Consider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How would it differ from the quantum probability?

Consider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How would it differ from the quantum probability?