(a) calculate the probability that an electron in the ground state of a hydrogen
ID: 1897234 • Letter: #
Question
(a) calculate the probability that an electron in the ground state of a hydrogen atom will be found at a greater distance from the nucleus than would be allowed classically.
(b) show that the uncertainty principle for delta x and delta p(sub x) is obeyed by an electron in the ground state of a hydrogen atom.
(c) make sketches of the regions in the x, z plane where the probability of finding an electron in a hydrogen atom ground state is large. Repeat for the 2S; the 2P, m=0; and the 2P, m=1 states.
Explanation / Answer
The electron in hydrogren has a binding energy of -13.6 eV. This is the energy it would take to remove the electron from the proton and place it at rest far away. This you know. This energy cost is not purely potential energy. The potential energy of the electron in the ground state of hydrogen is 2*-13.6 eV = -27.2 eV. The kinetic energy of the electron is + 13.6 eV. The total energy is +13.6 eV - 27.2 eV = -13.6 eV, which is the result you know. The fact that this energy is negative means the electron is bound to the proton, and you must supply energy to liberate it. The meaning of the binding energy in more detail is as follows. You want to take the electron moving with a kinetic energy of 13.6 eV and with a potential energy of -27.2 eV and move it far away to a final state at rest. Far away means no final potential energy. At rest means no final kinetic energy. You can think of it as putting your initial kinetic energy towards liberating the electron, but even after you use your 13.6 eV of initial kinetic energy, you still need to pay an additional 13.6 eV to get the total energy to zero i.e. an electron at rest far away. Hope this helps clarify the meaning of the binding energy for your students. I don't know how much you want to get into this, but KE can't equal PE because KE is positive while PE is negative. What is true in hydrogren is that the absolute value of PE is twice KE as I said above.
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