The Meaning Of the Results
In the original Stem-Gerlach experiment, two spots of silver were formed on the glass plate, not a vertical line of silver. This means that the component I-Lz along jj (and z) could not have any value between – J.L and +J.L as classical physics predicts. 0 instead, J.L:is restricted to only two values, one for each spot-on the glass. Thus, the original Stem-Gerlach experiment showed that 1.1.: is.quantized, implying (correctly) this iI is also. Moreover, because the angular momentum L of an atom is associated with iI, that angular momentum and its component L, are also quantized. With modem quantum theory, we can add to the explanation of the two-spot result in the Stem-Gerlach experiment. We now know that a silver atom consists of many electrons, each with a spin magnetic moment and an orbital magnetic moment. We also know that all those moments vectorially cancel out except for a single electron, and the orbital dipole moment of that electron is zero. Thus, the combined dipole moment iI of a silver atom is the spin magnetic dipole moment of that single electron. According to Eq, 41-13, that means that 1.1.: can hav.e only two components along the z axis in Fig. 41-8. One component is for quantum number m, = +1 (the single electron is spin up), and the other component is for quantum number ms = -1 (the single electron is spin down . Substituting into Eq, 41-13 gives us
Then substituting these expressions for 1.1.: in Eq. 41-17, we find that the force component F. deflecting the silver atoms as they pass through the magnetic field can have only the two values
which result in the two spots of silver on-the glass.
Sample Problem: In the Stem-Gerlach experiment of a beam of silver atoms passes through a magnetic field gradient dBldz of magnitude 1.4 T/mm that is set up along the z axis. This region has a length w of 3.5 em in the direction of the original beam. The speed of the atoms is 750 m/s. By what distance d have the atoms been deflected when they leave the region of the magnetic field gradient? The mass M of a silver atom is 1.8 x 1O-2~ kg.
SOLUTION: One Key Idea here is that the deflection of a silver atom in the beam is due to an interaction between the magnetic dipole of the atom and the magnetic field, because of the gradient dBldz. The deflecting forces directed along the field gradient (along the z axis) and is given by Eqs. 41-17. Let us consider only deflection in the positive direction of z; thus, we shall use F, = J-La(dBldz) from Eqs. 41-19.A second Key Idea is that we assume the field gradient doble: has the same value throughout the region through which the silver atoms travel. Thus, force component F, is constant in that region, and from Newton’s second law, the acceleration a: of an atom along the z-axis due to F: is also constant and is given byBecause this acceleration is constant, we can use Eq. 2-15 (from Table 2-1) to write the deflection d parallel to the z axis as.
Because the deflecting force on the atom acts perpendicular to the atom’s original direction of travel, the component v of the atom’s velocity along the original direction of travel is not changed by the force. Thus, the atom requires time t = w]v to travel through length w in that direction. Substituting w]v for t into Eq. 41-20, we find