# A question about writing a wave as the superposition of other waves

In quantum mechanics, if a wave function is the superposition of many wave functions, it can be written as $$\frac{1}{\sqrt{2\pi}}\int g(k)e^{i(kx-\omega t)}dk \, .$$ On page 24 of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloe, the following waves are superimposed: with wave numbers $$\{ k_0, k_0 - \Delta k/2, k_0 + \Delta k / 2 \}$$. However the expression that they write for it is $$\frac{g(k)}{\sqrt{2\pi}} \left( e^{ik_0x} + e^{i \left(k_0+\frac{\Delta k}{2} \right) x} + e^{i \left(k_0-\frac{\Delta k}{2} \right) x} \right) \, .$$ Shouldn't it be $$\frac{1}{\sqrt{2\pi}} \left( g(k_0)e^{ik_0x} + g(k_0 + \Delta k / 2) e^{i \left(k_0+\frac{\Delta k}{2} \right) x} + g(k_0 - \Delta k / 2) e^{i \left(k_0-\frac{\Delta k}{2} \right) x} \right) \, ?$$

• Should those $\Delta x/2$ be $\Delta k/2$? – Kyle Kanos Mar 22 at 16:39
• From the book: "... [the] amplitudes [of these plane waves] are proportional, respectively, to 1, 1/2 and 1/2." Thus, there is an overall factor of $g(k_0)$ out front (roughly speaking a normalization factor), and then the corresponding factors of 1, 1/2, and 1/2 are attached to each plane wave. For instance, what you called $g(k_0 + \Delta k/2)$ is equal to $g(k_0)/2$. – march Mar 22 at 17:23