Question

Calculate the probability that a particle in the first excited stated in a one-dimensional box of length a is found between 0 and 3a . Sketch the probability distribution. 4

Answer 1

The probability that a particle in the first excited state in a one-dimensional box of length (a) is found between 0 and 3a is (1/4).

Step-by-step explanation:

In quantum mechanics, the probability distribution of finding a particle in a one-dimensional box is determined by the wave function, typically represented by
`\(\psi(x)\)`. For the first excited state in a box of length (a), the wave function is given by
`\(\psi(x) = \sqrt{(2)/(a)} \sin\left((2\pi x)/(a)\right)\)`. The probability density function,
`\(|\psi(x)|^2\)`, provides the likelihood of finding the particle at a specific position.

To calculate the probability of the particle being located between 0 and 3a, we integrate the probability density function over this interval. For the first excited state, integrating from 0 to 3a yields a probability of (1/4). This result signifies that there is a one-in-four chance of finding the particle within the specified region.

This probability distribution aligns with the behavior of the wave function, emphasizing that the particle is more likely to be located near the center of the box. Such quantum phenomena underscore the probabilistic nature of particle behavior at the quantum scale, challenging classical intuitions about determinism and precise particle trajectories.