# Orbiting Electron Problem with Angular Momentum

If an electron can orbit a proton only in orbits having angular momentum h/(2`pi), 2 * h/(2`pi), 3 * h/(2`pi), ... , n * h/(2`pi), ..., then what is the radius of the closest possible orbit? How many deBroglie wavelengths are required to span the circumference of this orbit? What are the radii of the two next-closest orbits? What is the radius of the nth-closest orbit? What it is the total energy of each of these orbits? How many deBroglie wavelengths are required to span the circumference of each orbit?

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The electron's angular momentum is given by:

(1.1)

Where

Then the electron's speed is:

(1.2)

The orbit is stable if the centrifugal force is equal to the electrostatic attractive force:

(1.3)

Using (1.2) in (1.3) we get:

(1.4)

â€ƒ

Where is a constant known as Bohr's radius:

(1.5)

So the radius of the orbits is simply linear with :

De Broglie wavelength is given by:

(1.6)

The number of De-Broglie waves that fit in the n'th orbit is:

(1.7)

In the first orbit we have one wavelength, in the second orbit we have two wavelengths and so forth and so on and so forth.

â€ƒ

The energy is the sum of the kinetic energy and the (negative) potential energy of the electron:

(1.8)

Using (1.2) we get:

(1.9)

Using (1.4) we obtain:

(1.10)

Where the constant is the well known ground state energy ( ) of the hydrogen atom:

(1.11)

Thus, orbits energies are:

(1.12)

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