9 z e2 9 z 2 e2 the one body operator h0 gives

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Unformatted text preview: ÅÅÅÅÅÅÅ … 1 s 1 s] + 2 Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] - Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 2 s 1 s] r12 r12 r12 2 2 2 (e) We use the perturbation theory to work out the binding energy up to the first order in the Coulomb repulsion. We make use of the identities ¶ 1 r< l ÅÅÅÅÅÅÅ = ‚ ÅÅÅÅÅÅÅÅ1ÅÅ Pl Hcos q12 L Å r12 r > l+Å l= 0 and 4p Pl Hcos q12 L = ÅÅÅÅÅÅÅÅÅ1Å ⁄lm=-l Yl m* HW1 L Yl m HW2 L. ÅÅ 2 l+ The radial wave functions are R1 s HrL = a-3ê2 2 e-rêa r R2 s HrL = a-3ê2 ÅÅÅÅ1ÅÅÅÅÅ H1 - ÅÅÅÅÅaÅ L e-rê2 a Å è!!! 2Å 2 è !!! 6 R2 p HrL = a-3ê2 ÅÅÅÅÅÅÅÅÅÅ ÅÅrÅÅ e-rê2 a 12 a Here, a = aB ê Z , where aB = —2 ê Hme e2 L. ü 1 s 2 2 s configuration We first study the 1 s2 2 s configuration. Z e2 Z 2 e2 The one-body operator H0 gives simply E1 s + E1 s + E 2 s = ÅÅÅÅÅ9Å ÅÅÅÅÅaÅÅÅÅÅ = ÅÅÅÅÅ9Å ÅÅÅÅÅÅÅÅÅÅÅÅ . 8Å 8Å a BÅ Now we calculate the Coulomb repulsion terms. The first term is e2 Y1 s 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 s] r12 e = a-6 Ÿ d x 1 Ÿ d x 2 ÅÅÅÅÅÅÅ H2 e-r1 êa Y0 0 HW1 LL H2 e-r2 êa Y0 0 HW2 LL r12 Using the above identities, we find only l = m = 0 contributes, e2 Y1 s 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 s] r12 Ø Ø 2 2 2 1 = e2 a-6 Ÿ d x 1 Ÿ d x 2 ÅÅÅÅÅÅ 4 p Y0 0 HW1 L Y0 0 HW2 L r> Ø Ø H2 e-r1 êa Y0 0 HW1 LL H2 e-r2 êa Y0 0 HW2 LL 1 = e2 a-6 Ÿ r1 2 d r1 Ÿ r2 2 d r2 ÅÅÅÅÅÅ 16 e-2 r1 êa e-2 r2 êa r> 2 In[2]:= Out[2]= 2 1 2 IntegrateAIntegrateA16 r1 2 E-2 r1 êa r2 2 E-2 r2 êa ÅÅÅÅÅÅ , 8r2 , 0, r1 <, Assumptions Ø a > 0E, r1 8r1 , 0, ¶<, Assumptions Ø a > 0E 5 a5 ÅÅÅÅÅÅÅÅÅÅ 8 e 5e Hence Y1 s 1 s … ÅÅÅÅÅÅÅ … 1 s 1 s] = ÅÅÅÅ ÅÅÅÅÅÅ . r12 8a 2 2 e e r Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] = a-6 Ÿ d x 1 Ÿ d x 2 ÅÅÅÅÅÅÅ H2 e-r1 êa Y0 0 Hq1 , f1 LL I ÅÅÅÅ1ÅÅÅÅÅ H1 - ÅÅÅÅ2ÅaÅ L e-r2 ê2 a Y0 0 Hq2 , f2 LM Å è!!! r12 r12 2Å 2 Again only l = m = 0 contributes, e2 Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] r12 The second term is Ø 2 Ø Ø Ø 2 2 2 2 2 HW6.nb 6 e 5e Hence Y1 s 1 s … ÅÅÅÅÅÅÅ … 1 s 1 s] = ÅÅÅÅ ÅÅÅÅÅÅ . r12 8a 2 2 e e r Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] = a-6 Ÿ d x 1 Ÿ d x 2 ...
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This note was uploaded on 01/17/2014 for the course PHYSICS 221b taught by Professor Staff during the Winter '08 term at University of California, Berkeley.

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