Space Quantization and Atomic Dipole Moment

In quantum mechanics, we know the discreteness of the possible spatial orientations of the angular momentum of an atom (or another particle or system of particles, molecules) with respect to any arbitrarily selected axis (here z-axis).

Space quantization is manifested in that the projection Lz of the angular momentum L on Z-axis may only assume discrete values equal to an integer (0, 1, 2, …) or a half-integer (1/2, 3/2, 5/2, …) ml multiplied by the Planck constant h, Mz = mlh. (where ml is the magnetic quantum number) The other two projections of angular momentum Lx and Ly remain indeterminate, since, according to the main principle of quantum mechanics, only the magnitude of the angular momentum and one of its projections can simultaneously have exact values. For the orbital angular momentum, the quantity ml can assume the values 0, ±1, ±2, …, ±l, where l = 0, 1, 2, … determines the square of the momentum L (that is, its absolute magnitude): = l (l + 1)hbar square. For the total angular momentum J (orbital plus spin momentum), mj assumes values separated by unity— j to +j, where j determines the magnitude of the total momentum: Jsquare = j (j + 1)hbar square; it may be an integer or half-integer.

If an atom is placed in an external magnetic field B, then a well-defined direction in space—the direction of the field (which is taken as the z-axis)—appears. In this case, space quantization leads to quantization of the projection μB of the magnetic moment of the atom on the direction of the field, since the magnetic moment is proportional to the mechanical angular momentum (hence the name of m —magnetic quantum number). This leads to splitting of the energy levels of the atom in a magnetic field, since the energy of the atom’s magnetic interaction with the field, equal to μb B, (μb is Bohr’s magneton=9.27*10 raise power negative24 Joule/Tesla) is added to its energy. This particular phenomenon where in the presence of external magnetic field the energy level splits is known as ZEEMAN EFFECT.

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