Help with b through f What is the Hamiltonian (of the electrons) for He? Name or

Question

Help with b through f

What is the Hamiltonian (of the electrons) for He? Name or describe the individual terms. What is the electronic configuration of He in its ground state? In the ground state, the electrons couple to form a joint spin state |x> that can be written as 1/squareroot 2 (| - |). Explicitly determine the symmetry of this spin state and explain why |x> must be this particular state. Optional Bonus Problem: Now consider the first excited state of He. What orbitals do the electrons occupy (list their quantum numbers) and what are the possible joint spin states of these electrons? The exact ground state energy of He is -79.5 eV. With this knowledge, calculate the exact ionization energy for forming He^+ (i.e. the energy that it costs to remove one electron from He). Optional Bonus Problem: Is an atom of the isotope 3 2 He a boson or a fermion or neither? Briefly justify your answer.

Explanation / Answer

(b) In ground state

1s2

(c)

The total wavefunction for the electrons will be anti-symmetric since they are fermions and they must obey the Pauli exclusion principle.This forces the space part of the wavefunction to be anti-symmetric. The wavefunction for the electrons can be written as the product of the space and spin parts of the wavefunction.An anti-symmetric space wavefunction for the two electrons implies a larger average distance between them than a symmetric function of the same type. The probability is the square of the wavefunction, and from a simple functional point of view, the square of an antisymmetric function must go to zero at the origin. So in general, the probability for small separations of the two electrons is smaller than for a symmetric space wavefunction.

(d) 1s12s1.

Possible joint spin states

both down

both up

up and down (arrows towards each other)

up and down (arrows away from each other)

(e)

The helium ground state consists of two identical 1s electrons. The energy required to remove one of them is the highest ionization energy of any atom in the periodic table: 24.6 electron volts.

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