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Lecture24

Lecture24 - CHEM 356 Lecture 24 Fall 2009 1 Inuence of a...

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CHEM 356, Lecture 24, Fall 2009 1 Inﬂuence 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 𝐼 ﬂowing around a loop that encloses an area 𝐴 constitutes a magnetic dipole moment of magnitude ? (m) = 𝐼𝐴 . If the charge ? moves around the loop at speed 𝑣 in uniform circular motion, it takes ? = 2 𝜋?/𝑣 seconds to complete one loop, and since the current 𝐼 associated with this motion is given by 𝐼 = ? ? = ?𝑣 2 𝜋? , while the enclosed area is 𝐴 = 𝜋? 2 , we have a scalar magnetic moment of ? (m) = 1 2 ??𝑣 . 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 𝝁 (m) = 1 2 ? r × v = 1 2 𝑚 ? r × p or 𝝁 (m) = ? 2 𝑚 L . For an electron, we have charge ? = 𝑒 , and 𝝁 (m) = 𝑒 2 𝑚 e L .

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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 𝑉 field = 𝝁 (m) B . We normally choose B to lie along the 𝑧 –axis (or, equivalently, to define the 𝑧 direction), so that we may write B = 𝐵 0 k , which allows us to write the classical form for the dipole–field interaction as 𝑉 field = ? (m) ? 𝐵 0 = 𝑒𝐵 0 2 𝑚 e 𝐿 ? , which is replaced in quantum mechanics by the operator ˆ 𝑉 field = 𝑒𝐵 0 2 𝑚 e ? . The Hamiltonian for the interaction of a magnetic dipole with an external magnetic field can be written in the form = 0 + field = 0 + ˆ 𝑉 field , or = 0 + 𝑒𝐵 0 2 𝑚 e ? . With this Hamiltonian, the Schr¨ odinger equation for the H atom becomes: 𝜓 𝑛ℓ𝑚 = 0 𝜓 𝑛ℓ𝑚 + 𝑒𝐵 0 2 𝑚 e ?
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