Expanding Wavefunction in Infinite Well: Fourier Analysis

In summary, the conversation is discussing the wavefunction of a particle in an infinite well from -a to a, with an initial wavefunction of u_1^+(x;a) = cos(\pi x/ 2a)/\sqrt{a} for |x| < a. The question is whether Fourier analysis should be applied to obtain the wavefunction when the box is instantaneously expanded to [-b,b], and if so, whether the time dependence should be included.
  • #1
ehrenfest
2,020
1
I have an infinite well from -a to with a particle in its ground. The initial wavefunction is then

[tex]\psi(x) = u_1^+(x;a) = cos(\pi x/ 2a)/\sqrt{a}[/tex] for |x| < a.

In order to get the wavefunction for this particle when box that is instantaneously expanded to [-b,b] should I apply Fourier analysis via

[tex] a^{+}_n = 1/b \int_{-b}^{b}cos(\pi x/ 2a)/\sqrt{a}\cdot cos(\pi x/ 2b)/\sqrt{b}dx [/tex]

where a^{+}_n is the coefficient of the even wavefunction with that n in the expanded box?
 
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  • #2
Yes, then put in the time dependence.
 

Related to Expanding Wavefunction in Infinite Well: Fourier Analysis

1. What is the purpose of using Fourier analysis in studying the expanding wavefunction in an infinite well?

The purpose of using Fourier analysis is to break down a complex function, such as the expanding wavefunction in an infinite well, into simpler components that can be studied and understood more easily. This technique allows us to analyze the contributions of different frequencies to the overall function and gain insights into the behavior of the system.

2. How is the Fourier series used to express the expanding wavefunction in an infinite well?

The Fourier series is used to express the expanding wavefunction in an infinite well as a combination of sinusoidal functions with different frequencies, amplitudes, and phases. This series can be written as an infinite sum, where each term represents a specific frequency component of the wavefunction.

3. Why is it important to include all frequencies in the Fourier series of the expanding wavefunction?

It is important to include all frequencies in the Fourier series of the expanding wavefunction because each frequency contributes to the overall behavior of the system. Neglecting any frequency component would result in an incomplete representation of the wavefunction and could lead to incorrect conclusions about the system's behavior.

4. What is the significance of the Fourier coefficients in the expanding wavefunction in an infinite well?

The Fourier coefficients represent the amplitudes and phases of each frequency component in the expanding wavefunction. They provide valuable information about the relative contributions of each frequency to the overall behavior of the system and can be used to calculate important properties such as the probability density and energy of the particle.

5. How does the Fourier series of the expanding wavefunction change as the particle's energy increases?

As the particle's energy increases, the Fourier series of the expanding wavefunction will have more terms with higher frequencies. This is because the particle's higher energy allows it to occupy higher energy states, which correspond to higher frequencies in the Fourier series. As a result, the wavefunction will have more oscillations and a shorter wavelength, reflecting the increased energy of the particle.

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