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Particle in a Sphere

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  • Particle on a sphere is one out of the two models that describe rotational motion. A single particle travels on the surface of the sphere. Unlike particle in a box, the particle on a sphere requires angular momentum, \(J\).


    There is a vector that contains direction of the axis of rotation. The magnitude of angular momentum of the particle that travels around the sphere can be defined as:

    \[J = pr\]


    • \(p\) is linear momentum is the result of the mass and velocity of object (p=mv)
    • \(r\) is the radius of the sphere

    The faster a particle travels in a sphere the higher the angular momentum. In other words, if we increase the velocity of a particle we get an increase in angular momentum. Therefore this required stronger torque to bring the particle to stop. Particle of mass is not restrict to move anywhere on the surface of the sphere radius. The potential energy of the particle on a sphere is zero because the particle can travel anywhere on the surface of the sphere without a preference in location; the particle on the sphere is infinity. Furthermore, the wave-function needs to satisfy two cyclic boundary conditions which are passing over the poles and around the equator of the sphere surrounding the central point. Using the Schrödinger equation we are able to find the energy of the particle:

    \[E = l (l +1) (h/2?) 2(1/2I) \]

    with \(l = 0, 1, 2, 3, …\)

    We also know that the energy of the rotation of the particle is related to the classical angular momentum:

    \[E = \dfrac{J^2}{2I}\]

    \(I\) is the moment of inertia of the particle; heavy mass in a large radius path has a large \(I\). Because energy is quantized we can assume that these two equations can be compared with each other. Therefore the magnitude of the angular momentum is also limited to the values:

    \[J = \sqrt{L (L+1)} \hbar /2\]

    \(L\) is the orbital angular momentum quantum number.

    Considering motions in three dimensions, J has three components \(J_x\), \(J_y\), and \(J_z\), along x, y, and z – axis. The angular momentums of z-axis are quantized and have the values as \(J_z = m_l (h/2)\) with \(m_l =l, …, 1, 0, -1, …, - l\) Where \(m_l\) is the magnetic quantum number. The value of ml is restricted because of two cyclic boundary conditions, such that for ml equal to l there are 2l +1


    • Atkins, Peter and de Paula, Julio. Physical Chemistry for the Life Sciences. New York, N.Y.: W. H. Freeman Company, 2006. (361-362). Stenphen Berry, and Stuart A. Rosss, John Ross. Physical Chemistry. John Wiley and Sons 1980, R. (118-119).


    • An Nguyen