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Results tagged with fourier-transform
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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.
6
votes
Why does the mathematical constant $e$ enter into quantum mechanics so much?
It's not $e$ per se, but rather the exponential function
$$ \exp(x) \equiv \mathrm{e}^{x} = \sum_{n \in \mathbb{N}} \frac{x^n}{n!}$$
and the exponential functions appears naturally in various contex …
- 129k
7
votes
Accepted
Derivation of plane wave from inner product of position ket and momentum ket
Observe that
$$ p \langle x \vert p \rangle = \langle x \vert \hat{p} \vert p \rangle =-\mathrm{i} \partial_x \langle x \vert p \rangle$$
since, with $\hbar = 1$, $\hat p = -\mathrm{i}\partial_x$ whe …
- 129k
3
votes
Accepted
What am I REALLY doing when I take the Fourier transform of the momentum operator
($\hbar$ omitted in the following.)
That is not weird, it is one of the crucial properties of the Fourier transform $F(\bar{})$ that
$$ F(\partial_x f) = \mathrm{i}p F(f)$$
i.e. differentiation by …
- 129k
0
votes
Accepted
What is the difference between the momentum in the Fourier transform of a scalar field and t...
The $p$ in the Fourier transform of the (free real scalar) field
$$ \phi(x) = \int \left(a(\vec p)\exp(-\mathrm{i}px) + a^\dagger(\vec p)\exp(\mathrm{i}px)\right)\frac{\mathrm{d}^3p}{2p^0} $$
is a num …
- 129k
4
votes
Accepted
Motion of string fixed at both ends
$f$ is the initial condition for the string. If you have some differential equation (e.g. the wave equation) for the shape of the string varying in time, i.e. $\psi(x,t)$, you will require some initi …
- 129k
4
votes
Accepted
Are position and momentum spaces of a particle in classical mechanics related by Fourier tra...
In classical mechanics, position and momentum are independent variables on phase space and are not related by Fourier transformation at all.
Their Fourier relation in quantum mechanics arises from th …
- 129k
4
votes
Accepted
What does it mean to expand an operator into a Fourier series?
It is straightforwardly possible to generalize the notion of a Fourier transform of scalar functions $\mathbb{R}^n\to\mathbb{C}$ to Fourier transforms of functions $\mathbb{R}^n\to E$, where $E$ is a …
- 129k
41
votes
Accepted
To what extent can we use the informal version of the Dirac delta function in Physics?
The expression
$$ \int \delta(x)f(x)\mathrm{d}x = f(0)$$
is not wrong, you simply need to read the left-hand side of the equation as what a mathematician would write something like $\langle \delta, f\ …
- 129k
2
votes
Why is the position space free particle wavefunction a function of momentum?
No, the (elementary solution for the position representation of the) wavefunction of a free particle, $\psi(x) = \mathrm{e}^{\mathrm{i}px}$ is not an "explicit function of both" position and momentum. …
- 129k
2
votes
Accepted
Problem with momentum values in a QM problem
First, realize we are doing an approximation when we are evaluating the coefficient $A_p$ in the Fourier series
$$ \psi(x) = \sum_p A_p \psi_p(x)$$
by the integral
$$ A_p = \int_{-\infty}^\infty \psi_ …
- 129k
1
vote
Shifting momentum by a constant in the Schrodinger Equation
If $\phi_n(p)$ is an eigenfunction of $H$ in momentum space with eigenvalue $E_n$, then $\phi_n(p+\lambda)$ is an eigenfunction of $H'$ with eigenvalue $E_n -\frac{\lambda^2}{2m}$ (and vice versa), si …
- 129k
0
votes
Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\...
The step follows just by the usual relation that the Fourier transform of $p\cdot \phi(p)$ is $\partial_x \tilde{\phi}(x)$, i.e. multiplication by the variable becomes differentiation.
There is no ge …
- 129k
0
votes
Is the uncertainity principle explained by disturbances or only by the Fourier picture?
Neither of these explanations are actual explanations of the general quantum mechanical uncertainty principle. The "Fourier explanation" only covers observables that are canonically conjugate (i.e. ha …
- 129k
2
votes
Accepted
Klein-Gordon Hamiltonian in terms of Fourier transformed variables
The mode expansion is not a Fourier transform.
The Fourier transform of $\phi$ is
$$\tilde{\phi}(\vec p) = \int \phi(x) \mathrm{e}^{-\mathrm{i}\vec p \cdot \vec x}\frac{\mathrm{d}^3 x}{(2\pi)^3}$$
and …
- 129k
2
votes
Question About Momentum and Position Operators and the Postulates of Quantum Mechanics
Instead of explicitly postulating properties of "position space" or "momentum space" or talking about Fourier transforms, it is much more common in axiomatic treatments to - in the spirit of canonical …
- 129k