Homework for Periodic Systems

1. (DeltaFourier)
1. (repeat) Find a set of functions that approximate the delta function $\delta(x-a)$ with a sequence of isosceles triangles $\delta_{\epsilon}(x-a)$, centered at $a$, that get narrower and taller as the parameter $\epsilon$ approached zero.

2. Find the Fourier transform of $\delta_{\epsilon}(x-a)$.

3. Take the limit as $\epsilon\rightarrow 0$ to find the Fourier transform of a $\delta$ function.

2. (FourierGauss)

Consider the Gaussian wave function $f(x) = N e^{-x^2/2a^2}$.

1. Find the normalization constant $N$. Write a sentence describing the physical meaning of normalizing. (The identity $\int_{-\infty}^{\infty}e^{-u^2}du = \sqrt{\pi}$ may prove helpful.)

2. Find the Fourier transform of $f(x)$ by hand. You will need to “complete the square”. Sense-making: Discuss how changing the constant $a$ changes the shape of both $f(x)$ and its Fourier transform.

3. Show that the Fourier transform of $f(x)$ is also normalized. (This is true for any function and is known as Parseval's identity.) Write a sentence describing the physical meaning of normalizing in this case.

3. (VSpace)

Show that

\begin{align} i\int_{-\infty}^{\infty} \tilde f(k)^* \frac{d\tilde f}{dk} dk &= \int_{-\infty}^{\infty} x \left|f(x)\right|^2 dx \end{align}

where you should be sure to take note of the complex conjugates on each side of the equation.

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