Lecture 11 - Spin Optical spectra in the presence of...

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Unformatted text preview: Spin Optical spectra in the presence of magnetic field Splitting of energy levels due to Zeeman effect results in appearance of new lines in optical spectra. Indeed instead of transitions from a degenerate level, which are characterized by the same frequency, one can observe transitions from three different levels characterized by three different frequencies. Thus one spectral line in the presence of magnetic field is split in several (in the case shown in the figure, in three) Anomalous Zeeman Effect For each orbital quantum number, l , there are 2l+1 , states with different magnetic quantum numbers, m . Therefore, it is expected that in the presence of magnetic field each spectral line would split into 2l+1 lines, which is always an odd number However, in some experiments a line splits into even number of lines in the presence of magnetic field. How can this be? The only possible explanation is to allow l , and respectively m , take on half-integer values. These are allowed by algebraic properties of the angular momentum, but forbidden by the spatial dependence of the respective eigenfunctions. Thus, these values cannot correspond to the orbital angular momentum. Stern-Gerlah experiment In this experiment a beam of atoms thought to be in their ground state are sent through the inhomogeneous magnetic field, which separates atoms with different magnetic moments. Since in ground state l=0 , and there is no magnetic momentum, no reaction on the presence of field are expected. In reality the beam split in two. This again can be explained by allowing for half-integer values of m . But what m ? There is no orbital moment present here. We have to assume that electrons may have some additional angular momentum not related to their orbital motion. This intrinsic momentum is called SPIN. Spin of an electron qualitatively This angular momentum cannot be related to the orbital motion of electrons. This is evidenced by the fact that spherical harmonics are not defined for half-integer values of the orbital number. While commutation relation of operators of the angular momentum allow for the half-integer l , the additional requirement that the respective states are described by coordinate dependent wavefunctions exclude them. The existence of spin states is an additional intrinsic property, such as mass or charge. This property is responsible for electron’s interaction with magnetic field, and is characterized by a direction. Thus, unlike mass and charge, which are scalar quantities, and always have the same value, the new property is described by a vector, which can have different values of its components. The anomalous Zeeman effect and Stern-Gerlach experiments can be explained by assuming that an electron can be in a angular momentum state with only two allowed values of its z-component: 1 2 s m = ± Spin of electrons quantitatively Spin is an observable, and should, therefore be characterized by a hermitian operator, whose eigen values would present possible values of this observable,...
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This note was uploaded on 09/27/2010 for the course PHYSICS 365 taught by Professor Deych during the Spring '10 term at SUNY Stony Brook.

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Lecture 11 - Spin Optical spectra in the presence of...

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