Lectures22-24

Lectures22-24 - CHEM 350 and 24 March 1 3 and 5 2010 •...

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Unformatted text preview: CHEM 350 Lectures 22, 23, and 24, March 1, 3, and 5 2010 • Nuclear Spin States of the H 2 Molecule : see Sec. 14.5.3 in text – the H atom has 1 proton, spin–1 / 2, for which the corresponding spin states can be referred to as ‘spin–up’ and ‘spin–down’; we commonly represent these two spin states via the symbols ↑ , ↓ – the H 2 molecule has 2 protons: I a = I b = 1 / 2 – the total nuclear spin for an H 2 molecule is thus: I = I a + I b ,I a + I b- 1 , ··· , | I a- I b | = 1 , ↑↑ 1 √ 2 ( ↑↓ + ↓↑ ) ↓↓ : symmetric to nuclear interchange (spin triplet) 1 √ 2 ( ↑↓ - ↓↑ ) : antisymmetric to nuclear interchange (spin singlet) We refer to the triplet nuclear spin states of H 2 as ortho-hydrogen (oH 2 ) and the singlet nuclear spin state of H 2 as para-hydrogen (pH 2 ). Ortho means ”high spin”. • Pauli showed that the wavefunction for a collection of equivalent fermions must be antisym- metric to the exchange of any two of them, while the wavefunction for a collection of equivalent bosons must be symmetric to the exchange of any two of them. • Application to H 2 : 2 equivalent electrons and 2 equivalent protons requires that the overall (or total) wavefunction, which we can write in the form ψ total = ψ total ψ nuc . spin must be ANTISYMMETRIC under the exchange of the two protons, i.e., H 2 ( 1 Σ + g ) → ψ el even, ( ψ vib even, ψ trans even), ψ rot ( j ) → (- 1) j ψ rot ( j ) therefore, ψ total → (- 1) j ψ total is even for j even, and odd for j odd. Now, as ψ total (2 , 1) =- ψ total (1 , 2) for fermions (such as the proton), we may deduce that: – H 2 molecules in states with j odd must combine with the symmetric triplet of nuclear spin states – H 2 molecules in states with j even must combine with the antisymmetric singlet nuclear spin state • In the absence of an externally applied magnetic field, the nuclear spin states are strictly degenerate, so that the j = odd rotational states of H 2 have a nuclear spin statistical wieighting of 3, while the j = even rotational states have a weighting of 1. This nuclear spin statistical weighting has a profound effect both on the spectroscopy and low temperature thermodynamics...
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This note was uploaded on 02/28/2011 for the course CHEM 350 taught by Professor Prof.djasd during the Winter '10 term at Waterloo.

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Lectures22-24 - CHEM 350 and 24 March 1 3 and 5 2010 •...

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