Schrödinger wave equation for hydrogen atom pdf
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Sections starting with “Solution of the hydrogen radial wavefunction”, andNote 2, · The hydrogen atom wavefunctions. If we can solve for, in principle we know everything there is to know about the hydrogen atom. For Hydrogen-like atoms (He+ or Li++) Replace e2 with Ze2 (Z is the atomic number) Use appropriate reduced mass μ Text reference: Quantum Mechanics for Scientists and Engineers. Angular momentum Intrinsic spin, Zeeman effect, Stern-Gerlach experiment Energy levels and spectroscopic notation, The solution of the Schrödinger equation (wave equation) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially This equation gives us the wave function for the electron in the hydrogen atom. , · Schrödinger Equation for the hydrogen atom Potential for hydrogen atom: (r) V(r) (r) E (r) 2m r ZeV(r)h r r r r r ∴− ∇ψ + ψ = ψ πε =− Use polar Since the proton is much more massive than the electron, we will assume throughout this chapter that the reduced mass equals the electron mass and the proton is located at the , · Text reference: Quantum Mechanics for Scientists and Engineers. Sections starting with “Solution of the hydrogen radial wavefunction”, andNote: Section contains the complete mathematical details for solving the radial equation in the hydrogen atom problem We have all the eigenvalue/eigenvector equations, because the time independent Schrodinger equation is the eigenvalue/eigenvector equation for the Hamiltonian operator, i.e., the the eigenvalue/eigenvector equations are H fl flˆ> = E n fl flˆ>; L2 fl flˆ> = l(l+1)„h2 fl flˆ>; L z fl flˆ> = m„h Application of the Schrödinger Equation to the Hydrogen Atom The approximation of the potential energy of the electron-proton system is electrostatic: Rewrite the three-dimensional time-independent Schrödinger Equation. When we solved Schrödinger's equation in one dimension, we found that one quantum number was necessary to describe our systems Schrödinger Equation for the hydrogen atom Potential for hydrogen atom: (r) V(r) (r) E (r) 2m r ZeV(r)h r r r r r ∴− ∇ψ + ψ = ψ πε =− Use polar coordinates: x y z r P φ θ r s i n θ r cos θ Nucleus of = θ hydrogen atom` = φ θ = φ θ z r cos y r sin sin x r cos sinr L r r hydrogen atom is H; L2, and L z.