Question

In the case of a particle in a one-dimensional infinite potential well (box), what is the probability of finding the particle in the first half of the box for the ground state?

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{1}{4} \)
• D. \( 1 \)

Hint:

In quantum mechanics, the probability of finding a particle in a given region is proportional to the square of the wave function. For the ground state in a box, the probability is evenly distributed over the entire box for simple problems like this.

Answer 1

The Correct Option is A

Step 1: Problem Definition. For a particle in a one-dimensional infinite potential well (particle in a box), the probability density is the square of the wave function \( \psi(x) \). For the ground state, the wave function is: \[ \psi(x) = \sqrt{\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right) \] where \( L \) is the box length and \( x \) is the position within the box (0 to \( L \)). Step 2: Probability Calculation. The probability of finding the particle in a region is the integral of the probability density over that region. The probability of finding the particle in the first half of the box is: \[ P_{\text{first half}} = \int_0^{L/2} \left| \psi(x) \right|^2 dx \] Substituting the wave function: \[ P_{\text{first half}} = \int_0^{L/2} \left( \frac{2}{L} \sin^2 \left( \frac{\pi x}{L} \right) \right) dx \] Evaluating this integral yields: \[ P_{\text{first half}} = \frac{1}{2} \] Step 3: Result. The probability of finding the ground state particle in the first half of the box is \( \frac{1}{2} \). Answer: The probability of finding the particle in the first half of the box is \( \frac{1}{2} \).