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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Ignore the imaginary part of a solution when numerically integrating

I have the following equation: $$\frac{\pi\cot B}{A}=\int^{\pi}_0\left(\frac{\frac{b\cos x\sin^2B}{R}+\sqrt{-\frac{b^2}{R^2}\sin^2B\sin^2x+sin^2x+cos^2x\sin^2B}}{sin^2x+cos^2xsin^2B}\right)^2\cos xdx$$...
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How is Wick rotation an analytic continuation?

Wick rotation is formally described by the transformation $$t \mapsto it.$$ In many place it is stated more rigorously as an analytic continuation into imaginary time. I understand why we do it but ...
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Planar spin in two-dimensional CFT

I have several questions regarding the definition of planar spin. I was reading the big yellow book (by Di Francesco et. al.) Section 5.1.5 looks a little mysterious. Look at 5.25, which is the two-...
hossein mohammadi's user avatar
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How can the Rabi frequency be complex?

I've been doing some reading and came across a simple implementation of the Hadamard gate using Rabi oscillations of an atom in a laser field. However, the author mentions that it required the Rabi ...
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Conflicting definitions of vector conjugate in QM

Let $e$ be a finitely matrix representable operator. In physics, specially in quantum mechanics (QM), it is customary to define the conjugate operator $e^{\dagger}$, as the adjoint or the Hermitian ...
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Complex gaussian integral with a complex action and different source terms [duplicate]

I am trying to use the following Gaussian path integral identity $$\int D[\phi_1,\phi_1^*,\cdots,\phi_n,\phi_n^*] \exp(i\int z^\dagger D z+i\int f^\dagger z+z^\dagger g) = \det{D}^{-1}\exp(-i\int f^\...
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Possibility of complex EM waves

I'm currently studying Quantum Mechanics, and I have just been presented Schrödinger's (time dependent) equation. Of course, the first solution to said equation I've been taught is that of a (complex) ...
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Do all qubits have complex coefficients or are there “real qubits” as well?

I understand that a qubit is a quantum system with two basic states $|0\rangle$ and $|1\rangle$, so a general pure state of the qubit will be described by a linear combination $\lambda|0\rangle + \mu|...
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What is an imaginary gauge potential?

This paper considers a generalised Strum-Liouville equation, that is equations of the form $$ \left[-\frac{d}{dx}p(x)\frac{d}{dx}-\frac{i}{2}\left(\lambda_1(x)\frac{d}{dx}+\frac{d}{dx}\lambda_2(x)\...
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Why is psi square a possibility? [duplicate]

Is psi square just an assumption? Or there is a physical reason why they defined like that? My procedure is: It is intuitive for me to think possibility is proportional to energy distribution. ...
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Adjoint and index notation in Weyl field context

In the answer to a question I previously asked, the following manipulation was done but I don’t understand it$.$ $$ (U_{jm}\psi_m)^\dagger=\psi_m^\dagger U_{mj}^\dagger $$ aside from the context from ...
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Charge conjugated Dirac equation

I would very much like to understand the motivation behind the correlation between: $(i\partial\!\!/-eA\!\!/-m)\psi=0$ and $(i\partial\!\!/+eA\!\!/-m)\psi_c=0$ when dealing with the derivation of the ...
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Non-Hermitian Hamiltonian in the Heisenberg Picture

I am trying to study a system whose Hamiltonian, after some transformations can be written as \begin{equation} \hat{H} = \hat{N}_1(\omega-i\mu)+\hat{N}_2(\omega +i\mu)+\omega\hat{\mathbb{I}}, \end{...
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What is special about conformal field theory in 2d? [duplicate]

In the most of textbooks about CFT, the special case of 2d is noticed in which complex coordinates play important role and it reads some results like the conformal transformation of energy-momentum ...
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Beam splitter with complex parameter

A non-polarizing beam splitter can usually be described by a unitary operator such as $U=e^{i\theta(a^\dagger b+b^\dagger a)}$ given a parameter $\theta\in \mathbb R$ and a pair of independent modes $...
Quantastic's user avatar
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Wave amplitude as a complex number?

In section 1-3 An experiment with waves of The Feynman Lectures on Physics (https://www.feynmanlectures.caltech.edu/III_01.html) it says: "The instantaneous height of the water wave at the ...
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Standard Model irreducible representation notations

Does anyone know what the significance/meaning of the subscripts on the following notations for the irreps. in the Standard Model? an up, down left-hand fermion (quark) pair $(u,d)_L$ is denoted: $(\...
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Multiplying a force vector by rotated unit vector produces strange results [closed]

I'm by no means an expert in math, what I'm trying to do is to Isolate a force aligned with a vehicle in a game (specifically to do a directional friction). the equation is simple I take a vector $v_1$...
LemonJumps's user avatar
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Application of Cauchy residue theorem to Matsubara sums

For reference, this derivation is closely related to the discussion on pp. 169-173 of Altland and Simons. In quantum field theory (specifically when calculating free fermionic propagators via coherent ...
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Wick rotation and Exponential mapping of an "imaginary" differential operator acting on a real-valued wavefunction

The position shift operator $T^{a}$ (where $a \in \mathbb{R}$ ) takes a real valued wavefunction $\psi$ on $\mathbb{R}$ to its translation $\psi_{a}$, $T^{a} \psi(x)=\psi_{a}(x)=\psi(x+a)$. A ...
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Euler-Lagrange for Dirac Lagrangian - is $\bar \psi$ independent of $\psi$?

