7. The quantum mechanical model of the hydrogen atom can be calculated in the Born-Oppenheimer approximation, in which t

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7 The Quantum Mechanical Model Of The Hydrogen Atom Can Be Calculated In The Born Oppenheimer Approximation In Which T 1 (88.96 KiB) Viewed 22 times

7. The quantum mechanical model of the hydrogen atom can be calculated in the Born-Oppenheimer approximation, in which the nucleus is treated as stationary and taken as the origin of a spherical polar coordinate system. The radial component of the time-independent Schrödinger equation (TISE) for spherically symmetric electron wavefunctions can be written as, h2 d + V(r) (r) = Eur) 2mr2 dr dr where r is the electron position relative to the nucleus, m is the mass of the electron, E is the electron energy, (r) is the wavefunction with associated probability density, p = 4712 413, and V(r) is the Coulomb potential given by -1 e2 V(r) = 4πέο where e is the magnitude of the electron charge. in die on) VID}() n) = [8] (a) Show that y(r) = e-ra is a solution to TISE for the πα, hydrogen atom, derive an expression for a, in terms of fundamental constants and show that the energy of this state is E = 8περβο: (b) Determine the most probable value for the radius for this wavefunction in terms of ao. What is the quantity ao ? (c) Another solution to the hydrogen atom TISE is 4(r) = 4 Veng (2-6) ) (2-6) e-T/280. For this state determine the energy and the values of the quantum numbers n, I and m. [4] [8]