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Questions tagged [complex-numbers]

Numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.

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What is the physical interpretation of the Wick rotation?

What is the physical interpretation of the Wick rotation? How is it that we can just propose there's a new time coordinate tau? Are physicists saying time is modeled by an imaginary number? Isn't ...
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Derivation of Holomorphic Ward Identities in Franceso's CFT

In equation 5.37 of francesco's CFT he writes the Ward Identities for traslation symmetry in the language of holomorphic functions. He goes from \begin{equation} \frac{\partial}{\partial x^\mu} \...
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35 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
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42 views

Intensity of the resultant of two complex waves [on hold]

Suppose I have two waves: $Y_1= a_1e^{i(wt-kx1)}$ and $Y_2= a_2e^{i(wt-kx2)}$ I know by superposition $Y= Y_1+Y_2$ and intensity $(I) = |Y|² $ But how can I solve it. It seems hard for me to find the ...
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108 views

What is the physical interpretation of imaginary lengths?

My question is about the meaning of imaginary lengths, which occur often in the solution to various numerical problems in Physics. Generally imaginary quantities are discarded as nothing but ...
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38 views

2 in the Fermi’s Golden Rule

In the derivation of the Fermi's golden rule many authors expand periodic perturbation in this form $$\hat{V}=\hat{F} e^{-i \omega t}+\hat G e^{i \omega t}$$ However I do not understand the reason. ...
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Imaginary Capacitance and Imaginary Inductance interpretation [closed]

Today I came across with Complex Capacitances and Inductances for the first time: $$L(j\omega)=L_{real}+jL_{imaginary}$$ $$C(j\omega)=C_{real}+jC_{imaginary}$$ So I started looking at their meaning. ...
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18 views

LCR circuit , impedance, imaginary [duplicate]

What is the physical interpretation in behind impedance of capacitor and inductor? we know that resistance oppose the current in the same direction of voltage. But how capacitor and impedance oppose ...
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27 views

Need help understanding KVL in Phasor Notation

I may be missing some crucial observation but I'm getting stumped on this KVL proof for phasor notation. So to prove KVL in phasor notation I'd like to start off with what we know, KVL in a real ...
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26 views

Kramers-Kronig relations in susceptibility

I tried asking this question in the Math Stack Exchange but it got little attention since it was more focused on an application (and the notations provided were a little hard to understand), so ...
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2answers
96 views

Complex conjugated representation and its Young tableaux

This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions ...
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43 views

Infinitesimal parameter of Lorentz transformation

I'm working through the SUSY lecture notes by Lambert, and he does something which seems strange to me during the calculation of the Wess Zumino model. He says the spinor $\psi$ has the ...
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Complex Potential describing inviscid fluid flow

I'm a Mathematics student, working through a homework sheet for a Fluid Mechanics module. The question is given: Consider the flow described by the complex potential $$w=4z+\frac{8}{z}.$$ ...
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93 views

Schrödinger Equation with Imaginary Potential

I am trying to solve the following equation (in 1D) and stuck in the middle of the way. Here's the equation: $$i\frac{\partial\psi}{\partial t}=C\cdot\frac{\partial^{2}\psi}{\partial x^{2}}+iD\cdot\...
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69 views

What is complex frequency? [closed]

I am learning EE, and about complex frequencies, but what is its physical meaning? What is it used for? Why is it? And only happen in the laplace transform?
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102 views

Euler-Lagrange equations from a complex Lagrangian

I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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40 views

Four point function with complex momenta?

It is well known that the four-point function $$\int_{\mathbb{R}^{3,1}}\frac {d^4 q}{((q+p_1)^2-i\epsilon)((q+p_2)^2-i\epsilon)((q+p_3)^2-i\epsilon)((q+p_4)^2-i\epsilon)}$$ can be computed using the ...
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1answer
90 views

Orthonormal basis written in Dirac Notation

$\left\{ e _ { i } \right\}$ is an orthonormal basis which has the orthonormal condition as following: $$e _ { i } ^ { T } \cdot e _ { j } = \delta _ { i j }$$ In Dirac Notation where $| i \rangle = | ...
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58 views

Contour Integration in Schwartz

In Matthew Schwartz's QFT text, on page 39, he has the following contour integral: $$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$ This can be split into two terms, one ...
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75 views

Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $\psi$ and its "conjugate" $\psi^*$ (quotation marks theirs) has Lagrangian density: $$L=-\...
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33 views

Can we get rid of complex number? [duplicate]

Complex number is very weird to me, especially so when they appear in engineering and physics equations. It is possible to represent complex number as real matrix where $i = \begin{pmatrix} 0&1\\...
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1answer
50 views

Derivation for angular acceleration from quaternion profile

Given a profile of unit quaternions $q(t)$ that represents the orientation of a body over time, I like to get the angular acceleration $\dot \omega (t)$. I tried to find a formula myself, but I get ...
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62 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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1answer
36 views

Confused over the complex term in the simple harmonic wave equation

I am trying to derive the general equation of Lamb wave. My book says that $$y = A\exp(i(kx−\omega t))$$ is the general equation of simple harmonic wave propagating in +ve $x$ direction. but I am ...
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59 views

Is it necessary to learn complex analysis in order to learn classical electrodynamics?

I am reading "Classical electricity and magnetism, chapter 1" by Wolfgang K. H. Panofsky and Melba Phillips. I am having little trouble on page 13 and afterwards. It talks about singularity, poles, $2^...
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Why doesn't Feynman take the real part of a complex EM wave before finding the intensity?

