1. For a particle executing simple harmonic motion (SHM), the total mechanical energy is the sum of kinetic and potential energies:

$E = K + U$

2. In SHM, if the amplitude is $A$ and force constant is $k$, then:

$U = \frac{1}{2}kx^2$

and the kinetic energy is:

$K = \frac{1}{2}k(A^2 - x^2)$

because the speed at displacement $x$ is such that:

$v^2 = \omega^2(A^2 - x^2)$

3. Therefore, total energy becomes:

$E = \frac{1}{2}kx^2 + \frac{1}{2}k(A^2 - x^2)$

$E = \frac{1}{2}kA^2$

4. Since $A$ and $k$ are constants for a given SHM, the total energy is constant and does not depend on displacement $x$.

5. Now evaluate the options:

• A: independent of $x$ ✅
• B: $\propto x^2$ ❌
• C: $\propto x$ ❌
• D: $\propto x^{1/2}$ ❌

Hence, the correct answer is:

$\boxed{A}$