Inverse Fourier Transform of X(w): Real Values for x(t)

In summary, the conversation discusses the inverse Fourier transform of a given equation and the resulting imaginary values of x(t). It is mentioned that this is expected in quantum mechanics problems, but a book provides a different solution with real values and a negative sign. Assistance is requested in understanding how to obtain this solution.
  • #1
hula
3
0
X(w) = 1/(j*(w*hbar-Ek)+(hbar/T2)) - 1/(j*(w*hbar+Ek)+(hbar/T2))

The inverse Fourier transform of the above equation using MATLAB will obtain the following:

x(t) = 2*j/hbar*heaviside(t)*sin(t/hbar*Ek)*exp(-t/T2)

We can see that the values of x(t) are all imaginary values, however this shouldn't be the case, should have real values for x(t) instead.

Does anyone knows what should be the correct inverse Fourier transform?

Thanks!
 
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  • #2
The result given by Matlab will be correct, assuming you have adopted the same sign conventions when setting up the analytic FT. In particular the antisymmetry of X results in imaginary x.

Also, this looks like a QM problem, and as far as I know there's no reason you can't have an imaginary solution: it just means the time evolution is 90 degrees out of phase with that of a real solution.
 
  • #3
I've found an answer from a book which states that

x(t) = -2/hbar*heaviside(t)*sin(t/hbar*Ek)*exp(-2*pi/T2*t)

I'm wondering how to get this answer, because it's a bit different from what I got from MATLAB (the difference is that the solution obtained from the book is in real values, and now there's a negative sign in front, and lastly the exponential term has an additional 2*pi)

Hope that someone would be able to assist me in my queries! Thanks in advance!
 

1. What is the inverse Fourier transform of X(w)?

The inverse Fourier transform of X(w) is a mathematical operation that converts a function in the frequency domain, X(w), back to its original form in the time domain, x(t).

2. Why are real values important for x(t) in the inverse Fourier transform?

Real values are important because they represent the physical quantities of a signal in the time domain. Inverse Fourier transforms of complex-valued functions can result in imaginary components that do not have a physical meaning.

3. How is the inverse Fourier transform of X(w) calculated?

The inverse Fourier transform of X(w) can be calculated using the formula: x(t) = (1/2π)∫X(w)exp(iwt)dw, where exp(iwt) is the complex exponential function and dw represents the change in frequency.

4. What is the significance of the real part and imaginary part in the inverse Fourier transform?

The real part of the inverse Fourier transform represents the actual signal in the time domain, while the imaginary part represents the phase shift of the signal. Together, they give a complete representation of the signal in both the time and frequency domains.

5. Can the inverse Fourier transform be applied to any function in the frequency domain?

Yes, the inverse Fourier transform can be applied to any function in the frequency domain, as long as the function satisfies certain mathematical conditions, such as being absolutely integrable. This allows for a wide range of applications in signal processing and other scientific fields.

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