Final answer:
The most likely positions to find a particle in an allowed state are represented by the probability density |ψ(x,y)|^2. This value provides the likelihood of finding the particle at any given location in a two-dimensional space, as opposed to other expressions that do not indicate probability density.
Step-by-step explanation:
The question asks for the most likely positions to find a particle if it is in an allowed state, given several expressions. In quantum mechanics, the probability density, which gives the likelihood of finding a particle at a given location, is represented by the square modulus of the wave function, |ψ(x,y)|^2. This is because the wave function ψ itself can be complex, and the square modulus gives a real, positive value that can be interpreted as a probability density. Therefore, the correct expression that represents the most likely positions to find a particle, given these options, is a) |ψ(x,y)|^2, as this defines the probability density for a particle's position in a two-dimensional space.
For a particle in a box, the highest probability densities are typically found where the wave function has the largest amplitude, which, depending on the state of the particle (ground state, first excited state, etc.), might be at different points within the box. The other expressions provided, such as (b) (x+y), (c) 2xy, and (d) x^2 + y^2, do not relate to the probability density of finding the particle at a specific location.