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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Complex physical quantities

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, ...
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Quantum probabilities of a projection - does it require two different definitions…?

Say I have a wave-function $$ |\psi \rangle=\pmatrix{a_1+ib_1\\a_2+ib_2} $$ where of course $\langle \psi |\psi \rangle=1$. I can get the probability of a given state as follows: $$ \begin{align} \...
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Inner product of $\langle \phi | \psi \rangle$ gives a complex value - why/meaning?

Say I have the following two wave-functions: $$ |\psi\rangle= \pmatrix{a_1+i b_1\\ a_2+ib_2 } $$ $$ |\phi\rangle= \pmatrix{c_1+i d_1\\ c_2+id_2 } $$ Since these are unit vectors of the Hilbert space, ...
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Complex fourier transformation function [duplicate]

can the FT function can be a complex function ? and if yes what does it mean because in all cases i came across till now the FT function only had a real part
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Meaning of complex Fourier transformation function

what does it mean, if the Fourier transformation function of an electromagnetic wave is complex? I know that normally the FT function $f(k)$ shows the wavenumbers that are involved in the wave. but i ...
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Wave equation and skin-depth

I initially posted this question to the math stack overflow, but realized that it is really a physics question This is a follow-up question to one of my unanswered questions https://math....
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76 views

Solution to differential equation

If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$ Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\...
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Why not just complex conjugate bras and kets instead of Hermitian conjugate?

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate ...
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Electric field/intensity for complex envelope

When calculating the propagation of a pulse I can either use an envelope-based or a carrier-based approach. For the carrier-based approach I can define my (focused) pulse with the central frequency $\...
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(Non-)Hermiticity of Dirac operator

I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ ...
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How the Lorentz algebra bracket relations change during complexification

There have been a lot of questions in the past about the subtleties of the Lorentz algebra. In particular the usage of the real Lie algebra $\mathfrak{so}(1,3;\Bbb R)$ and it's complexification which ...
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Mysterious path integral divergence after Hubbard-Stratonovich transformation

Let us consider common gaussian path integral over some complex random field $\displaystyle \Psi (\mathbf{r})$: \begin{equation*} N=\int D\Psi ^{*} D\Psi \ \exp\left( -\int d^{n} r\ \Psi ^{*}\hat{K} \...
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Residue of the Fermi Distribution Function

In the "Lecture notes on many-body theory" by Michele Fabrizio, it is stated: How we do show that the Fermi distribution function $f(z)$ has residue $-T$? In the examples on Wikipedia, the ...
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How to decide if a mass matrix is real or complex?

In the lagrantian in particle physics, how to decide if a mass matrix is real or complex? As I know, a Dirac field is complex while a Majorana field is real. Does a complex mass matrix correspond to ...
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Complex time theories with spacetime $\mathbb{R}^3\times\mathbb{C}$

Are there any well-developed (string?..) theories assuming that, what we perceive as a (3+1) Minkowskian manifold, is a projection/compactification of a 5-dim spacetime, locally obtained via ...
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Gauge theory on complex manifolds

recently I'm pondering on defining gauge theory on complexified space. However, I found that it is hard to make electromagnetism (which will be the simplest gauge theory) work well on complex ...
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Deriving the complex dynamic modulus in a maxwell model

In the context of the Maxwell model for viscoelastic materials, how would one derive the complex dynamic modulus? It is defined as the ratio of the stress to the strain.
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Dielectric constant and conductivity in Maxwell's macroscopic equation

In my experimental physics lecture we are looking at the Maxwell equations in matter (macroscopic Maxwell equations) and there is a point where we jump from $$ \nabla \times \vec B = \mu_0 \biggl( \...
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Is the time dimension naturally linked to the real axis?

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, ...
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Gaussian integral with respect to Grassmann variables

Let $A$ be an antisymmetric matrix of even dimension $n$ and $\theta$ be a column vector consisting of $n$ Grassmann variables $\theta_i$. Then the solution of the integral $$\int d\theta_1\dots d\...
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In the Wick rotated path integral, are the paths functions of an imaginary time variable?

Consider the following action: \begin{equation} S=\int_{-\infty}^{\infty}[\frac{1}{2} \dot x^2(t)-V(x(t))]dt. \end{equation} I promoted $t$ to a complex parameter, and calculate the action over the ...
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Expression for the causal retarded potential for $t<0$ must give $0$ but my calculation produces a nonzero result. What's the mistake?

This question was previously asked here in the Mathematics StackExchange but using a slightly different notation. But I did not find the answer I was looking for or rather got two very different ...
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QED Lagrangian - real or complex?

I'm confused about the use of complex numbers in the QED Lagrangian: $$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e\bar{\psi}\gamma^{\mu}A_\mu\psi.$$ ...
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Solving coupled propogation equations for EM waves

I have recently come across a set of partial differentials that describe the propogation of two coupled EM fields, in a 2D system currently being investigated. In their most general form they are $\...
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Does swapping “$i$” with “$-i$” change a physical theory?

Mathematically speaking, "$i$" and "$-i$" are the two roots of the equation $x^2+1=0$ and it seems to me at least that there is no obvious way of distinguishing between them. Thus, ...
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Do we necessarily need real number for quantum? [closed]

In the quantum mechanics, one asked if the complex number was necessary? A typical answer was that it was not, or that it's simple direct product of real numbers. However, consider rational number to ...
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Complex exponential method of solving differential equations

In the twenty third Feynman lecture, the solution of the following differential equation is discussed: $$ \frac{d^2 x}{dt^2} + \frac{kx}{m} = \frac{F}{m}$$ AFter 'complexifying' this differential ...
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How to find the input impedance of a woodwind instrument? (playing frequency of a woodwind instrument)

I'm trying to reproduce the model described in this paper https://hal.archives-ouvertes.fr/file/index/docid/683477/filename/clarinette-logique-8.pdf. The logical clarinet is a succession of 18 ...
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The use of complex fields in electromagnetism

Virtually all treatments of electromagnetic wave propagation, and in particular of monochromatic plane waves, use basic complex analysis to simplify calculations. I am comfortable with these ...
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Derivative of a complex potential for the $\lambda \Phi^{4}$-model

A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given ...
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Phasor transformation to sinus or cosinus?

