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lecture08_umn

# lecture08_umn - Lecture 8 Parity Spectroscopic Transitions...

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Lecture 8 Parity Spectroscopic Transitions Tunneling September 28 2009 McQuarrie and Simon pp. 67-70. problem 3-32 pp. 102-104.

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Parity Often in quantum mechanics we face a fairly simple question with respect to a given integral: Is it zero or can it be non-zero ? It can be surprisingly easy to answer this question by taking advantage of a property known as parity. If it is the case that a function of a given variable changes sign when the variable changes sign but does not change in magnitude , i.e., (8-1) we say the function has ‘odd’ parity . If, on the other hand, it neither changes in magnitude nor in sign , i.e., (8-2) we say that it has ‘even’ parity . [Note that f ( x ) = 0 is a rather odd beast that has both even and odd parity.] " f x ( ) = f " x ( ) f x ( ) = f " x ( )
Integrating a even/odd function Parity is a useful quality because any integral that equally spans either side of zero may be non-zero if the function being integrated has even parity, and it will be zero if the function being integrated has odd parity. If these statements are not intuitively obvious, quick graphical examples should make things clear. x f ( x ) exactly cancelling areas for odd function

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Parity Operator From inspection of eq. 8-1 it is clear that an odd function must pass through the origin, and that the areas on the left and right sides of the origin will exactly cancel one another. For an even function (not shown above), the area on one side will exactly equal (not cancel) the area on the other side, so the integral may be non-zero. A simple example of a trivial even function is f ( x ) = C where C is a constant (i.e., no dependence on x ). In that case, the area defined by the integral from a to a is simply 2 aC , which is not zero except in the boring case of C = 0. Note that an arbitrary function does not have to have any parity at all . For instance, f ( x ) = e x is neither even nor odd. We can actually regard parity as having an associated operator, call it Π , where the operator replaces every coordinate variable with its negative . That is, (8-3) " f x , y , z ( ) = f # x , # y , # z ( )
Parity Operators and the Hamiltonian From generalizing eqs. 8-1 and 8-2, it should be clear that odd and even functions are eigenfunctions of the parity operator having eigenvalues 1 and 1, respectively. [These are also the only possible eigenvalues if the function f is required to be differentiable at x =0.] Now, let us consider whether the parity operator commutes with the Hamiltonian for the particle in a box. Put differently, does [ H , Π ] = 0? For the particle in a box, the Hamiltonian is simply the kinetic energy operator. Let us evaluate the commutator for an arbitrary function (of one variable, for simplicity).

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Commutator of H and Π (8-4) The kinetic energy operator and the parity operator commute . Since the eigenfunction of one Hermitian operator is also an eigenfunction for all Hermitian operators with which the first one commutes, it is required that the particle-in-a-box wave functions be either even or odd (the only two possibilities for Π ).
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lecture08_umn - Lecture 8 Parity Spectroscopic Transitions...

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