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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How should I use the complex permittivity of a material?

Here: $$\epsilon = \epsilon' + j *\epsilon''$$ I understand that the first part ($\epsilon'$) is the relative permittivity of a material, while the second part $\epsilon'' = \frac{\sigma}{\epsilon_0\...
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Complex scale factor for a model in cosmology [closed]

I was calculating the scale factor versus the time but I just realized that a scale factor is a complex number in my model. What is the mean complex number for the scale factor?
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The Partition Function of $0$-Dimensional $\phi^{4}$ Theory

My question is related with this question. Several years ago, I posted an answer to the question, and the author of the reference removed the link permanently, now I have no clue what's going on. In ...
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Arbitrarity of $i$ in the propagator

My question is simple: how arbitrary can the factor in front of the propagator be? What I mean by that is, if we call the wave operator $K$ and the propagator $G$, I've seen different books use ...
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Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function?

Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function instead of the one with $e^{ikx}$? Both are the solutions but the one with $e^{ikx}$ is seldom used.
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1D bound state for a real potential

The prof says: "for 1Dimensional bound states with a real potential, the wave function is real, up to a phase". The proof goes like this: 1D bound states are never degenerated. So $\Psi_{...
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Singularities of $f(z)=\dfrac{z}{\sin\pi z^{2}}$? [migrated]

Consider the function $f(z) = \dfrac{z}{\sin\pi z^{2}}$. It has a simple pole at $z=0$. There are also 4 poles at $z=\pm\sqrt{n}$ and $z=\pm i\sqrt{n}$, where $n\in Z^{+}$. What is the order of the ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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Questions on the Zakharov-Shabat inverse scattering paper

I am trying to work through the Zakharov and Shabat paper on inverse scattering for the nonlinear Schrodinger equation (PDF). I am stuck on section 2. Problem 1. I need to know how to reconstruct $\...
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2D rotation dynamics/control systems as a complex number

I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode. I'm having trouble defining the ...
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Applying Kramers-Kronig to the real part of conductivity results in a real and imaginary term

Following Tinkham's Introduction to Superconductivity, the complex conductivity in the two-fluid approximation is defined as \begin{equation} \sigma_i(\omega)\equiv\sigma_{1i}(\omega)-i\sigma_{2i} \...
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Real and Imaginary part for particle polarisability

I am trying to determine the real and imaginary parts from the following equation for the polarisability of particles $\Lambda_{MG}(\lambda)$: $$\Lambda_{MG}(\lambda) = \frac{\epsilon(\lambda) - \...
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Why don't we chose $\det(H)$ in winding number density?

Hello i have a short question on the winding number in chiral systems. If we have a chiral system described by a hamiltonian like this $$ H = \left [ \begin{array}{cc} 0 & K \\ K^{\dagger} & 0 ...
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Why are the electromagnetic wave equation is represented in a complex number? [duplicate]

The wave function $$ f(x,t) = A \sin (\vec{k}\cdot\vec{x} - \omega t+ \phi) $$ Can graphically describe the linear 2d wave propagation. Why this equations is written in this form: $$ f(x,t) = A [\cos ...
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What is the fundamental reason for the imaginary unit in Heisenberg's commutator relations?

The well known Heisenberg commutator relation $$[p,q]=\cfrac{\hbar}{i} \cdot \mathbb{I}$$ introduces the imaginary unit $i$ into quantum mechanics. I ask for the deeper reason: Why does the ...
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Separating a complex number into real and imaginary equations

I'm trying to find motion equations in a rotating frame, using the Cauchy-Riemann equations. I've currently arrive at the function $$\eta(t)=e^{-i\Omega t}\left[ x_0+v_{x,0}t + i(v_{y,0}+\Omega x_0)t ...
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6 votes
2 answers
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Why does time reversal symmetry requires a real Hamiltonian?

