Introduction:
In quantum mechanics, the probability of finding a particle within a certain region is determined by its wave function. The wave function describes the spatial distribution of the particle's probability density. In the case of a particle in a one-dimensional box, the wave function can be determined using the Schrödinger equation.

Particle in a One-Dimensional Box:
A one-dimensional box is an idealized system where a particle is confined to a region of space between two infinitely high potential energy barriers. The width of the box is given as 10 Å. The wave function of the particle in the box can be obtained by solving the time-independent Schrödinger equation.

State of Least Energy:
The state of least energy corresponds to the ground state of the system. In this state, the wave function is at its lowest possible energy level. The ground state wave function for a particle in a one-dimensional box is given by:

ψ(x) = sqrt(2/L) * sin((nπx)/L)

where L is the width of the box and n is the quantum number.

Calculating the Probability:
To calculate the probability of finding the particle within an interval of 1 Å at the center of the box, we need to integrate the square of the wave function over that interval. Since the particle is at the center of the box, the interval would be from -0.5 Å to 0.5 Å.

To simplify the calculation, we can substitute the values into the ground state wave function:

ψ(x) = sqrt(2/10) * sin((nπx)/10)

Now, we square the wave function to obtain the probability density:

|ψ(x)|^2 = (2/10) * sin^2((nπx)/10)

To find the probability of the particle being within the interval -0.5 Å to 0.5 Å, we integrate the probability density over that interval:

P = ∫[-0.5 Å, 0.5 Å] (2/10) * sin^2((nπx)/10) dx

This integral can be evaluated using standard techniques, such as substitution or trigonometric identities, to obtain the probability of finding the particle within the given interval.

Conclusion:
By evaluating the integral, we can determine the probability of finding the particle within an interval of 1 Å at the center of the box. The specific value of this probability will depend on the quantum number n and the mathematical calculations involved.