When the width \(L\) is halved, the energy increases by a factor of four due to \(E \propto \frac{1}{L^2}\). This corresponds to the electron having higher energy levels when confined to a smaller space.
This concept is pivotal in understanding the principles underlying quantum wells, nanostructures, and the behavior of materials at the atomic scale.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9ba852f-104a-46ab-8acb-a0c81c1ec59c%2F42b745b7-73de-4e8c-9fa6-9dde51bb577c%2Fxdcnoy3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Quantum Mechanics: Particle in a Box
**Problem Statement:**
An electron of mass \(9.11 \times 10^{-31}\) kg in the ground state is trapped inside a one-dimensional box of width \(L = 1 \times 10^{-10}\) meters (about the size of an atom). In other words, the potential energy \(U(x) = 0\) inside the box (between \(0 \leq x \leq L\)), but \(U = \infty\) outside the box (for \(x < 0\) and \(x > L\)).
If the width of the box is reduced by half, how many electron volts (eV) will be the energy of the electron inside the box?
This demonstrates that the particle-in-a-box model works approximately for electrons in a single-electron atom, assuming no ionization.
### Explanation
In quantum mechanics, the particle-in-a-box model (also known as the infinite potential well) provides insights into the behavior of particles, like electrons, in a confined space. The model illustrates that the energy levels are quantized, meaning electrons can only occupy specific energy levels.
For an electron in a one-dimensional box of width \(L\):
- The energy levels are defined as:
\[
E_n = \frac{n^2 h^2}{8mL^2}
\]
where:
- \(E_n\) is the energy of the electron at level \(n\),
- \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \, \text{m}^2 \, \text{kg/s}\)),
- \(m\) is the mass of the electron,
- \(L\) is the width of the box,
- \(n\) is a quantum number (n = 1, 2, 3,...).
- When the width \(L\) is halved, the energy increases by a factor of four due to \(E \propto \frac{1}{L^2}\). This corresponds to the electron having higher energy levels when confined to a smaller space.
This concept is pivotal in understanding the principles underlying quantum wells, nanostructures, and the behavior of materials at the atomic scale.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
