Explanation:

Given Potential Energy Function:
V = 1/2mw^2(x^2 + 4y^2)

Corresponding Eigenvalue:
To find the corresponding eigenvalue, we need to solve the Schrödinger equation for the simple harmonic oscillator using the given potential energy function. The Schrödinger equation for the simple harmonic oscillator is given by:
HΨ = EΨ
Where H is the Hamiltonian operator, Ψ is the wave function, E is the energy eigenvalue, and the Hamiltonian operator for the simple harmonic oscillator is:
H = -ħ^2/2m (∂^2/∂x^2 + ∂^2/∂y^2) + 1/2mw^2(x^2 + 4y^2)
Solving the Schrödinger equation will give us the corresponding eigenvalues for the simple harmonic oscillator.
In this case, the potential energy function is given as V = 1/2mw^2(x^2 + 4y^2), which can be used to determine the Hamiltonian operator and subsequently solve the Schrödinger equation to find the corresponding eigenvalues.
By solving the Schrödinger equation with the given potential energy function, we can determine the eigenvalues associated with the simple harmonic oscillator system.
Therefore, the corresponding eigenvalues can be obtained by solving the Schrödinger equation with the given potential energy function.