To find the energy of an electron in a one-dimensional infinitely high potential box of width 1, we can use the Schrödinger equation for a particle in a box. The Schrödinger equation for a particle in a one-dimensional box is given by:

-((ħ^2)/(2m)) * (∂^2ψ/∂x^2) + V(x)ψ = Eψ

In this case, the potential energy inside the box is zero, and outside the box it is infinitely high. Therefore, the potential energy function V(x) is:

V(x) = {
0, 0 ≤ x ≤ 1
∞, x < 0="" or="" x="" /> 1
}

Since the potential energy is zero inside the box, the Schrödinger equation simplifies to:

-((ħ^2)/(2m)) * (∂^2ψ/∂x^2) = Eψ

We can solve this second-order differential equation to find the possible energy levels of the electron. The general solution to this equation will be a combination of sine and cosine functions. However, since the potential energy is infinitely high outside the box, the wavefunction must be zero at x = 0 and x = 1.

Therefore, the boundary conditions for the wavefunction are:

ψ(0) = 0
ψ(1) = 0

Using these boundary conditions, we can find the allowed energy levels by solving the equation -((ħ^2)/(2m)) * (∂^2ψ/∂x^2) = Eψ subject to the boundary conditions.

The solutions to this equation will give us the energy eigenvalues for the electron in the box. The energy levels will be quantized, meaning they will take on discrete values. The lowest energy level (the ground state) will have the lowest energy, and the energy levels will increase as we move to higher energy states.

To find the specific energy values, we need to solve the differential equation and apply the boundary conditions. The solutions will be in the form of a wavefunction ψ(x), and the corresponding energy levels E will be determined by the solutions.

Note: The specific solutions and energy levels will depend on the chosen unit system, such as the units used for the Planck constant (ħ) and the mass of the electron (m).