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Unformatted text preview: 8-24 P1sr 4a3r2e2r afor hydrogen ground state, U r ke2ris potential energy (Z1) UU r P1sr dr 4ke2a3re2r adr 4ke2a3a22zezdzwhere z2rake2a 2 13.6 eV 27.2 eV.To find K, we note that KUE ke22a 13.6 eVso, Kke2a 13.6 eV. 8-25 The most probable distance is the value of rwhich maximizes the radial probability density P r rR r 2. Since P r is largest where rR r reaches its maximum, we look for the most probable distance by setting d rR r drequal to zero, using the functions R r from Table 8.4. For clarity, we measure distances in bohrs, so that rabecomes simply r, etc. Then for the 2sstate of hydrogen, the condition for a maximum is ddr2rr2er222r122rr2er2or 46rr2. There are two solutions, which may be found by completing the square to get r325or r35bohrs. Of these r355.236agives the largest value of P r , and so is the most probable distance. For the 2pstate of hydrogen, a similar analysis gives ddrr2er22r12r2er2with the obvious roots r(a minimum) and r4(a maximum). Thus, the most probable distance for the 2pstate is r4a, in agreement with the simple Bohr model. 8-26 The probabilities are found by integrating the radial probability density for each state, P(r), from rto r4a. For the 2sstate we find from Table 8.4 (with Z1for hydrogen) P2sr rR2sr 28a1ra22ra2er aand P8a1ra22ra2er adr4a....
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