I Parity Selection Rules: I'm Confused

ergospherical
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I'm confused by the discussion in section §30 (Parity of a state), page 98 of Landau's QM. The functions ##\psi_u## and ##\psi_g## are odd an even states respectively. If ##f## is a true scalar, then it should remain unchanged by inversion of the co-ordinates. Writing ##q' = -q##, then its matrix element at position ##(u,g)## is\begin{align*}
f_{ug} = \int \psi_u^*(q) \hat{f}(q) \psi_g(q) dq =-\int [-\psi_u^*(q')] \hat{f}(q') \psi_g(q') dq' = \int \psi_u^*(q') \hat{f}(q') \psi_g(q') dq'
\end{align*}however it is written in the text that ##f_{ug} = -f_{ug}##. What did I mis-understand?
 
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What is ##q##? What are the limits of all those integrals?
 
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Gaussian97 said:
What is ##q##? What are the limits of all those integrals?
Oh yes, thanks, there should be an extra minus sign due to inverting the limits of the third integral.
 
ergospherical said:
I'm confused by the discussion in section §30 (Parity of a state), page 98 of Landau's QM. The functions ##\psi_u## and ##\psi_g## are odd an even states respectively. If ##f## is a true scalar, then it should remain unchanged by inversion of the co-ordinates. Writing ##q' = -q##, then its matrix element at position ##(u,g)## is\begin{align*}
f_{ug} = \int \psi_u^*(q) \hat{f}(q) \psi_g(q) dq =-\int [-\psi_u^*(q')] \hat{f}(q') \psi_g(q') dq' = \int \psi_u^*(q') \hat{f}(q') \psi_g(q') dq'
\end{align*}however it is written in the text that ##f_{ug} = -f_{ug}##. What did I mis-understand?
Where does the additional ##-## sign after the 2nd equality sign come from? This should be absent since ##f(q)=f(-q)## by assumption, if I understand right what you mean by "true scalar", i.e., a scalar under rotations AND parity.
 
ergospherical said:
Oh yes, thanks, there should be an extra minus sign due to inverting the limits of the third integral.
No, why? You have
$$\mathrm{d}^3 q' =\left | \mathrm{det} \frac{\partial(q')}{\partial q} \right| \mathrm{d}^3 q = |-1| \mathrm{d}^3 q=\mathrm{d}^3 q.$$
 
I have taken ##dq’ = -dq##, but in changing variables must also reverse the limits of the integral
\begin{align*}
f_{ug} = \int_{-\infty}^{\infty} \psi_u^*(q) \hat{f}(q) \psi_g(q) dq &=-\int_{\infty}^{-\infty} [-\psi_u^*(q')] \hat{f}(q') \psi_g(q') dq' \\
&= \int_{\infty}^{-\infty} \psi_u^*(q') \hat{f}(q') \psi_g(q') dq' \\
&= -f_{ug}
\end{align*}
 
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