which has the lowest ground state energy, an electron trapped in a one dimensional box of height 10^-6m or one with the length on 10^-6m
The ground state energy of an electron trapped in a one-dimensional box is given by the formula:
E = (n^2 * h^2) / (8 * m * L^2)
where:
E = energy
n = quantum number (1 for ground state)
h = Planck's constant (6.626 x 10^-34 J*s)
m = mass of electron (9.11 x 10^-31 kg)
L = length of the box
Since both the height and the length of the boxes are equal to 10^-6m, the only difference between the two scenarios is the dimension in which the electron is confined (height vs. length).
As the energy depends on the square of the length of the box in the denominator, the one-dimensional box with the length of 10^-6m will have a lower ground state energy compared to the box with the height of 10^-6m.