We normally think of the one-dimensional well as being horizontal. Suppose it is vertical; then the potential energy of the particle depends on x because of the presence of the gravitational field. Calculate the first-order correction to the zero-point energy, and evaluate it for an electron in a box on the surface of the Earth. Account for the result. Hint. The energy of the particle depends on its height as m g h, where g=9.81 m s⁻². Because g is so small, the energy correction is small; but it would be significant if the box were near a very massive star.
a) Δ E = m² g²/2²; Δ E ≈ 10⁻³⁸ J
b) Δ E = m g²/2²; Δ E ≈ 10⁻³⁸ J
c) Δ E = m² g²/²; Δ E ≈ 10⁻³⁸ J
d) Δ E = m g²/²; Δ E ≈ 10⁻³⁸ J