Answer to Question #167989 in Quantum Mechanics for Sherine Horo

Question

Show that the orbit nearest to the nucleus in the hydrogen atom accommodates one electron waves. Given that the energy of the electron in the orbit is 13.6 eV, h=6.625x10-34Js, m=9.11x10-31kg, Bohr radius r1 =5.292x10-11 m.

Answer 1

The angular momentum for the electron in the first Bohr orbit is

$$mvr_0=\frac{h}{2\pi}.$$

Rearrange and notice there is de Broglie wave:

$$2\pi r_0=\frac h{mv}=\lambda.$$

Substitute the values and get

$$\lambda=2\pi r_0=333\text{ pm}.$$

Thus, the orbit nearest to the nucleus in the hydrogen atom accommodates one electron wave.