Lithium is the third-simplest element, with Z = 3 and three electrons. a) Let the three...

 

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Lithium is the third-simplest element, with Z = 3 and three electrons. a) Let the three electrons be in the single-particle hydrogenlike states n, n and nc. What is the formula for the energy of the state, neglecting the electron-electron interactions? Write the answer as a number in eV times the dependence on n, nB and no. b) What values of n, ng and nc are appropriate for the ground state of lithium? What is the corresponding energy, according to the previous formula? If you included the electron- electron repulsion, would this energy go up or down? c) In the ground state, what can we say about the spin quantum numbers of the three electrons? What is the net spin quantum number Stot (Stot = 0, 1/2, 1,...) of the three electrons? What about the land m quantum numbers for the three electrons? Give the total degeneracy of the ground state. d) Assuming the unpaired electron has spin up, write down the ground state as a superpo- sition of kets of the form In m n m nc m), with each spin given as m, t. Here we are suppressing possible values of land m for each electron. Hint: the result should obey the exchange postulate for any pair of electrons, so it will be a sum of several terms. Lithium is the third-simplest element, with Z = 3 and three electrons. a) Let the three electrons be in the single-particle hydrogenlike states n, n and nc. What is the formula for the energy of the state, neglecting the electron-electron interactions? Write the answer as a number in eV times the dependence on n, nB and no. b) What values of n, ng and nc are appropriate for the ground state of lithium? What is the corresponding energy, according to the previous formula? If you included the electron- electron repulsion, would this energy go up or down? c) In the ground state, what can we say about the spin quantum numbers of the three electrons? What is the net spin quantum number Stot (Stot = 0, 1/2, 1,...) of the three electrons? What about the land m quantum numbers for the three electrons? Give the total degeneracy of the ground state. d) Assuming the unpaired electron has spin up, write down the ground state as a superpo- sition of kets of the form In m n m nc m), with each spin given as m, t. Here we are suppressing possible values of land m for each electron. Hint: the result should obey the exchange postulate for any pair of electrons, so it will be a sum of several terms.