To calculate the lowest three energy levels for an electron confined to a one-dimensional box, we can use the formula:
En = (n^2 * h^2) / (8mL^2)
where En is the energy level, n is the quantum number (1, 2, 3...), h is the Planck's constant (6.626 x 10^-34 m^2 kg / s), m is the mass of the electron (9.11 x 10^-31 kg), and L is the length of the box (1.50 x 10^-9 m).
For n = 1:
E1 = (1^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E1 = 1.941 x 10^-17 J
For n = 2:
E2 = (2^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E2 = 7.765 x 10^-17 J
For n = 3:
E3 = (3^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E3 = 1.739 x 10^-16 J
Now, to calculate the wavelength of light necessary to promote the electron from the ground state (n = 1) to the first excited state (n = 2), we can use the formula:
螖E = E2 - E1
螖E = (7.765 x 10^-17 J) - (1.941 x 10^-17 J)
螖E = 5.824 x 10^-17 J
Now, we can use the formula E = hc / 位 to calculate the wavelength of light:
螖E = hc / 位
位 = hc / 螖E
位 = (6.626 x 10^-34 * 3 x 10^8) / (5.824 x 10^-17)
位 = 1.02 x 10^-6 m
位 = 1020 nm
Therefore, the wavelength of light necessary to promote the electron from the ground state to the first excited state is 1020 nm.