sp2 states and Dirac Notation

This post contains a video which describes and uses Dirac notation. It shows a calculation of the expectation values of x and y which explains how sp2 hybridization permits the construction of three states at 120 angles to each other. It also includes a contour plot, in figure 1, that shows the nature of the probability density for one of the three sp2 states. Each sp2 state is equivalent to the other two, but with its orientation rotated by 120 degrees. That is why each of the three in-plane sp2 states has the same amount of \(\psi_{2s}\) mixed in. Different amounts of \(\psi_{2x}\) and \(\psi_{2y}\) are mixed in to create the angles (orientations) you will see in this video.

 Figure 1. Contours of constant probability density are show for the x-oriented sp2 state.
The important part is the dark area in the right where the PD is large. Within that region is where the expectation value or \(\vec r\) lies.

    These states are essential to understanding the nature of the upper rows of the periodic table, the structures on which life is based, most or all bonding involving C, N and O (including \(H_{2}O\), the 2D semiconductor graphene...
   With these states we can understand the origin of many important molecular structures and compounds, and trace the nomenclature and rules of bonding, such as Lewis dot structures, back to their origins in the Schrodinger equation.


* At around 25:20 I should have said sp hybridization states are responsible for all bonding involving carbon, nitrogen and oxygen. (Not just sp2). sp2 is a particular example of sp hybridization, albeit, the most important one. In water, for example, O forms a different sp hybrid state, which combines with the H atom ground state to form the OH bond in water.

This next video shows integration: