Upon what definitions do we base the claim that the and states of equations are related to x and y just as is to .
It is proven by converting Cartesian to spherical polar coordinates.
A polar coordinate is one of two numbers that identify a point in a plane based on its distance from a fixed point on a line and the angle that line makes with the fixed line.
States and are of the same shapes as and they are all equivalent. The only difference between these states is the orientation of coordinates. These wave functions are defined based on spherical polar coordinates. .
The transformation between Cartesian and spherical polar coordinates is
The wave functions of the 2px, 2py and 2pz states are given as
Hence, from this its prove that the states 2px, 2py and 2pz depends on angles and which are the same as that described in Cartesian coordinates for X, Y, and Z .
The bonding of silicon in molecules and solids is qualitatively the same as that of carbon. Silicon atomic states become molecular states analogous to those in Figure 10.14. and in a solid, these effectively form the valence and conduction bands. Which of silicon's atomic states are the relevant ones, and which molecular state corresponds to which band?
As we see in Figures 10.23, in a one dimensional crystal of finite wells, top of the band states closely resemble infinite well states. In fact, the famous particle in a box energy formula gives a fair value for the energies of the band to which they belong. (a) If for n in that formula you use the number of anitnodes in the whole function, what would you use for the box length L? (b) If, instead, the n in the formula were taken to refer to band n, could you still use the formula? If so, what would you use for L? (c) Explain why the energies in a band do or do not depend on the size of the crystal as a whole.
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