Fund Quantum Mechanics Lect &amp; HW Solutions 93

# Fund Quantum Mechanics Lect &amp; HW Solutions 93 - 5.8...

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Unformatted text preview: 5.8. MATRIX FORMULATION 75 So there are two energy eigenvalues: E 1 = J − L and E 2 = J + L. In the first case, since according to the equations above ( J − E 1 ) a 1 − La 2 = 0 = ⇒ La 1 − La 2 = 0 it follows that a 1 and a 2 must be equal, producing the eigenfunction a 1 parenleftBig ψ S 1 + ψ S 2 parenrightBig = a 1 ( ψ l ψ r + ψ r ψ l ) and normalization shows that a 1 = 1 / √ 2, assuming you take it real and positive, (and as- suming that ψ l and ψ r are really adjusted to be orthonormal as assumed in this particular question). So the first energy eigenstate is eigenfunction: 1 √ 2 ( ψ l ψ r + ψ r ψ l ) eigenvalue: E 1 = J − L Similarly the second energy eigenstate is eigenfunction: 1 √ 2 ( ψ l ψ r − ψ r ψ l ) eigenvalue: E 2 = J + L. Comparing with section 5.3, the first eigenstate can be recognized as the ground state in which the nuclei share the electrons symmetrically. The second eigenstate is the excited-energy antisymmetric state in which the electrons share the electrons antisymmetrically. Other statesantisymmetric state in which the electrons share the electrons antisymmetrically....
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