Consider a particle in a one-dimensional rigid box of length a. Recall that a rigid box has U (x) = 00 for a <0 and a> a, and U(2) = 0 for 0 < a < a. Suppose the particle is in the second excited state (two states above the ground state). (The next 2 problems below pertain to this information.)
Consider a particle in a one-dimensional rigid box of length a. Recall that a rigid box has U (x) = 00 for a <0 and a> a, and U(2) = 0 for 0 < a < a. Suppose the particle is in the second excited state (two states above the ground state). (The next 2 problems below pertain to this information.)
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#### Explanation:
In quantum mechanics, when a particle is confined in a one-dimensional rigid box (also known as an infinite potential well), the energy levels and corresponding wave functions are quantized. For a box of length \( L \), the wave function for the \( n \)-th state is given by:
\[
\psi_n(x) = \sqrt{\frac{2}{L}} \sin{\left( \frac{n \pi x}{L} \right)},
\]
where \( n \) is a positive integer representing the quantum number of the state.
In the second excited state (\( n = 3 \)), the wave function becomes:
\[
\psi_3(x) = \sqrt{\frac{2}{L}} \sin{\left( \frac{3 \pi x}{L} \right)}.
\]
The probability density function, which determines the probability of finding the particle at a position \( x \), is given by:
\[
|\psi_3(x)|^2 = \left( \sqrt{\frac{2}{L}} \sin{\left( \frac{3 \pi x}{L} \right)} \right)^2 = \frac{2}{L} \sin^2{\left( \frac{3 \pi x}{L} \right)}.
\]
Using this probability density function, we can deduce the positions where the particle is most likely to be found, known as the most probable positions \( x_{\text{mp}} \).
#### Key Points:
- The particle in the second excited state of a one-dimensional rigid box has quantized energy levels and wave functions.
- The wave function for the second excited state (\( n = 3 \)) is \( \psi_3(x) \).
- The probability density function is \( |\psi_3(x)|^2 = \frac{2}{L} \sin^2{\left( \frac{3 \pi x}{L} \right)} \).
To find the most probable positions, we seek the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aaf72ae-8073-4cb4-ae75-7d87fdc3f506%2F74e37c22-d429-48d5-acde-0613d6499feb%2Fqdgg9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Most Probable Positions in the Second Excited State of a One Dimensional Rigid Box
In this educational section, we will discuss the concept of the most probable positions \( x_{\text{mp}} \) of a particle in the second excited state of a one-dimensional rigid box.
For detailed formulas and mathematical background, please refer to the provided document:
[Formulas.pdf](#)
#### Explanation:
In quantum mechanics, when a particle is confined in a one-dimensional rigid box (also known as an infinite potential well), the energy levels and corresponding wave functions are quantized. For a box of length \( L \), the wave function for the \( n \)-th state is given by:
\[
\psi_n(x) = \sqrt{\frac{2}{L}} \sin{\left( \frac{n \pi x}{L} \right)},
\]
where \( n \) is a positive integer representing the quantum number of the state.
In the second excited state (\( n = 3 \)), the wave function becomes:
\[
\psi_3(x) = \sqrt{\frac{2}{L}} \sin{\left( \frac{3 \pi x}{L} \right)}.
\]
The probability density function, which determines the probability of finding the particle at a position \( x \), is given by:
\[
|\psi_3(x)|^2 = \left( \sqrt{\frac{2}{L}} \sin{\left( \frac{3 \pi x}{L} \right)} \right)^2 = \frac{2}{L} \sin^2{\left( \frac{3 \pi x}{L} \right)}.
\]
Using this probability density function, we can deduce the positions where the particle is most likely to be found, known as the most probable positions \( x_{\text{mp}} \).
#### Key Points:
- The particle in the second excited state of a one-dimensional rigid box has quantized energy levels and wave functions.
- The wave function for the second excited state (\( n = 3 \)) is \( \psi_3(x) \).
- The probability density function is \( |\psi_3(x)|^2 = \frac{2}{L} \sin^2{\left( \frac{3 \pi x}{L} \right)} \).
To find the most probable positions, we seek the
Transcribed Image Text:**Quantum Mechanics: Particle in a 1D Rigid Box**
Consider a particle in a one-dimensional rigid box of length \( a \). Recall that a rigid box has \( U(x) = \infty \) for \( x < 0 \) and \( x > a \), and \( U(x) = 0 \) for \( 0 \le x \le a \). Suppose the particle is in the second excited state (two states above the ground state).
(The next 2 problems below pertain to this information.)
**Resources**
[Formulas.pdf (Click here-->)
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