Phys 2D soln6

Phys 2D soln6 - 1 The /r2 term originates from the angular...

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1 The λ/r 2 term originates from the angular momentum of the electron moving through the Coulomb potential. U eff diverges as r 0 because 1 /r 2 goes to zero faster than the Coulomb term. The boundary conditions on this wavefunction are that it must go to zero at r = 0 and r = . This is the lowest energy state for = 1 because it only goes to zero at 0 and and nowhere in between. The principle quantum number for this state is n = 2. The probability density is symmetric about the z-axis and is invariant under rotations about the z-axis. See the figure below. 2 The allowed energies will all be multiples of E 0 π 2 ~ 2 / 2 mL 2 . The easiest thing to do is try different combinations of n 2 1 + n 2 2 + n 2 3 / 4 and see which 5 are the lowest. E n 1 n 2 n 3 deg. 9 E 0 / 4 1 1 1 1 3 E 0 1 1 2 1 17 E 0 / 4 1 1 3 1 21 E 0 / 4 1 2 1 2 2 1 1 6 E 0 1 1 4 3 2 1 2 1 2 2 3 Use the expectations values, h L 2 i = ~ 2 ( + 1) , h L z i = ~ m , and the constraint m = - ‘, - + 1
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Phys 2D soln6 - 1 The /r2 term originates from the angular...

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