Fund Quantum Mechanics Lect & HW Solutions 77

Fund Quantum Mechanics Lect & HW Solutions 77 -...

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5.2. THE HYDROGEN MOLECULE 59 general, it is just not worth the trouble for the electrons to stay away from the same position: that would reduce their uncertainty in position, increasing their uncertainty-demanded kinetic energy. Answer: The repulsive potential is the term e 2 4 πǫ 0 1 | vectorr 1 vectorr 2 | and when vectorr 1 = vectorr 2 , you are dividing by zero. 5.2.1.2 Solution hmola-b Question: Note that the total kinetic energy term is simply a multiple of the six-dimensional Laplacian operator. It treats all Cartesian position coordinates exactly the same, regardless of which direction or which electron it is. Is this still the case if other particles are involved? Answer: The kinetic energy of the two electrons is ¯ h 2 2 m e parenleftBigg 2 ∂x 2 1 + 2 ∂y 2 1 + 2 ∂z 2 1 + 2 ∂x 2 2 + 2 ∂y 2 2 + 2 ∂z 2 z parenrightBigg . and the terms within the parentheses are the six-dimensional Laplacian. But if other particles
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Unformatted text preview: would be involved, they would have a diFerent value of ¯ h 2 / 2 m and the total operator would no longer be the normal Laplacian. All else being the same, heavier particles would have less kinetic energy. But, of course, all else is usually not the same. ±or example, the atoms in noble gases have the same kinetic energy at the same temperature regardless of atom mass. 5.2.2 Initial approximation to the lowest energy state 5.2.2.1 Solution hmolb-a Question: If electron 2 does not aFect where electron 1 is likely to be, how would a grey-scale picture of the probability of ²nding electron 1 look? Answer: It would be a blob around the left proton, exactly like the one of a lone hydrogen atom at that position....
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