Chapter 2: Q26P (page 77)

What is the Fourier transform $δ (x)$ ? Using Plancherel’s theorem shows that $δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$ .

Short Answer

Expert verified

The Fourier transform for the given function is

$δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$

Step by step solution

Define the Schrodinger equation

A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

Determine the Fourier transform

By using the Fourier transform,

$F (k) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (x) e^{ikx} dx$

Substitute $f (x) = δ (x)$ into equation

$F (k) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} δ (x) e^{ikx} dx$

$\int_{- \infty}^{\infty} δ (x) e^{ikx} dx = 1$

$F (k) = \frac{1}{\sqrt{2} π}$

$\begin{matrix} f (x) = δ (x) \\ = \frac{1}{\sqrt{2} π} \int_{- \infty}^{\infty} \frac{1}{\sqrt{2} π} e^{ikx} dk \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk \end{matrix}$

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What is the Fourier transform $δ (x)$ ? Using Plancherel’s theorem shows that $δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$ .

Short Answer

Step by step solution

Define the Schrodinger equation

Determine the Fourier transform

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Short Answer

Step by step solution

Define the Schrodinger equation

Determine the Fourier transform

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Most popular questions from this chapter

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What is the Fourier transform $δ (x)$ ? Using Plancherel’s theorem shows that $δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$ .