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# 37 - Physics 6210/Spring 2007/Lecture 37 Lecture 37...

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Physics 6210/Spring 2007/Lecture 37 Lecture 37 Relevant sections in text: § 5.7 Electric dipole transitions Our transition probability (to ﬁrst order in perturbation theory) is P ( i f ) 4 π 2 α m 2 ¯ 2 fi N ( ω fi ) |h n f ,l f ,m f | e - i | ω fi | 1 c ˆ n · ~ X ˆ e · ~ P | n i ,l i ,m i i| 2 , where α = q 2 ¯ hc 1 137 is the ﬁne structure constant and N ( ω ) = ω 2 c | A ( ω ) | 2 is the energy per unit area per unit frequency carried by the EM pulse characterized by the vector potential in the radiation gauge with frequency components A ( ω ). We now want to analyze the matrix element which appears. This factor reﬂects the atomic structure and characterizes the response of the atom to the electromagnetic wave. Let us begin by noting that the wavelength of the radiation absorbed/emitted is on the order of 2 πc/ω fi 10 - 6 m , while the atomic size is on the order of the Bohr radius 10 - 8 m . Thus one can try to expand the exponential in the matrix element: h n f ,l f ,m f | e - i | ω fi | 1 c ˆ n · ~ X ˆ e · ~ P | n i ,l i ,m i i = h n f ,l f ,m f | (1 - ip | ω fi | 1 c ˆ n · ~ X + ... e · ~ P | n i ,l i ,m i i . The ﬁrst term in this expansion, if non-zero, will be the dominant contribution to the matrix element. Thus we can approximate h n f ,l f ,m f | e - i | ω fi | 1 c ˆ n · ~ X ˆ e · ~ P | n i ,l i ,m i i ≈ h n f ,l f ,m f | ˆ e · ~ P | n i ,l i ,m i i , which is known as the electric dipole approximation . Transitions for which this matrix ele- ment is non-zero have the dominant probability; they are called electric dipole transitions . We shall see why in a moment. Transitions for which the dipole matrix element vanishes are often called “forbidden transitions”. This does not mean that they cannot occur, but only that the probability is much smaller than that of transitions of the electric dipole type, so they do not arise at the level of the approximation we are using. If we restrict attention to the electric dipole approximation, the transition probability is controlled by the matrix element h n f ,l f ,m f | ~ P | n i ,l i ,m i i . To compute it, we use the fact that [ ~ X,H 0 ] = i ¯ h ~ P m , 1

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Physics 6210/Spring 2007/Lecture 37 and that H 0 | n,l,m i = E n | n,l,m i . We get h n f ,l f ,m f | ~ P | n i ,l i ,m i i = m i ¯ h h n f ,l f ,m f | ~ XH 0 - H 0 ~ X | n i ,l i ,m i i = imω fi h n f ,l f ,m f | ~ X | n i ,l i ,m i i . Now perhaps you can see why this is called a dipole transition: the transition only occurs according to whether or not the matrix elements of the (component along ˆ e of the) dipole moment operator, q ~ X , are non-vanishing. Selection rules for Electric Dipole Transitions We have seen that the dominant transitions are of the electric dipole type. We now consider some details of the dipole matrix elements h n f ,l f ,m f | q ˆ e · ~ X | n i ,l i ,m i i| . In particular, we derive necessary conditions on
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37 - Physics 6210/Spring 2007/Lecture 37 Lecture 37...

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