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Since the original \(\Psi _{P_{+1}}\) and \(\Psi _{P_{+1}}\) were both solutions of the Schrödinger wave equation, their combinations are also solutions, and so we can visualize atomic orbitals as shapes along the x,y,z axes. All three quantum numbers influence the ultimate shape.
31 Ιαν 2024 · A complete solution to the Schrödinger equation, both the three-dimensional wavefunction and energy, includes a set of three quantum numbers (n, l, ml). The wavefunction describes what we know as an atomic orbital; it defines the region in space where the electron is located.
As shown in Figure \(\PageIndex{5}\), the phase of the wave function is positive for the two lobes of the \(dz^2\) orbital that lie along the z axis, whereas the phase of the wave function is negative for the doughnut of electron density in the xy plane.
Any disturbance that complies with the wave equation can propagate as a wave moving along the x-axis with a wave speed v. It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.
Wave functions, and thus orbitals, are functions of three coordinates. One option for visualization is to hold two variables constants and plot the variation of ψ with the third variable. The wave function plot shown at the bottom of this page sets y = z = 0 and plots how ψ varies with x.
11 Ιαν 2024 · The vertical axis measures the displacement of the medium from the equilibrium point, which in the case of the red dot on the spring coil for the longitudinal wave in Figure 1.2.3 is the center of the horizontal dotted red lines.
The hydrogen atom Hamiltonian is by now familiar to you. You have found the bound state spectrum in more than one way and learned about the large degeneracy that exists for all states except the ground state. We will call the hydrogen atom Hamiltonian H(0) and it is given by p2 e2. H(0) = − . (2.1.1) 2m r.
