Fourier transform. Impulse representation.

In summary, the identity ##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1## holds, but the presence of ##\hbar## in the transform is necessary for it to be true.
  • #1
LagrangeEuler
717
20
##\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int^{\infty}_{-\infty}dx\psi(x)e^{-\frac{ipx}{\hbar}}##. This ##\hbar## looks strange here for me. Does it holds identity
##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1##?
I'm don't think so because this ##\hbar##. So state in impulse space is not normalized.
 
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  • #2
LagrangeEuler said:
Does it holds identity
##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1##?

Yes, it does. The ##\hbar## in the transform is required in order to make this true. You may be thinking of the standard Fourier transform:
$$A(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{\psi(x)e^{-ikx}dx}$$
Changing variables from k to ##p = \hbar k## introduces the extra ##\hbar## in the constant factor.
 

Related to Fourier transform. Impulse representation.

1. What is the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is used in various fields such as mathematics, physics, and engineering to analyze signals and systems.

2. What is the impulse representation of the Fourier transform?

The impulse representation of the Fourier transform is a way of representing a signal or function as a sum of weighted impulses. It is useful in understanding the frequency content of a signal and is often used in signal processing applications.

3. How is the Fourier transform related to the frequency domain?

The Fourier transform is a mathematical tool that converts a function from the time domain to the frequency domain. This means that it can represent a signal or function in terms of its constituent frequencies, making it easier to analyze and manipulate.

4. What is the difference between Fourier transform and Fourier series?

The Fourier transform and Fourier series are two related but distinct concepts. The Fourier transform is used for continuous signals or functions, while the Fourier series is used for periodic signals or functions. In other words, the Fourier transform is applicable to a wider range of signals than the Fourier series.

5. What are some practical applications of the Fourier transform?

The Fourier transform has a wide range of practical applications, including signal processing, image processing, data compression, and solving differential equations. It is also used in other fields such as physics, chemistry, and finance to analyze and understand complex data and systems.

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