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![**Wave Function and Probability Calculation for a Quantum Particle**
The wave function for a quantum particle is given by:
\[ \psi(x) = \sqrt{\frac{a}{\pi (x^2 + a^2)}} \]
for \(a > 0\) and \( -\infty < x < +\infty\).
**Objective:**
Determine the probability that the particle is located somewhere between \(x = -a\) and \(x = +a\).
**Explanation:**
1. **Wave Function Overview**: The given wave function \(\psi(x)\) describes the probability amplitude of finding the quantum particle at a position \(x\).
2. **Parameter \(a\)**: The parameter \(a\) should be greater than zero.
3. **Probability Calculation**: To find the probability of the particle being between \(x = -a\) and \(x = +a\), we integrate the square of the wave function (which gives the probability density) over the interval \([-a, +a]\).
Please proceed with the integral calculation to determine the specific probability.](https://content.bartleby.com/qna-images/question/3fe8677b-d2b4-4cf1-b1ed-08820154fcb5/59a321e8-5af9-49f2-9964-4e64f43ec5ae/lbbn88_processed.png)
Transcribed Image Text:**Wave Function and Probability Calculation for a Quantum Particle**
The wave function for a quantum particle is given by:
\[ \psi(x) = \sqrt{\frac{a}{\pi (x^2 + a^2)}} \]
for \(a > 0\) and \( -\infty < x < +\infty\).
**Objective:**
Determine the probability that the particle is located somewhere between \(x = -a\) and \(x = +a\).
**Explanation:**
1. **Wave Function Overview**: The given wave function \(\psi(x)\) describes the probability amplitude of finding the quantum particle at a position \(x\).
2. **Parameter \(a\)**: The parameter \(a\) should be greater than zero.
3. **Probability Calculation**: To find the probability of the particle being between \(x = -a\) and \(x = +a\), we integrate the square of the wave function (which gives the probability density) over the interval \([-a, +a]\).
Please proceed with the integral calculation to determine the specific probability.
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