I have some problems understanding the consequences of time reversal symmetry. If Hamiltonian $H$ is symmetric under time reversal, it satisfies: $$ \mathcal{T} H \mathcal{T}^{-1} = H \quad \mathrm{...
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3 votes
1 answer
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Anti-holomorphic contribution to 2d conformal algebra

I am reading Ginsparg's notes on 2D-CFT, and I am deeply confused about why Ginsparg states after (1.8) that the conformal algebra for 2d Euclidean space consists of two copies of the Witt algebra. My ...
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On the physics of representations of the Lorentz group over real vs. complex vector spaces

After reviewing this question as well as this one, I am left with some confusion, mainly about the nature of complex and real representations of the Lorentz group and how we do physics. I understand ...
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"Proper Branch Cuts" in A Physical Problem

I am solving the following acoustic problem. Consider a two-dimensional duct bounded by two parallel walls separated by a height of $H$ as shown. The duct is filled with air. A vibrating piston(...
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Two Liouville's theorem

Within the context of Hamiltonian mechanics and phase spaces I have learnt that the phase space distribution function is constant, in other words, that the "volume" of any region is constant,...
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Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$

Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom. We start by accepting the ...
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Converting the complex Wigner function to its real form in terms of the quadrature operators

I noticed something that bugged me recently, the Wigner function which is defined for one mode in the complex plane as $$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{...
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Path integral with double integration involving the free particle case

Suppose we have the path integral: \begin{equation} Z=\int \mathcal{D}x\mathcal{D}y\,\exp\left[-\frac{a}{2}\int_0^1 dt\,\left(\,\dot{x}(t)^2-\,\dot{y}(t)^2\right)\right]. \end{equation} The ...
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Solving a complex Gaussian Integral for Path Integral Formalism [duplicate]

I am trying to solve the following integral, $$\int dz_1d\bar{z}_1\cdots dz_nd\bar{z}_n\:\exp(-\sum_{i,j}\bar{z}_i A_{ij}z_j),$$ where $A_{ij}$ is an $n\times n$ hermitian matrix. I know how to do ...
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Second linearly independent solution of Airy Differential equation

The Airy differential equation is $$ \frac{d^2y}{dx^2}=xy. $$ After Fourier transforming the equation, we get $$ y=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\left(kx+\frac{k^3}{3}\right)}dk. $$ Here $k$...
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Klein-Gordon Hamiltonian in terms of Fourier transformed variables

The Klein-Gordon Hamiltonian density is a function of four complex variables $\psi , \psi ^* , \pi , \pi ^*$. Suppose we make the change to Fourier transformed variables. Then the Fourier expansions ...
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Does the Dirac Spinor live in the complexification of the Lorentz group?

In this question I learned that when working with quaternionic representations of (the double cover of) our relevant orthogonal group, we cannot avoid working in complex vector spaces. However it is ...
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Interference pattern double slit - exponential electric field

So I am trying to prove that the intensity received at a screen from a light point source through two slits is $I(x)=cos^2(\frac{\pi ax}{\lambda D})$. I know that the electric field at screen is $E(x)=...
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Computing $\frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha)\exp(-|\lambda|^{2}/2)d^{2}\lambda$

This came up when attempting to do a routine calculation of Wigner function of the vacuum state $$ \frac{1}{\pi^{2}} \int_{\mathbb{C}} \exp(\lambda\alpha^{*}-\lambda^{*}\alpha-|\lambda|^{2}/2)d^{2}\...
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Imaginary functions in metric elements

I have solved Einstein's equations for a specific black hole, starting from a general metric with some unknown functions, after solving differential equations I had one imaginary solution. What does ...
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How can I reconcile the apparent all-zeroes diagonal of commutators with the canonical commutation relation? [duplicate]

Out of interest I was trying to derive some properties of commutators in quantum mechanics. I found that, in my calculations, will have an all-zeroes main diagonal. I must be making a mistake, because ...
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3 answers
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Why we take complex current into effect why not just the real part and similarily power formula why we use complex conjugate of current?

In a RLC circuit having a AC source . The actual current flowing in any branch will be the real part of the complex current. The imaginary component has no important role. It is just there. Now that ...
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Why the intensity of a wave is $ \Psi \Psi^*$?

