Wave+functions


 * 1s orbital**
 * [[image:1s_vis.PNG]]

Figure 1a || The lowest energy level in an atom is the 1s energy level. The 1s level is usually depicted as a sphere around the nucleus. Why is it depicted this way? What does this depiction actually tell us about the electrons? The current model of the atom describes electrons as having a wave property. To better understand what this means let’s take a look at the wave function for the electron along the x-axis line which has been hi-lighted yellow in the figure to the left. ||


 * [[image:1s_wave.PNG]] || Figure 1b

Figure 1c ||

The wave function has its show in figure 1b, peaks at the nucleus. Such peaks are called the anti-nodes. The node is where the wave has value of zero. The wave does not actually have a value of zero anywhere in the picture, but as one moves farther and farther from the nucleus the value of the wave function gets closer and closer to zero so we say that it has a node at infinity. The wave function tells us the probability of finding an electron at that position. Where the square of the wave function is greater the likelihood of finding an electron is also greater. Based on this, an electron in the 1s orbital is most likely to be in or near the nucleus and is least likely to be very far away. This distribution can be represented two-dimensionally for the x-y plane as a circle that is dark at the center and gets lighter as you move towards the edges. The darker the region the more likely it is that an electron is there.

Figure 2a || The next energy level is the 2s energy level which is often depicted as a sphere within a sphere. To see what this means about the location of the electron we will start by again looking at just the linear wave function along the x-axis ||
 * 2s orbital**
 * [[image:2s_vis.PNG width="399" height="360"]]

This wave is more complicated than the wave function for the 1s orbital. It still has an anti-node at the nucleus and a node at infinity but it also has another node and anti-node between the nucleus and infinity. It is important to remember that the probability of finding an electron is the highest where the square of the wave function is the greatest. So it is more likely that the electrons will be located at one of the nodes (weather they are positive or negative) than anywhere else. The wave function does have a much larger absolute value at the nucleus that at the outer anti-node so the electrons in this orbital are still more likely to be at or near the nucleus than anywhere else. This wave function is only along the x-axis. Figure 2c represents the function in the x-y plane. As in the previous picture darker areas represent areas where the electrons are more likely to be.
 * [[image:2s_wave.PNG width="433" height="417"]] || Figure 2b ||
 * figure 2c ||  ||

<span style="font-family: Verdana,Geneva,sans-serif;">Imagining a three dimensional version of the wave function is more difficult. The picture below attempts to depict the three dimensional wave function with a corner removed.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 14.6667px;">The anti-nodes are location where the electrons are likely to be and the nodes are areas where the electrons are highly unlikely to be. So there is a spherical surface around the nucleus where electrons are not likely to be. This is called spherical nodal surface.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">**2p orbital** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">The next highest energy level is the 2p orbital. The 2p orbital actually contains 3 sub orbitals that each contain 2 electrons.
 * <span style="font-family: Verdana,Geneva,sans-serif;">[[image:2p_vis.PNG]] || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">This picture shows all 3 sub orbitals. One sub orbital includes the two regions on the x axis, a second orbital includes the two regions on the y axis, and the third sub orbital includes the two regions on the z axis.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">To analyze this shape we will start by looking at the sub orbital around the x-axis and again look at the wave function along the x axis. || Figure 3a <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">For the p x sub orbital their is a node at infiity and another at the y-z plane. Since the p orbital has a node at the nucleus the electrons in this orbital are more likely to be away from the nucleus than near it, especially when compared to the s orbitals. The picture below shows the three dimensional shape of the <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">px orbital. It also shows the nodal surface at the y-z plane.
 * [[image:2p_wave.PNG width="525" height="401"]] || Figure 3b ||
 * Figure 3c ||  ||

If you go back and look at the first picture of p orbitals you should be able to see that the p y orbital has a nodal surface at the z-x plane, and the p z orbital has a nodal surface at the x-y plane.