Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 44(x) = sin(x) a (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)
Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 44(x) = sin(x) a (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)
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![**Text Explanation for Educational Website**
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**Quantum Mechanics: Particle in a One-Dimensional Box**
Consider a particle in the \( n = 1 \) state within a one-dimensional box of length \( a \), with infinite potential at the walls. The normalized wave function for this system is given by:
\[
\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right)
\]
**Task:**
(a) Calculate the probability of finding the particle between \( \frac{a}{2} \) and \( a \).
*Hint: It may be helpful to visualize the box and sketch the probability density.*
**Graphical Explanation:**
The wave function \(\psi(x)\) describes the quantum state of the particle in terms of its position \(x\) within the box. In this scenario:
- The graph of \(\psi(x)\) is a sine wave that starts at 0, increases to a maximum, and then decreases back to 0 from \(x = 0\) to \(x = a\).
- The probability density, given by \(|\psi(x)|^2\), represents the likelihood of finding the particle at a certain position \(x\). In this case, it will be a squared sine wave, indicating regions where the particle is more or less likely to be found.
To calculate the probability, integrate \(|\psi(x)|^2\) from \(\frac{a}{2}\) to \(a\).
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Transcribed Image Text:**Text Explanation for Educational Website**
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**Quantum Mechanics: Particle in a One-Dimensional Box**
Consider a particle in the \( n = 1 \) state within a one-dimensional box of length \( a \), with infinite potential at the walls. The normalized wave function for this system is given by:
\[
\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right)
\]
**Task:**
(a) Calculate the probability of finding the particle between \( \frac{a}{2} \) and \( a \).
*Hint: It may be helpful to visualize the box and sketch the probability density.*
**Graphical Explanation:**
The wave function \(\psi(x)\) describes the quantum state of the particle in terms of its position \(x\) within the box. In this scenario:
- The graph of \(\psi(x)\) is a sine wave that starts at 0, increases to a maximum, and then decreases back to 0 from \(x = 0\) to \(x = a\).
- The probability density, given by \(|\psi(x)|^2\), represents the likelihood of finding the particle at a certain position \(x\). In this case, it will be a squared sine wave, indicating regions where the particle is more or less likely to be found.
To calculate the probability, integrate \(|\psi(x)|^2\) from \(\frac{a}{2}\) to \(a\).
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