PS7-08-Solution - Chem 442 Homework Set#7 DUE Thursday 1...

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Chem 442 Homework Set #7 DUE: Thursday, November 13, 2008 1) McQ&S 7-20 () 22 01 23 4 2 0 0 2 0 0 2 2 11 12 1 2 ˆˆ ), 6 2 4 Harmonic oscillator wavefunctions. 1 E 2 2 sin E 8 ' )2 ' , 2 e dk b aH x H x x mdx n da bH H b x a nx aa nh ma e cH e H mr r r γ ψ ω π =− + = + = ⎛⎞ =+ ⎜⎟ ⎝⎠ = < < = = ∇ +∇ + = = = = = () ( ) () 0 0 2 1 0 1 2 where 1 represents a hydrogen-like wavefunction for electron 1 and 2 is a hydrogen-like wavefunction for electron 2. E4 ' c o s 2 Hy H e R nn e dH H er ψψ εθ = + ∇ − = = = 0 2 2 2 0 2 0 drogen atom wavefunctions 1 E c o s 2 Rigid Rotor wavefunctions=Y , E1 2 H m l R n eH H I JJ I με θ ψθφ = = = = Problem Set #7 - Solutions - 1 -
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2) McQ&S 7-22 () ( ) 10 1 0 1 0 /2 1 2 1 / ˆ Recall: 2 ˆ ,s i n 2 22 ˆ sin sin 1 2 sin 1 cos 2 2 sin 2 a a aa a EH an x Hb x a nx b bn x b n x Ed x a a ba n x Ex a ψψ π ψ ππ = ⎛⎞ =< < = ⎜⎟ ⎝⎠ == =+ ∫∫ d x d x 2 1 1 / 2 sin 2 sin a a Ea a n n n ba b E a =− + 3) McQ&S 7-23 1 0 1 3/2 0 / ,, 1 ,0 1 0 3/2 3/2 2 1 // 2 00 0 2 1 2/ 3 3 0 1 ˆ ˆ cos 11 = for the ground state ˆ cos sin cos sin ra nlm He r e a Ee e r e r d r e d e r d r a E εθ d d ε θθ φθ θ −− = = = =⋅ = = ∫∫∫ φ 0 0 because cos sin d 0 = 4)a) Using 2 2 cf cf Φ= + where ( 2 2 12 and ) f xLx f xLx = as a trial function, apply the variational treatment to calculate an upper bound to the lowest two energy states of a particle in a box.
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PS7-08-Solution - Chem 442 Homework Set#7 DUE Thursday 1...

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