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Unformatted text preview: CHEM 356, Lecture 19, Fall 2009 1 Raising and lowering operators. The Cartesian component operators are not always those best-suited for use in specific quantum mechanical problems, especially if the system being considered has spherical symmetry. Another set of three operators that are linear combinations of the Cartesian component operators, defined specifically by ˆ + ≡ ˆ ? + ˆ ? ; ˆ − ≡ ˆ ? − ˆ ? ; ˆ ≡ ˆ ? , is more relevant to systems having spherical symmetry: it is often referred to as the set of spherical components of ˆ J . – The operator ˆ + is referred to as a raising operator , while the operator ˆ − is referred to as a lowering operator . The two operators ˆ + , ˆ − are together also often referred to as ladder operators . ∙ The spherical components of ˆ J obey the commutation relations: [ ˆ ? , ˆ + ] − = ℏ ˆ + , [ ˆ ? , ˆ − ] − = − ℏ ˆ − , [ ˆ + , ˆ − ] − = 2 ℏ ˆ ? . (1) These commutation relations can be derived from the definitions given earlier, together with the commutation relations for the ?, ?, ? –components of angular momentum op- erators. ∙ We can also invert the defining relations for ˆ + and ˆ − to give ˆ ? and ˆ ? in terms of ˆ + and ˆ − : ˆ ? = 1 2 ( ˆ + + ˆ − ) , ˆ ? = 1 2 ( ˆ + − ˆ − ) . ∙ The operator ˆ J 2 can be written equivalently in terms of ˆ ? , ˆ ? , ˆ ? or in terms of ˆ + , ˆ − and ˆ ? : i.e., ˆ J 2 ≡ ˆ 2 ? + ˆ 2 ? + ˆ 2 ? = 1 2 ( ˆ + ˆ − + ˆ − ˆ + ) + ˆ 2 ? , Note that ˆ J 2 is simply defined only in the Cartesian representation. It would have been incorrect to have assumed that ˆ J 2 could be written as the sum of the squares of its three spherical components. CHEM 356, Lecture 19, Fall 2009 2 ∙ Using the spherical component expression for ˆ J 2 , together with the commutator of ˆ + , ˆ −...
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- Fall '09
- pH, J2, ladder operators, Cartesian component operators, 1 10 J, spherical components