The equation, lmatter = h/mv, is due to deBroglie and expresses the fact that particles
can have wave-like properties. Here lmatter is the wavelength (in meters) of the matter-wave where
Planck's constant, h = 6.626x10-34 J-s ; m is the mass of the particle (in kg) ;
and v is the velocity (in m/s).
Regular objects (like a baseball) have masses of the order of 1 kilogram so that their lmatter is much
smaller than the size of the object itselft. Thus we don't notice the wave-like properties of ordinary
matter. For very small objects like electrons or protons whose masses are more like 10-30 kg, we can
get lmatter of the order of the "size" of the electron itself and so this wave-like property becomes
very noticeable and important.
Now recall Figure 6.20 on page 204 of Brown-Lemay Bursten which shows the electron density of
the 1s, 2s, and 3s orbitals of the hydrogen atom. This is the picture of the 3s wavefunction vs r with its
two nodes
If we square the above we get the electron density which is always non-negative:
We can think of this wave function as a standing matter wave by analogy to a light wave.
Recall that a light wave is cosine function that looks like the following:
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