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# Particle in a ring

1. 1. References
2. 2. Contributors

One type of rotational motion in quantum mechanics is a particle in a ring. An important aspect of this is the angular momentum J which includes a vector with a direction that shows axis of rotation1. The particle’s magnitude of angular momentum that is traveling along a circular path of radius r is classified as J=pr where p is the linear momentum at any moment.

When the particle of mass m travels in the horizontal radius r, the particle has purely kinetic energy since potential energy is constant and is set to zero everywhere. The energy with regards of angular momentum can be expressed as:

E = J2z/2mr2 (J = angular momentum in z-axis and mr2 is the particle's moment of inertia I on the z-axis).

A particle has a moment of inertia I when traveling along a circular path. I is defined by m (mass) multiplied by r2 (radius squared). The heavier particle in the top picture has a large moment of inertia on the central point while the lighter particle in the lower picture has a smaller moment of inertia while traveling on the path of the same radius1. For the heavier particle, the I is large and therefore, the particle's energy can be expressed by:

E = J2z/2I

We then use the de Broglie equation to quantize the energy of rotation. This is done by expressing the angular momentum in wavelengths:

Jz = pr = hr/λ

where p is the linear momentum and h is Planck's constant (6.626 x 10-34 Js). It can also be written:

λ = h/p

With this equation, de Broglie postulated that there is a wave correlated with the electron via wavelength. He had done this to explain Bohr's model of the Hydrogen atom, in which the electron is only allowed permitted to orbit from the nucleus at certain distances.2

### References

1. Atkins, Peter, and Julio de Paula. Physical Chemistry for the Life Sciences. New York. Oxford University Press. 2006, p 358.
2. http://www.hull.ac.uk/chemistry/spectroscopy/deBroglie_wave.php?res=high&plotType=particle%20on%20a%20ring

### Contributors

• Stephanie Montenegro (UCD)