This is from The Feynman Lectures vol 2 chapter ch 33 on reflection of complex EM waves on metal. In the above I would expect $\frac{I_r}{I_i} = \frac{Re(E'_0)^2}{Re(E_0)^2}$, rather then $\frac{...
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1answer
55 views

Pure imaginary Schroedinger wave function

I know that the solutions to the time-dependent Schrodinger equation are always linear combinations of the form $$ \Psi(x,t)=\sum_n c_n e^{-iE_nt/\hbar} \psi_n(x) $$ If $ \Psi(x,0) $ is PURELY ...
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61 views

Action of complex conjugation on Hamiltonian

Consider a finite-dimensional non-relativistic QM system with hamiltonian $H$. Let $K$ denote the complex conjugation operator. What does $K H K$ simplify to, if the system is: (a) spin-zero; (b) spin-...
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34 views

But how exactly do you calculate the Joukowsky Airfoil, within a minimal margin of error?

After reading a fair bit of theory around the uses of conformal mapping to solve for the forces of lift acting on a wing, or a 2D cross section of the wing, in relation to the angle of attack. However ...
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1answer
160 views

Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
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1answer
51 views

When can viscosity be ignored?

I am working on understanding the motion of a solid ball in an infinite elastic (or possibly viscoelastic) medium when subjected to a sinusoidal driving force. The frequency-dependent complex wave ...
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1answer
57 views

Scalar particles are described by a real scalar field or by a complex one?

Well, in the title is already stated my main question. I know you can use a complex scalar field to describe two real scalar fields, by using just one that involves both of them. But, in the modern ...
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77 views

Reflectivity with Complex Refractive Index

I would like to ask a followup question on a previous post found here. Starting from the general expression for reflectivity: $$R = \bigl\lvert\frac{n_1-n_2}{n_1+n_2}\bigr\rvert^2$$ and substituting ...
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1answer
30 views

Phasor rotation direction in electromagnetism

In electromagnetism, we often define a field propagating in the $z$ direction over time $t$, with wavenumber $k$ and angular frequency $\omega$ as $\text{cos}(-kz+\omega t + \phi)$. Choosing $z = 0$ ...
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63 views

How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate

If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean? When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
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146 views

Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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Interpretation of the magnetic potential ($A$-field) in the quantum mechanical probability of current

The probability of current in quantum mechanics when the is a magnetic potential, A, is defined as: $$\boldsymbol j=\frac{1}{2m}(\psi^*\hat{\boldsymbol p} \psi-\psi\hat{\boldsymbol p}\psi^* -2q{\...
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74 views

How does the Dirac operator change under adjoint transformation?

I'm trying to demonstrate an identity $$\int \overline{\psi}D\phi = \int \overline{D\psi}\phi$$ by substituting in the dirac operator as $D = i\gamma^{a}\partial_{a}$ and $\overline{\psi} = \psi^{\...
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67 views

Some basics about Bracket Notation

I'm trying to prove something. Sorry this post is so long but I wanted to keep things as basic as possible so people have an easier time understanding. Let's assume we have a quantum system $\rho$ ...
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1answer
44 views

Electrodynamics : Problem with Notations for fields $\vec{(\vec{r},t)}$ and $\vec{B}(\vec{r},t)$(complex and real notations)

I'm sutyding a course on electrodynamics and am stuck on a few lines I can't make sense of. The professor uses $$\vec{E}(\vec{r},t) = \vec{U_0} cos (\vec{k}\cdot \vec{r} - \omega t + \phi)$$ (so far, ...
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66 views

Complex solutions to an Underdamped Oscillator

In many of the books talking about damped simple harmonic motion, the underdamped oscillator is treated as follows: Newton's second law says $$m\ddot{x} + r\dot{x} + sx = 0 $$where s is stiffness ...
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1answer
62 views

The hermitian conjugate of anti-linear operator

Some quantum mechanics books tell us that the definitions of hermitian are If $\langle\psi|A\phi\rangle=\langle B\psi|\phi\rangle$ for linear operators, then $B=A^\dagger$ If $\langle\psi|C\...
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Solution to Klein-Gordon equation: real field condition and other questions

Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. ...
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Would the time dependent wave equation of a free particle be the same in 4D Euclidean spacetime as it is in 4D Minkowski spacetime?

In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there ...
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99 views

Value of complex angle in Snell's law

According to Snell's law: $$\sin \theta_t = \displaystyle \frac{n_1}{n_2} \sin \theta_i$$ where $\theta_i$ is the incidence angle in medium 1 and (after the ray crossed the interface between medium ...
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1answer
38 views

Forced Oscillations and Complex Representation

An oscillating force $F \cos \omega t = \Re\{Fe^{i\omega t}\}$, where $F$ is real, is applied to a mass $m$ on the end of a spring with spring constant $k$. The displacement, $x$, of the particle can ...
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204 views

Lagrangian density for a complex scalar field

I am taking a course on classical field theory, and am not entirely sure as to what motivates the for of the Lagrangian density for a complex scalar field. In my lecture notes, this is first ...
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1answer
33 views

Complex permittivity of mixture

I have been reading a few papers regarding the estimation of the permittivity $\epsilon_{mix}$ of a mixture of materials of permittivities $\epsilon_1$ and $\epsilon_2$ (I am not a phisicist or a ...
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125 views

Must a classical Lagrangian or a Hamiltonian be a real function?

$\bullet$ Is it fair to assume that the classical Hamiltonian or Lagrangian of a system (a particle or a field) is always a real-valued function? $\bullet$ If not, can you provide counter-examples? ...