Question

Mvr = nh/2π, prove this equation.

a) Apply the principles of classical mechanics to derive the equation.
b) Utilize the angular momentum formula to prove the given equation.
c) Use the de Broglie wavelength equation to establish the relationship.
d) Apply the laws of quantum mechanics to derive the given equation.

Answer 1

The equation mvr = nh/2π is proven by establishing the relationship between centripetal and Coulomb forces, defining angular momentum, and applying Bohr's quantization condition to equate mvr with multiples of h/2π.

Step-by-step explanation:

The equation mvr = nh/2π is derived from the quantization of angular momentum in the Bohr model of the hydrogen atom. Here's how you can prove the equation following the provided instructions:

1. First, attribute the circular motion of the electron to the balance between Coulomb force and centripetal force.
2. Then, know that angular momentum L = mvr for a particle of mass m moving with speed v at a radius r from the origin.
3. The condition for quantization of angular momentum introduced by Bohr states that angular momentum can only take on discrete values, specifically multiples of h/2π, where h is Planck's constant. Hence, L = nh/2π for n being integers (1, 2, 3, ...).
4. By equalizing the expression for angular momentum mvr and the quantized angular momentum nh/2π, you establish the relationship mvr = nh/2π.

This equation was fundamental in allowing Bohr to calculate the energy levels of the hydrogen atom, thereby explaining its emission spectrum.