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Chapter 6 Quantum Theory of the Hydrogen Atom. 6.1 Schrödinger's Equation for the Hydrogen Atom Today's lecture will be all math. Advice: grit your teeth and bear it. Now that we have discovered a "new" theory (quantum mechanics as exemplified by Schrödinger's equation) we ought to test it out on something.
Calculating Distances in Real Space. In Cartesian coordinates, the distance, r, between the points P and Q is given by Pythagoras' rule as: r2 = ( XP − XQ) 2 + ( YP − YQ) 2 + ( ZP − ZQ) 2 = Δ X2 + Δ Y2 + Δ Z2. In vector notation, the points P and Q may be represented by the vectors. P = XPi + YPj + ZPk.
15 kwi 2023 · The Lennard-Jones Potential is a mathematical model that explains interactions between two atoms at a certain distance. It consists of two terms that describe attractive and repulsive force between …
We can show that the angular part of the wave function produces the planar node in the xy plane when we convert it from spherical into Cartesian coordinates (Eq. 1.2.15). Equation 1.2.15 Angular part of the wave function of 2p z in cartesian coordinates. The function becomes zero when z=0.
ˆH(r, θ, φ)ψ(r, θ, φ) = Eψ(r, θ, φ) employs the same kinetic energy operator, ˆT, written in spherical coordinates. For the hydrogen atom, however, the distance, r, between the two particles can vary, unlike the diatomic molecule where the bond length was fixed, and the rigid rotor model was used.
6 THE HYDROGEN ATOM 2 Figure 1: The relationship between polar and Cartesian coordinates. The Laplacian operator, which in Cartesian coordinates takes the form ∇2 = ∂ 2 ∂x 2 + ∂ ∂y + ∂2 ∂z2, (9) is now written as ∇2 = 1 r 2 ∂ ∂r r2 ∂ ∂r! + 1 r sinθ ∂ ∂θ sinθ ∂ ∂θ! + 1 r2 sin2 θ ∂2 ∂φ2. (10) We now re ...
The simplest atom is hydrogen, consisting of a single proton as the nucleus about which a single electron moves. The electrostatic force attracting the electron to the proton depends only on the distance between the two particles.
