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Fund Quantum Mechanics Lect &amp; HW Solutions 73

# Fund Quantum Mechanics Lect &amp; HW Solutions 73 - 4.5...

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4.5. THE HYDROGEN MOLECULAR ION 55 Now evaluate the expectation energy: ( E ) = ( ax ( x ) | H | ax ( x ) ) = | a | 2 (Bigg x ( x ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ¯ h 2 2 m 2 ∂x 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x ( x ) )Bigg You can substitute in the value of | a | 2 from the normalization requirement above and apply the Hamiltonian on the function to its right: ( E ) = 30 5 ¯ h 2 m ( x ( x ) | 1 ) The inner product is by definition the integral integraltext 0 x ( x ) d x , which was given to be 3 / 6. So the final expectation energy is ( E ) = ¯ h 2 10 2 mℓ 2 versus ¯ h 2 π 2 2 mℓ 2 exact. The error in the approximation is only 1.3%! That is a surprisingly good result, since the parabola ax ( x ) and the sine a sin( πx/ℓ ) are simply different functions. While they may have superficial resemblance, if you scale each to unit height by taking a = 4 /ℓ 2 and a = 1, then the derivatives at x = 0 and are 4 /ℓ respectively π/ℓ , off by as much as 27%. If you go the next logical step, approximating the ground state with two functions as
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