^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

^{3}

^{*}

The spinning period for a free electron and the periods of spin and orbital motion of the electron in an atomic state have been calculated. We have shown that for a free electron the spinning period is:
*(T*
_{s}
*)*
_{free}=1.9×10
^{-20}s. But in the atomic case we show that, both the spin and the orbital periods depend on the quantum numbers
*n, m*
_{l}
*, m*
_{s} and the effective Landé-g factor,
*g*
^{*} which is a function of the quantum number
* l* of the atomic state
given in Dirac notation. We have also calculated these periods for the ground state and some excited states—hydrogen and hydrogen-like atoms. For atomic states the approximate values of spinning period are
and the related orbital periods are:
*(T*
_{0}
*)*
_{atomic}=(10
^{-16}-10
^{-15})s. Therefore atto-second processes which are related to the pulse of 10
^{-18} s will filter the orbital motion of the electron but will be long enough to detect the details of the spin motion, such as flip-flops.

To calculate the periods of spin and orbital motions of an electron in an atomic state

where

To calculate the spin period of an electron, we will use the magnetic top model which was first introduced by Barut et al. [

From Equation (1) the z-component of magnetic moment associated with the spinning motion is:

To proceed further, we calculate the intrinsic magnetic moment of electron with a semiclassical, magnetic top model which was first introduced by Barut et al. [

In the magnetic top model, the spin angular momentum of electron is produced by the spinning of the electronic charge (−e) which is assumed to be uniformly distributed inside a sphere of a radius R. We denote the spin angular frequency of the rotating charged sphere by

In the presence of the magnetic field

If we compare Equation (2) and Equation (4) we can write:

where

Let us consider the equatorial velocity of this spinning sphere,

Which defines the radius of electron as below:

For a free electron

Substitution of Equation (8) in Equation (5b) gives us the spinning period for a free electron:

which is in good agreement with the semiclassical calculation of Olszewski [

For an electron in an atom, we cannot calculate the radius directly from Equation (7), because we need to know the effective values of

In the following section we find an expression for the period of orbital motion,

When we take the ratio of the periods given in Equation (5b) and Equation (14), we find:

Substituting

It is known that when there is no quantum entanglement, for a free electron, the Landé-g factor is equal to 2. For an electron in an atom the Landé-g factor is given by:

which varies is in range of

If we calculate the effective g-factor,

spinning period of electron in

From Equation (1) the z-component of the total magnetic moment is:

Spinning periods | Periods of orbital motion |
---|---|

where

Now we find another expression for

If we compare Equation (12) and Equation (13) we write:

where we replace

Now we can find the values of

With these replacements Equation (14) becomes:

We note that the quantum number (l) gets involved through the effective Lande-g factor,

For example, for the ground state of hydrogen atom

Similarly for the state

where we put:

Authors from Miami University acknowledge financial support from the National Science Foundation (Grant No. NSF-PHY-1309571).

ZiyaSaglam,MesudeSaglam,S. BurcinBayram,TimHorton, (2015) The Spinning Period of a Free Electron and the Periods of Spin and Orbital Motions of Electron in Atomic States. Journal of Modern Physics,06,2239-2243. doi: 10.4236/jmp.2015.615229

Let us denote the uniform charge density by

where

The current element

The magnetic moment element of this band current will be:

Integrating over the spherical shell gives us the magnetic moment of this shell,

If we substitude

Substituting (A-I) in (A-VI) we find: