Conservation of Angular Momentum in Bohr's Model of the One-Electron Atom

The conservation of angular momentum is a fundamental principle in physics. In this article, we will delve into the concept of angular momentum in the Bohr's model of the one-electron atom and explore how angular momentum is conserved within this theoretical framework.

Introduction to Angular Momentum

Angular momentum is a measure of the amount of rotational motion in an object. It is given by the cross product of the position vector and the linear momentum vector:

[ L r times p ]

To understand the conservation of angular momentum, we need to follow the mathematical derivation based on Newton's second law of motion.

Mathematical Derivation of Angular Momentum Conservation

To prove the conservation of angular momentum, let's start with the definition and apply the necessary mathematical principles.

The time derivative of angular momentum is given by:

[ frac{dL}{dt} frac{d(r times p)}{dt} ]

Using the product rule for differentiation, we can further break it down:

[ frac{dL}{dt} frac{dr}{dt} times p r times frac{dp}{dt} ]

Given that velocity is the time derivative of position, we have:

[ frac{dL}{dt} v times p r times frac{dp}{dt} ]

Substituting the definition of linear momentum, which is momentum as the product of mass and velocity:

[ p mv quad text{(where m is the mass of the particle)} ]

We get:

[ frac{dL}{dt} m(v times v) r times frac{dp}{dt} ]

Note that the cross product of any vector with itself is zero:

[ v times v 0 ]

This simplifies our expression to:

[ frac{dL}{dt} r times frac{dp}{dt} ]

By Newton's second law, force is equal to the mass times the acceleration:

[ F ma m frac{dv}{dt} ]

Substituting this into our equation:

[ frac{dL}{dt} r times F ]

By definition of torque, which is the cross product of the position vector and the force vector:

[ M r times F ]

Therefore, the change in angular momentum is:

[ frac{dL}{dt} M ]

To conclude, whenever the net torque ( M 0 ), the angular momentum ( L ) is conserved. Thus, angular momentum is conserved in the absence of an external torque.

Experimental Evidence and Bohr's Model of the One-Electron Atom

The Bohr model of the one-electron atom, which assumes that the electron moves in circular orbits around the nucleus, also supports the conservation of angular momentum. In this model, the radius of the electron's orbit is quantized, meaning it can only have certain discrete values. When the electron transitions between two orbits, it emits or absorbs a photon, which conserves both the energy and angular momentum of the system.

Therefore, according to Bohr's model, the angular momentum of the atom is conserved during these transitions. This is consistent with the fundamental principle that angular momentum is conserved in the absence of an external torque.

Conclusion

Angular momentum is a deeply rooted concept in physics that is mathematically proven to be conserved. In the context of the Bohr model of the one-electron atom, the conservation of angular momentum is an essential feature. This model provides a clear example of how angular momentum is conserved in the absence of an external torque.

For more detailed information and experiments related to angular momentum conservation, you can refer to the following sources:

Sacchetti, A. (2022). Why do scientists claim that conservation of angular momentum has never been violated? Bohr Model - Wikipedia

Feel free to explore these resources for a deeper understanding of the topic.