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The following is a quote from the Haifa Lectures (Mendel Sachs)

But if both slits are open, the wave function for the electron penetrating screen S1 is the superposition of states, $(\psi_1 + \psi_2)$, so that the probability density for the electron wave reaching screen S2 is $\left|\psi_1 + \psi_2\right|_2 = \left|\psi_1\right|_2 + \left|\psi_2\right|_2 + (\psi_1^*\psi_2 + \psi_1\psi_2^*)$. The first two terms above are the partial probability densities that the electron will pass through slit s1 or slit s2. The third term (the cross product) is the ‘interference’ part of the scattering. It shows up on screen S2 as a diffraction pattern.

I understand that this explains the diffraction pattern produced by electrons, and I know (in a general way) what $\psi_1$ and $\psi_2$ are. However I run into trouble with the next bit of math [$\left|(\psi_1 + \psi_2)\right|_2 = \left|\psi_1\right|_2 + \left|\psi_2\right|_2 + (\psi_1^*\psi_2 + \psi_1\psi_2^*)$.] especially the last part.

Could someone please offer a guide for the mathematically challenged?

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The complex Euclidean norm $\left| \cdot \right|_2$ considered here is calculated by $$ \left| \varphi \right|_2 = \varphi\varphi^* $$ where $\varphi\in\mathbb{C}^n$ and the $^*$ denotes complex conjugation.

Then, since $$ \left(\psi_1 + \psi_2\right)^* = \psi_1^* + \psi_2^* $$ one has \begin{align} \left|\psi_1 + \psi_2\right|_2 &= \left(\psi_1+\psi_2\right) \left(\psi_1+\psi_2\right)^* \\&= \left(\psi_1+\psi_2\right) \left(\psi_1^*+\psi_2^*\right) \\&= \psi_1\psi_1^* + \psi_1\psi_2^* + \psi_2\psi_1^* + \psi_2\psi_2^* \\&= \left|\psi_1\right|_2 + \psi_1\psi_2^* + \psi_2\psi_1^* + \left|\psi_2\right|_2 \end{align} which is the claimed result.

Did that help you, or does the problem lie elsewhere?

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  • $\begingroup$ Thanks, that helps a little, but I'm afraid I'm too mathematically challenged for it to answer my question completely. $\endgroup$ – Bill S. Feb 14 '15 at 18:26
  • $\begingroup$ Thanks, that helps a little, but I'm afraid I'm too mathematically challenged for it to answer my question completely. For example, what is the significance of the vertical lines in|ψ|? φ∈Cn means nothing to me, and what is the significance of the asterisks? $\endgroup$ – Bill S. Feb 14 '15 at 18:37
  • $\begingroup$ @BillS. Sorry for the delayed answer. $\mathbb{C}$ denotes the complex numbers, the asterisk stands for complex conjugation. The vertical bars denote the "absolute value" or "magnitude" of a variable (see the previous links). $\mathbb{C}^n$ is the set of $n$-dimensional vectors; for an introduction to that concept, see the very closely related but simpler case of real vectors. $\endgroup$ – RQM Feb 20 '15 at 10:55

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