Fund Quantum Mechanics Lect & HW Solutions 73

Fund Quantum Mechanics Lect & HW Solutions 73 -...

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Unformatted text preview: 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 x ( x ) d x , which was given to be 3 / 6. So the final expectation energy is ( E ) = h 2 10 2 m 2 versus h 2 2 2 m 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( x/ ) are simply different functions. While they may have superficial resemblance, if you scale each to unit height by taking...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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