Reference no: EM13960418

The Bohr atom derived from de Broglie'.r relation. Here is another development of Bohr's results for hydrogen. based directly onthe de Broglie relation. If a de Broglie wavelength can be associated with an electron in orbit. then it seems reasonable to suppose that the circumference of an orbit be equal to an integral number of wavelengths. Otherwise (one might argue) the electron would interfere destructively with itself. Leaving to one side the mongrel-like nature of this argument (which employs both words like "orbit" and words like "wavelength"). apply it to re derive Bohr's results for hydrogen using the following outline or some other method.

(a) Calculate the speed. and hence the magnitude of the momentum. of an electron in a circular orbit in a hydrogen.

(b) Use the de Broglie relation to convert momentum to wavelength.

(c) Demand that the circumference of the circular orbit be equal to an integral number of de Broglie wavelengths.

(d) Solve for the radii of permitted orbits and calculate the permitted energies. Check that they conform to Bohr's results. as veri?ed by experiments.

(e) As an alternative method. omit all mention of forces and simply use the de Broglie relation and the condition of integer number of wavelengths in a circumference to show directly that the angular momentum rp (for the circular case) equals an integer times It/Zn'. This is just the condition that we showed earlier leads most simply to the Bohr results. (You should keep in mind the fact. noted in Section