# Explanation of terminology in equation used for energy calculation using Hartree–Fock method

I found in the literature that the energy of the Hartree–Fock method is given by the following equation:

$$E_\mathrm{HF} = \int (\Psi^* \hat{H}\Psi)\,\mathrm d\tau$$

The term $$\Psi$$ is the wave function, but what would the term $$\Psi^*$$ be?

Does anyone have a simple of explanation of what this may be?

• Something with an asterisk in QC usually refers to a complex conjugate. See Wikipedia and a related question on Physics.SE. – andselisk Jul 27 '20 at 12:06
• I had seen it on wikipedia but thought there would be another explanation for that (physical meaning or something like that). Thanks for the references @andselisk – Emerson P L Jul 27 '20 at 12:16
• I highly recommend that you review the basic math of quantum mechanical operators before tackling the HF method. It will make a lot more sense that way. – Andrew Jul 27 '20 at 21:15
• @Andrew would you have review articles in the area to kindly recommend reading? – Emerson P L Jul 28 '20 at 9:44
• Levine's Quantum Chemistry textbook has a great section just on the math that only assumes familiarity with standard calculus. – Andrew Jul 28 '20 at 12:52

For every $$x$$ and $$t,$$ $$\Psi(x,t)$$ is a complex number. $$\Psi^*$$ is the conjugate of that number, no more, no less. The reason it seems like sometimes it's only the $$t$$ part that gets conjugated is simply that often it is the only part of the wavefunction that is complex. Let's use an example:
$$\Psi = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)\mathrm e^{-\mathrm iE_n t}.$$
We want to calculate $$\Psi^*$$. Well, since the conjugate of the product of two numbers is the product of their conjugates (that is, $$(zw)^* = z^* w^*),$$ let's do it step by step.
First we need to conjugate $$\displaystyle\sqrt{\frac{2}{a}}$$, but since it's a real number, it is equal to its conjugate. So we leave it alone and move on. Now we need to conjugate $$\displaystyle\sin\left(\frac{n\pi x}{a}\right),$$ but again, this is a real number, because $$\sin x$$ is real whenever $$x$$ is real. The last part is $$\mathrm e^{-\mathrm iE_n t}$$. This is actually complex, so we need to conjugate it, and its conjugate is $$\mathrm e^{\mathrm iE_n t}.$$ So putting it all together, we have
$$\Psi^* = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)\mathrm e^{\mathrm iE_n t}.$$