In my EM waves lecture, our lecturer somehow explained the way we make phasor transformation of a particular function such as $$A\cos(\omega t-( \alpha +\beta z))u_{y}$$ converted into phasor form of $...
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Kramers-Kronig relations for geometric series

Suppose $\phi(z)$ is an analytic function in the upper complex plane, so it satisfies the Kramers-Kronig relations, i.e. \begin{equation} \Re\phi(w) = \frac{1}{\pi}\int \frac{\Im\phi(x)}{x-w} dx \end{...
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Quantum Mechanics Griffiths Problem [closed]

I was doing problem no. $4.4$ from Griffiths Third Edition. I cannot understand one thing related to the solution of the normalization of $Y_2^1$. $$Y_2^1 = -\sqrt{\frac{15}{8\pi}}e^{i\phi}sin\theta ...
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Imaginary part of complex non-Hermitian Hamiltonians

I have just started studying the $PT$-symmetric and non-hermitian hamiltonians. But I am not able to interpret the imaginary part of the hamiltonian. If Hamiltonian is basically the total energy of ...
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Are gaussian integrals also valid for complex constants? [duplicate]

In the WP article about propagators, there is an integral solved as: $$K(x,x';t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk\,e^{ik(x-x')} e^{-\frac{i\hbar k^2 t}{2m}}=\left(\frac{m}{2\pi i\hbar t}\right)...
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Real and imaginary parts of derivative of complex velocity in fluids

Just by chance, I noticed that for an ideal fluid which has a complex velocity say $w=u-iv$, that its derivative with respect to $z=x+iy,$ when written out in real and imaginary parts, looks like $$\...
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Confusion About Complex and Scalar Fields

I have been reading QFT for the Gifted Amateur, on page 110, it reads: "For the case of the real scalar field...it is Hermitian. For our complex scalar field there is no reason why the field ...
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Problems deriving the Quantum Hamilton-Jacobi equation

This is my first question at Physics SE so please be kind. I am well versed in the etiquette over at Math SE, but not so much here. Anyway, I thought this question was better suited to this site ...
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Modeling a 3D Wing — Joukowsky Airfoils and Prandtl Lifting-Line Theory

I am trying to calculate the distribution of lift over the span of a rectangular, non-swept wing with a (constant) Joukowsky airfoil cross-section. The wing is rectangular in the sense that the chord ...
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Fourier transform of a real initial wavefunction

Consider the initial wavefunction given by: $$ \Psi (x,0) = \sin(k_0 x).$$ I've been taught that in order to time evolve a wavepacket one must first find the momentum space representation of the ...
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Transition amplitude integral and causality

I was trying to prove that Quantum Mechanics violates causality. To do that, I started by computing the transition amplitude between the fixed position $x_0$ and an arbitrary position $x$, during a ...
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Is this definition of complex wave number in dispersive media correct?

In Griffith's Introduction to Electrodynamics (4th edition, p.421), the complex wave number in the section on dispersive media is defined as $\tilde{k}=\sqrt{\tilde{\epsilon}\mu_0}\omega$. Why is the ...
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Why is the Quantum Electrodynamic Hamiltonian complex?

For reference, this is from Shankar's QM book, with the Hamiltonian $$H=\frac{1}{2m}\left(\textbf{P}\cdot\textbf{P}-\frac{q}{c}\textbf{P}\cdot\textbf{A}-\frac{q}{c}\textbf{A}\cdot\textbf{P}+\frac{q^2}{...
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Does an imaginary expectation value correspond to 0? [closed]

If an expectation value is purely imaginary, then the real component is obviously 0. Because expectation values are real quantities, does this mean that the expected value must be 0? I feel like this ...
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Calculate a rotational speed for an object spinning in 3 axes

I've got an object spinning in 3 axes and I'm tracking it with a motion capture system. For each timepoint, depending on how I export the data, I either get 4 columns of data for the quaternion ...
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1answer
52 views

Solving the 2nd order linear differential equation in $LC$ oscillator [duplicate]

I was reading Halliday and Resnick and it had given the differential equation $$L\frac{d^2q}{dt^2} +\frac{1}{C}q=0$$ Now when I attempted to solve this equation, I got $q=Q(\cos \omega t+i\sin \omega ...
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Complex notation in harmonic oscillator

For a simple harmonic oscillator, $$x(t) = A \cos(\omega t).$$ We can also write $x(t)$ as: $$x(t) = C_1 e^{i\omega t} + C_2 e^{-i\omega t}.$$ Why is it necessary that the coefficients $C_1$ and $C_2$ ...
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Question regarding step potential

We are learning about step potential in class. I have completely understood that the behavior of the wave function representing the particle, can have different responses depending on the energy of ...
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71 views

Why could $i\hat i$ in complex quaternions be identified as $\sigma_x$?

In the complex quaternions algebra $\mathbb{C}\otimes\mathbb{H}$, there're 8 elements: 1, $i$, $\hat i$, $\hat j$, $\hat k$ (quateronic ijk), $i\hat i$, $i\hat j$, $i\hat k$. The last three objects ...
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Conjugate complex of linear operators in quantum mechanics

I'm pretty new to quantum mechanics (I would like to understand it broadly as an hobbyist). I'm trying to reading Principles of Quantum Mechanics by Dirac. I've found difficult to understand a ...

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