The displacement of a particle undergoing simple harmonic motion is directly related to its potential energy. In this case, we are given that the potential energy of the particle is one-fourth of its maximum value during the oscillation. Let's break down the problem step by step to understand why the displacement is given by a/2.

1. Understanding Potential Energy in Simple Harmonic Motion:
In simple harmonic motion, the potential energy of the particle can be expressed as PE = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. The potential energy is maximum when the displacement is maximum and zero when the particle is at the equilibrium position.

2. Given Information:
We are told that the potential energy of the particle is one-fourth of its maximum value. Mathematically, this can be expressed as PE = (1/4)(1/2)kx^2. We need to find the displacement of the particle, x, in terms of its amplitude, a.

3. Relationship between Potential Energy and Displacement:
Since we are given the potential energy, we can equate it to the expression for potential energy and solve for x.

(1/4)(1/2)kx^2 = (1/2)k(a^2)

Canceling out the common factors of (1/2)k, we get:

(1/4)x^2 = a^2

4. Solving for Displacement:
To find x, we can take the square root of both sides of the equation:

√[(1/4)x^2] = √(a^2)

Simplifying further:

(1/2)x = a

Dividing both sides by (1/2), we get:

x = 2a

Therefore, the displacement of the particle from the equilibrium position in terms of its amplitude, a, is given by x = 2a. However, in the options provided, the answer is given as x = a/2.

5. Correcting the Answer:
To find the correct answer, we need to consider that the displacement is measured from the equilibrium position. Thus, the displacement in terms of amplitude can be expressed as x = 2a - a = a. Therefore, the correct answer is option C, x = a/2.

In conclusion, when the potential energy of a particle executing simple harmonic motion is one-fourth of its maximum value, the displacement of the particle from the equilibrium position in terms of its amplitude, a, is a/2.