In here at the bottom, it says the intensity of a wave is the wave phasor times it's conjucate $$ I(x) = \Psi \Psi^* = |\Psi |^2$$ But when I compute the intensity of an electromagnetic wave in c.g.s, ...
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Convergence of an oscillatory integral to real number

I have a physical model, where the long-time behavior of the system can be described by $$C(t)=\frac{1}{2\pi}\int_{-\pi}^\pi\mathrm{d}k\,\mathrm{e}^{-t\omega(k)}$$ with $\omega(k)\geq0$ and $\omega(t)\...
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Bras and kets question

If a state $|\psi\rangle$ can be written as a linear combination of the orthonormal states $|\phi_{n}\rangle$ as: $|\psi\rangle=\sum_{n=1}^{\infty}c_{n}|\phi_{n}\rangle$ then is it valid to write: $\...
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Connection between Möbius transformation and equipotential lines in 2D electrostatics

Recently, while reading about the Möbius transformation, I found this picture: As written on the Wikipedia page, this represents an hyperbolic Möbius transformation. Does this remind you of anything? ...
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Complex Vectors: $E$ & $H$ field coordinate transform

We are given an electric field (or magnetic field) described by complex phasors for the three Cartesian components, $$ \vec{E} = E_x \hat{\imath}+E_y \hat{\jmath}+E_z \hat{k} $$ with $$ E_x = E_{x0} e^...
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9 votes
2 answers
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Path integral for complex scalar field

I am taking a QFT course which focuses on the path integral formulation. At a certain point, I was confused because we saw that, when integrating over complex Grassmann fields for fermions, we defined ...
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Feynman rule with complex coupling

Suppose I have 3 complex scalar fields and an interaction term $$ \mathcal{L}\supset -g \chi \phi_1 \phi_2 + {\rm h.c.} $$ where $g$ is a complex constant whose phase I cannot get rid of by field ...
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Real part of electromagnetic wave

If I want to calculate $(\Re(\vec{E}))^2$ of $\vec{E}=\vec{E}_0\mathrm{e}^{\mathrm{i}(\vec{k}\vec{x}-\omega t)}$ how get I $|\vec{E}_0|^2\cos^2(\vec{k}\vec{x}-\omega t)$? I would say that $\Re(\vec{E}...
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Symmetry Factor for Feynman diagram of complex field

When studying Feynman diagram I have been told that when treating a complex scalar field we use pertubation which is normalize with factor $1/n$ (where $n$ is the number of terms in the perturbation) ...
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Quaternion formulation for parallel transport along a curve on a 2-sphere

One image that is often used to illustrate curvature in general relativity is the triangle on a 2-sphere, made out of great circle arcs. At the end of a geodesic transport along this triangle, the ...
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Why particularly probability density is defined as $|\Psi|^2=\Psi \Psi^{*}$?

It may be a stupid question, but why particularly for probability density expression $k~|\Psi|^2 = k~\Psi^{*}\Psi$, it's assumed that $k=1$? As it is now, then in a complex plane probability density ...
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I am stuck with standing waves mathematical treatment when I am using a complex function

$$f=Ae^{i\omega t}\big(e^{-ikz}-e^{ikz}\big) = -2iAe^{i\omega t}\sin(kz)$$ This is the displacement equation I got. As it is complex, I used the real part to represent the nth normal mode of standing ...
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Find Voltage using Complex Impedance

I am trying to calculate the Voltage at the top right segment ($V_{out}$ in the diagram). Importantly, I'm trying to calculate this voltage using complex impedance. I know in the complex impedance ...
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4 answers
151 views

What's the meaning of a complex momentum in classical mechanics?

I'm looking at a section of Griffiths and Schroeter's Introduction to Quantum Mechanics, pp. 355. It states a straightforward set of equations that got me thinking about the exact way in which complex ...
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How does this experiment "rule out real-valued standard formalism of quantum theory"?

I came cross this paper: https://arxiv.org/abs/2103.08123 To be frank I don't understand most of it, but the summary seems a bit shocking. I found it rather strange. There have been multiple ...
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In a 2D complex vector space, are there more than 3 orthonormal bases that are all "45 degrees" from each other?

If I understand correctly, if a quantum spin is measured/prepared in the $+x$ direction, then there is a .5 probability of subsequently measuring it in the $+y$ direction and .5 probability of ...
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4 votes
3 answers
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Appearance of $e^{i(kx-\omega t)}$ in the derivation of Schroedinger Equation

One of the derivations of Schroedinger's Equation that I came across assumed the wave function for the particle to be $$\Psi = e^{i(kx-\omega t)} ,$$ and I hardly understand why. This equation implies ...
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