Final answer:
To reach the second excited state (n=3), an additional 12.32 eV must be added to the particle's initial ground state energy. To reach the third excited state (n=4), an additional 23.10 eV is required.
Step-by-step explanation:
The energy levels of a particle in an infinite potential well are quantized and can be described by the following formula for the energy of the nth level: En = (n2 × h2) / (8 × m × L2), where h is Planck's constant, m is the mass of the particle, and L is the width of the well. Given that the energy of the ground state (n=1) is 1.54 eV, we can use the formula to find the energies of the second excited state (n=3) and the third excited state (n=4).
To find the amount of energy needed to go from the ground state to the third state, we need to calculate E3 and subtract E1 (ground state energy) from it.
Similarly, to find out the energy required to reach the fourth state, we calculate E4 and subtract E1. The factor n2 indicates that the energy is proportional to the square of the quantum number n. Therefore, E3 will be 9 times E1 and E4 will be 16 times E1.
For n=3: E3 = 9 × 1.54 eV = 13.86 eV, so the energy to be added is 13.86 eV - 1.54 eV = 12.32 eV.
For n=4: E4 = 16 × 1.54 eV = 24.64 eV, so the energy to be added is 24.64 eV - 1.54 eV = 23.10 eV.