Question

Consider a trial wavefunction for an electron in H atom in the following form •(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! for a>0 A nEN ne-andc= +1 Integration of function f(r, 0,6) in spherical coordinates: $*$*$* $(1,0, 6)r? sin ødødødr f(,0,)ra sin døddr

Answer 1

And we have to consider that a toial wave function for an election in the atoms in the following form $(8), seº1 Then it is toial wave function Let's first netralize for this wave function is 0(8) - Are-os 0<x< Here a represents Normalization constant, Such that Søer) Ø(8) der-1 all In this can we sub the values Aº $55 giesmes. gising dni do dø = 1 In above cauation do, 0(0) can be substitude that means $(8) = A.8.628 do : oʻsino do do do by using Then solve Gamma the above cauation Integrals, we get

2 UTA" JW84.23.20do si And again using gomma Integrals LT A ] - (1 - JF, :) 3) GTA 5 = 1 (41 = 4*3*2x1 = 34] To get it becomes À value the 4t is go to another side and divided, then we get > A = (3x) Fun -0 24 x 47 Now, for <VIHW > Now we can solve it, by using Hamiltonian of H-atom H 42 - Ke au Ta en til ) de te amo

Now, | < > : [이 > + 이응.19] - [이 sin Estress afrasasay + IRE - Saselfgs + se gs3 ] - amuey as and 4TTA Ho - 28 c6 2 | 나 CS GWA re? 3! - Am E () - Put value of a ] : ) QT gㅆ 4 · <> [ ]

Horce the solution can be found