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2.3: Complex Functions

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  • The concepts of complex conjugate and modulus that we discussed above can also be applied to complex functions. For instance, in quantum mechanics, atomic orbitals are often expressed in terms of complex exponentials. For example, one of the \(p\) orbitals of the hydrogen atom can be expressed in spherical coordinates (\(r,\theta,\phi\)) as

    \[\psi(r,\theta,\phi)=\dfrac{1}{8 \sqrt{a_0^5 \pi}} r e^{-r/2a_0} \sin\theta e^{i \phi} \nonumber\]

    We will work with orbitals and discuss their physical meaning throughout the semester. For now, let’s write an expression for the square of the modulus of the orbital (Equation \(2.2.2\)):

    \[|\psi|^2=\psi \psi^* \nonumber\]

    The complex conjugate of a complex function is created by changing the sign of the imaginary part of the function (in lay terms, every time you see a +\(i\) change it to a -\(i\), and every time you see a -\(i\) change it to a +\(i\)). Therefore:

    \[ \begin{align*} |\psi|^2 &=\left(\dfrac{1}{8 \sqrt{a_0^5 \pi}} r e^{-r/2a_0} \sin\theta e^{i \phi}\right)\left(\dfrac{1}{8 \sqrt{a_0^5 \pi}} r e^{-r/2a_0} \sin\theta e^{-i \phi}\right) \\[4pt] &=\dfrac{1}{64 a_0^5 \pi} r^2 e^{-r/a_0} \sin^2\theta \end{align*} \nonumber\]

    Notice that \(\psi \psi^*\) is always real because the term

    \[e^{+i \phi} e^{-i \phi}=1.\]

    This has to be the case because \(|\psi|^2\) represents the square of the modulus, and as we will discuss many times during the semester, it can be interpreted in terms of the probability of finding the electron in different regions of space. Because probabilities are physical quantities that are positive, it is good that \(\psi \psi^*\) is guaranteed to be real!

    Confused about the complex conjugate? See how to write the complex conjugate in the different notations discussed in this chapter in this short video: lcry7ma​​​​​​​

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