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HatomWavefunctionEPSoln

# HatomWavefunctionEPSoln - 00 8 from 0 to n and ¢ from 0 to...

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ME 501: Hydrogen Atom Wavefunction Example Problem The time-independent parts of the wavefunction If/ nfm (r,8, ¢) for the n = 2, I! = 1, me = + 1 state of the I hydrogen atom is given by r exp ( -r / 2a o ) . . If/211(r,8,¢) = - r;; I sm8 exp(+l¢) 8 1T as 2 o R (r)= r exp( -r / 2a o) 0211(8) = -~ 3 sin8 211 2.f6 a~12 81T The Bohr radius a o = (41T Co li 2 ) / (me e 2 ) = 0.52918 X 10- 10 m and the electron spin is neglected (Note: In these formulas the factor Co is the dielectric permittivity, not an energy). (a) Show that the wavefunction given above is normalized. Recall that for spherical coordinates dV = r2 sin8 drd8d¢ and that r can take on values from 0 to

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Unformatted text preview: 00, 8 from 0 to n, and ¢ from 0 to 2n. The following definite integral relation will be needed r n -bxdx n! x e =-b n + 1 (b) Calculate the expectation value of the z-component of angular momentum, (Lz)' where Lz,op = -iii aa¢, and the expectation value of the square of total angular momentum (L2), where L~p = _liZ [_._I-~(sin8~) + + a Z z ]. [Hint: recall that 1= sin 2 8 + cos 2 8. Also, little sm 8 a 8 a 8 sm 8 a ¢ or no integration is required in part (b), you can use the results of part (a)]. 2 II S 'i':tl 'f'l-ll cd V 1 tf~1} ~ R ~il P~I C{ l \ Sf ace. ~~I\ P~tI ~ 1 /J . 1./ / I . . . .-'....
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HatomWavefunctionEPSoln - 00 8 from 0 to n and ¢ from 0 to...

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