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The problem asks to:

> Find the probabilities of finding a particle trapped in a box of length \( L \) in the region from \( 0.45L \) to \( 0.55L \) for the ground and first excited state.

This is a quantum mechanics problem where we need to calculate the probability of finding a particle in a specific region within a one-dimensional "particle in a box" setup. Here’s how we can approach the solution:

### Step-by-Step Solution

1. **Wave Functions for a Particle in a Box**:  
   For a particle in a one-dimensional box of length \( L \), the wave functions \( \psi_n(x) \) for the allowed energy levels \( n \) are given by:
   \[
   \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right)
   \]
   where \( n = 1 \) for the ground state and \( n = 2 \) for the first excited state.

2. **Probability Density**:  
   The probability density for finding the particle at a position \( x \) is \( |\psi_n(x)|^2 \).

3. **Probability of Finding the Particle in the Interval \( [0.45L, 0.55L] \)**:  
   The probability \( P \) of finding the particle in the interval \( [a, b] \) (here \( a = 0.45L \) and \( b = 0.55L \)) is given by:
   \[
   P = \int_{0.45L}^{0.55L} |\psi_n(x)|^2 \, dx
   \]

4. **Ground State Calculation (n = 1)**:
   Substitute \( \psi_1(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) \) into the probability formula:
   \[
   P_{\text{ground}} = \int_{0.45L}^{0.55L} \frac{2}{L} \sin^2\left(\frac{\pi x}{L}\right) dx
   \]

5. **First Excited State Calculation (n = 2)**:
   Similarly, for the first excited state, substitute \( \psi_2(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{2 \pi x}{L}\right) \):
   \[
   P_{\text{first excited}} = \int_{0.45L}^{0.55L} \frac{2}{L} \sin^2\left(\frac{2 \pi x}{L}\right) dx
   \]

6. **Evaluating the Integrals**:
   Both integrals involve the function \( \sin^2\left(\frac{n \pi x}{L}\right) \), which can be simplified using trigonometric identities. The results will yield the probabilities for the specified region in both states.

Would you like a detailed calculation of these integrals?

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### Additional Questions to Expand Understanding:

1. How do the probabilities change if we choose a different interval within the box?
2. What is the physical meaning of the ground and first excited states in this context?
3. How would the probabilities differ for a particle in a box with a different length \( L \)?
4. Can you explain how the particle's wave function changes with higher energy states?
5. How does the probability distribution reflect the wave-like behavior of particles?

### Tip:
For integrals of trigonometric functions like \( \sin^2(x) \), using trigonometric identities (e.g., \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)) can simplify the calculation.

Find the probabilities of finding a particle trapped in a box of length L in the region from 0.45L to 0.55L for the ground and first excited state.

Learn how to find the probabilities of locating a particle in a specific region of a box of length L for the ground and first excited states. This problem involves wave functions, probability densities, and integration over a defined interval. Ideal for undergraduate students studying quantum mechanics.

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