Question

Quantum chemistry help(not sure if I should post it in the physics section instead)

Answer 1

prove that the 1s orbita for a one electron hydrogen atom give by \[\psi(x,y,z)=e ^{\frac{ -r }{ a }}\] is the solution to the TISE, where\[r=\sqrt{x ^{2}+y ^{2}+z ^{2}} \] and a is the born radius of the hydrogen atom.
Then, relate a and E, the ground state energy of hydrogen atom, to natural constants, by using matching r dependence i.e. given that A/r + B = C/r + D, require A=C and B=D

Answer 2

I got the answer to the first part, so the wavefunction is a solution par is proven

Answer 3

However, I cannot find what is the matching r depedence, the ground state energy eigenvalue does have an 1/r term and a constant term, but what would the RHS of my equation be. So I know what A and B are from the energy eigenvalue but where should I get C and D?

Answer 4

@whpalmer4 , pls help

Answer 5

@sweetburger , any idea?

Answer 6

@pooja195 , pls help me, now progress posted below

Answer 7

I just learned a new equation called virial theorem which states, in the context of one electron hydrogen atom, that 2T=-V, where T and V are kinetic energy and potential energy respectively

Answer 8

KE is \[-\frac{ \hbar ^{2} }{ 2m }*\left( -\frac{ 1 }{ 2ar } +\frac{ 1 }{ a ^{2} }\right)\]

Answer 9

hbar is h/2pi, m is mass of a electron, a and r are defined as before

Answer 10

PE is just the usual Columbic potential

Answer 11

If I plug those into the virial theorem, and let r=a since I want to solve for bohr radius, I got a negative number. Negative radius?!