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Question: How does the energy difference between two energy level and the energy of the incoming photon match in order for the electron in the hydrogen atom to absorb and be excited to the next energy level? If the energy of the photon was greater than (but not great enough to excite the electron in the first to the third energy level for example) the energy difference between energy level 1 and 2, then that photon won't be absorb. So in exciting the electron to the next energy level, the energy of the incoming photon must be a single value that corresponds to each transition. However, in the photoelectric effect, in removing the electron, why isn't the energy of the photon limited to a distinct number that must match exactly with the work function? Instead, the photon's energy can be equal or GREATER than the work function. Because following the logic of the electron transition within the energy level, photon greater than the work function shouldn't be absorb at all.
Answer: In the first case of excitation of H atom, the energy of the incoming photon is smaller than the ionization energy of H atom. The photon can only excite the electron in H atom from lower energy state to higher energy state. Thus H atoms can only absorb photons with fixed energy. But if the energy of the incoming photon is larger than (not necessary equal to) the ionization energy of H atom (Ie), the electron in H atom can also be ionized, I.E. the electron is excited to a energy state with quantum number n = infinity. In the second case of photoelectric effect, the energy of the incoming photon must be large enough to ionize the metal atoms at the surface in order to knock off electrons from the metal surface. This is the photoelectric effect--ejection of electrons from metal surface by shining light on metal. The criterion of energy of the incoming photon is that it must be larger than the ionization energy. But it does not need to be a fixed value. The excess energy is converted into kinetic energy for the electron.
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