Final answer:
The minimum energy to excite an electron from the ground state in a one-dimensional box is 3/4 of the ground state energy. However, all given options neglect the factor of 3 and are incorrect, yet option (b) is closest and refers to the energy of the first excited state itself.
Step-by-step explanation:
The minimum energy required to excite an electron from the ground state in a one-dimensional box model can be calculated using the energy difference between the first excited state and the ground state. This model falls under the principle of quantum mechanics, where energy levels are quantized and given by the formula En = n2h2/8mL2, where En is the energy of the state with quantum number n, h is Planck's constant, m is the mass of the electron, and L is the width of the box.
For the ground state, n = 1, and for the first excited state, n = 2. Therefore, the energy required to go from n = 1 to n = 2 is E2 - E1 = (4h2/8mL2) - (h2/8mL2) = 3h2/8mL2. This simplifies to 3/4 of the ground state energy. After converting the ground state energy into the mentioned units, we can find the numeric answer, which is not given here as we are not provided with numeric values for the ground state energy or the width of the box in the question. The options provided all neglect the factor of 3, thereby all of them being incorrect. If one needs to choose, the closest option to the concept would be (b) ℎ2/2mL2, which corresponds to the energy of the first excited state itself, not the difference we are seeking.