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Results tagged with fourier-transform
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user 21270
A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.
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Delta Function Identity in Peskin & Schroeder
In Notation and Conventions of their QFT textbook (page no. xxi), Peskin and Schroeder mentions the following identity:
$$\int d^4x \, e^{ik\cdot x} = (2\pi)^4 \delta^{(4)}(k).$$
They define the Fouri …
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Derivation of Equation 2.27 from Peskin & Schroeder
In Section 2.3, Peskin & Schroeder discusses the quantization of real scalar field in Schrodinger picture. He writes Eq. (2.25) as follows.
$$\phi(\textbf{x}) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sq …
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Expression for Delta Function in Quantum Mechanics
For a free particle, we can derive the following well-known relation:
$$\langle k|k'\rangle = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(k-k')x} \, dx = \delta(k-k'). \tag{1.10.33}$$
Reference: …
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Delta Function Identity in Peskin & Schroeder
Given that $$f(x) = \int \frac{d^4k}{(2\pi)^4} e^{-ik\cdot x} \, \tilde{f}(k) \tag 1$$
and,
$$\tilde{f}(k) = \int d^4 x \, e^{ik\cdot x} f(x). \tag 2$$
Let $$\tilde{f}(k) = \delta(k) = \left\{ \begin …
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