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Unformatted text preview: CHEM 356, Lecture 24, Fall 2009 1 Inuence of a Magnetic Field upon the Energy Levels of H. We consider the model of the H atom as an electron revolving around a proton in a circular orbit (essentially the Bohr model), and enquire about the electrodynamic consequences of this model of a moving charge. Magnetic Moment of the Electron. Let us begin by considering the motion of a charge ? around a closed loop of radius ? . A current owing around a loop that encloses an area constitutes a magnetic dipole moment of magnitude ? (m) = . If the charge ? moves around the loop at speed in uniform circular motion, it takes ? = 2 ?/ seconds to complete one loop, and since the current associated with this motion is given by = ? ? = ? 2 ? , while the enclosed area is = ? 2 , we have a scalar magnetic moment of ? (m) = 1 2 ?? . What about the vector form, since a dipole moment must have a vector character? We know that the position vector r is perpendicular to the velocity v for a circular loop, so that we can write a vector form for the dipole moment as (m) = 1 2 ? r v = 1 2 ? r p or (m) = ? 2 L . For an electron, we have charge ? = , and (m) = 2 e L . CHEM 356, Lecture 24, Fall 2009 2 Potential (Interaction) Energy. The potential energy of interaction of a magnetic dipole with a magnetic field B is given by field = (m) B . We normally choose B to lie along the axis (or, equivalently, to define the direction), so that we may write B = k , which allows us to write the classical form for the dipolefield interaction as field = ? (m) ? = 2 e ? , which is replaced in quantum mechanics by the operator field = 2 e ? . The Hamiltonian for the interaction of a magnetic dipole with an external magnetic field can be written in the form = + field = + field , or = + 2 e ?...
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