Question

If the particle is in the allowed state, what is the expected position to find the particle? Find an expression in terms of Ψ and n. In the box below, report the expected position to find the particle, for n.

a. ⟨x⟩=Ψ⋅n
b. ⟨x⟩=n/Ψ
c. ⟨x⟩=Ψ n
d.⟨x⟩=n−Ψ

Answer 1

The expected position to find a particle in an allowed quantum state in a box is in the middle of the box, demonstrated by = L/2 for any quantum number n. This results from the symmetry of the system and holds true for all allowed energy states.

Step-by-step explanation:

When discussing the expectation value of the position for a particle in a quantum state, we refer to the average position at which one might expect to find the particle if it is in an allowed energy state within a confining potential, such as a particle confined to a box. The wave function, Ψ, along with the principal quantum number, n, allows us to calculate this expectation value.

In quantum mechanics, specifically in the context of a particle in a box, the expectation value of the particle's position is calculated by integrating the product of the position variable, x, and the probability density, which is the square of the normalized wave function, over the entire space of the box. For a particle in the nth quantum state of a box of width L, the expectation value of position, , in a symmetrical potential, is always in the middle of the box, which is L/2. However, this is irrespective of the value of n and holds for all allowed energy states. This location corresponds to the highest probability density for finding the particle as depicted by the wave function in the ground state.