Question

The quantum-mechanical treatment of the particle in a box system predicts the existence of quantized energy levels. Assume that you have an electron in a box with a length of dx = 1 nm. For the first excited state (quantum number of the system is n = 2), what are the points (x1 and x2) in space (from x=0 nm (left boundary of the box) to x=1 nm (right boundary of the box)) where the probability of finding the electron is maximal?

Answer 1

In the first excited state (n=2) of a quantum particle in a box, the points where the probability of finding the electron is maximal are at x1 = 1/3 nm and x2 = 2/3 nm from the left boundary within a 1 nm box.

Step-by-step explanation:

The quantum-mechanical treatment of a particle in a box with quantized energy levels indicates that for the first excited state (n=2), the probability density of finding an electron at a particular location has a specific pattern. For the first excited state of a particle in a 1 nm box, the probability distribution has two peaks, which correspond to the locations where the probability of finding the electron is maximal. This distribution is zero at the exact midpoint (x=0.5 nm) of the box and reaches its maxima at one-third and two-thirds of the box length.

Thus, the points in space where the probability of finding the electron is maximal for the first excited state (n=2) are x1 = 1/3 nm and x2 = 2/3 nm from the left boundary of the box (x=0 nm).