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Neothilic
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- TL;DR Summary
- So I am confused on the steps to find out how you would get to having the second order differential operator to k^2 in the exponent.
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Charles Link said:The operator ## \partial^2_x ## is to the left of the ## dk ## integral. The only thing that is of importance here is the ## e^{-ikx} ## term in the integrand. If the operator were by itself, (not in an exponential), I think you can see you get ##-k^2 e^{-ikx} ## when it operates on this term. The effect of the ## \partial^2_x ## operator is ## -k^2 ##. The same thing applies when the operator is in an exponential.
Charles Link said:I think you have the basic idea. You need to operate on ## u_o(x) ## with ##e^{it \partial^2_x } ## though, and the result is ## e^{-it k^2} ##.
The Fourier Integral of the Schrodinger Equation is a mathematical representation of the time-dependent Schrodinger equation in quantum mechanics. It allows for the calculation of the wave function of a quantum system at any given time, given the initial conditions.
The Fourier Integral is derived from the time-dependent Schrodinger equation. It is a solution to the equation, allowing for the calculation of the wave function at any given time.
The Fourier Integral is an essential tool in quantum mechanics as it allows for the calculation of the wave function, which describes the behavior of quantum systems. It also allows for the prediction of the probabilities of different outcomes of quantum experiments.
The Fourier Integral is calculated by taking the Fourier transform of the initial wave function and multiplying it by the time-evolution operator. The resulting function is then inverse Fourier transformed to obtain the wave function at any given time.
The Fourier Integral has various applications in quantum mechanics, including the study of atomic and molecular systems, quantum computing, and quantum cryptography. It is also used in the analysis of quantum phenomena, such as quantum tunneling and quantum entanglement.