In Peskin and Schroeder (3.34) they write the Dirac Lagrangian: $$ L_{Dirac} = \bar \psi (i \gamma^\mu \partial_\mu - m ) \psi $$ where $\bar \psi = \psi^\dagger \gamma^0$. Then, they write: "The ...
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Fourier transform of Green function using residue theroem

I want to compute the Fourier transform of a Green function in $k$-space : $$ G^R_{n,m}(\omega)=\int_0^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(n-m)}}{\omega+i\eta-\epsilon_k} $$ By substituting $\omega$ and ...
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Can $\int e^{ix^2 t} dx$ be defined without going to imaginary time? [duplicate]

The motivation is trying to define the path integral, where at some point we get the integral $$ Z=\int e^{iS}Dx $$ which is then taken to imaginary time $$ Z_E=\int e^{-S_E}Dx $$ such that $Z_E$ can ...
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Help With Complex Numbers in Polar Form in the Context of AC Voltages

I'm learning about AC right now, and I think I've got an OK grasp of complex numbers (not in their polar form, though), but I don't understand this step in a derivation at all: $$V_i=V_0e^{j\omega t}$$...
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Does the completeness relation for vector bosons hold also in this form?

Consider a generic Spin-1 vector field (massless) $A^{\mu}$, we know that the solution for its equations of motion can be built as $$A^{\mu}(x)=∫ d^{3}kN_{k}(\epsilon^{\mu}(\stackrel{\rightarrow}{k},\...
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How do I show that a transformation is a symmetry of the Lagrangian? [closed]

Hello All I have the following question see part (b). I have already done part a and my answer is $L=\frac{m}{2}\dot{z}\bar{\dot{z}}-\frac{k}{2}z\bar{z}$. I have no idea how to go about part b, I ...
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What is meant by dimension of the defining representation (and adjoint representation)?

A linear representation of a classical Lie group $G$ is defined by $\rho:g \to GL(V)$ where $g \in G$ is a group element and $V$ is the representation space. The dimension of the representation space(...
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How do I write angular displacment in quaternion form? [closed]

Work done is given by $W = F_ids_i$ and in angular displacement moment form it is $W = \tau_i d\theta_i$. I want to get the work done in using quaternion instead. Like $W = \tau_i q_i$ I cannot figure ...
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Absolute values when normalising the wavefunction

My question concerns Problem 1.4 from Griffiths. I understand the general working of the problem (given below), and derived the correct result for the normalisation constant $A$, but I am troubled by ...
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Correct frame for angular velocity in quaternion's kinematic

I am reading a paper where the quaternion's kinematic is used, unfortunately the description of the angular velocity does not match with how it's computed, so I have a doubt on which frame $\omega$ is ...
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Complex solution in equation of motion [duplicate]

When I tried to solve some motion questions, I got complex numbers for time, displacement, etc. And my teacher said my answer was correct. Is it possible to have a complex number solution in the ...
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Cosmology: Covariance between Gaussian distributions for complex spherical harmonics coeficients

In the context of the computation of a variance about $a_{\ell m}$ spherical coefficients of Legendre, I am faced to an issue : There is a term $\langle \text{Re}(a)\text{Im}(a)\rangle$ that appears ...
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Quantum Mechanics without Complex Numbers in a multipartite setting

I was fairly convinced that usual QM formalism didn't necessitate the use of complex numbers and that ultimately they're just a matter of convenience and utility rather than anything fundamental. This ...
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Poisson noise on $a_{\ell m}$ complex number: real or complex?

In a cosmology context, when I add a centered Poisson noise on $a_{\ell m}$ and I take the definition of a $C_{\ell}$ this way : $C_{\ell}=\dfrac{1}{2\ell+1} \sum_{m=-\ell}^{+\ell} \left(a_{\ell m}+\...
guizmo133's user avatar
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196 views

Why using real wave functions instead of complex ones?

I have already seen similar questions asked in the site (like this or this), but I don't feel that my question has been fully addressed. I understand that orbitals $np_x$ and $np_y$ are linear ...
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Why there are no Dirac $B $ boson monopoles?

This differs from an earlier post in that the writing is new and hope better, it uses MathJax, and it gives some details about the particle that poses a problem for certain commonly used theories of ...
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Complex numbers in quantum mechanics [duplicate]

Are complex numbers used in the way the are in quantum mechanics for convenience sake? Or are they fundamental to quantum mechanics. In other words can quantum mechanics be completely described ...
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Polchinski's doubling trick for extending open string theory to the whole complex plane

Open string theory can be described on the upper-half complex plane. To simplify the description of open string theory, Polchinski asserts (eq. 2.6.28 in his Vol. I String Theory book) that it is ...
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Proving that the interference current integrated over a small cone does not depend on the angle of the cone

I'm studying quantum mechanical scattering and I have gotten to $$\psi=\psi_{in}+\psi_{scattering}=e^{ikrcos\theta}+f(\theta,\phi)\frac{e^{ikr}}{r}$$ and when calculating the current, i get three ...
Gioele Chr's user avatar
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Derivation for the interference of two plane waves

I am trying to understand the derivation for the intensity of two interfering waves. In my textbook, I see this: I am confused by everything on the first line, namely: It seems like we are ...
Victor M's user avatar
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Validity condition for Wick rotation?

I'm reading page 193 of section 6.3 of the QFT textbook by Peskin and Schroeder. There are two integrals that we need to evaluate for the calculation in this section. (here, $\Delta>0$) $$\int\frac{...
Function's user avatar
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Why does Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ is frequently written with a factor of $i$?

The tensor Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ has the matrix tensor $\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^\mu,\gamma^\nu\right]$. I can understand that the factor of $\frac{1}{2}$ is a ...
JavaGamesJAR's user avatar
1 vote
1 answer
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Function with two complex variables [closed]

I have a project in an advanced mathematical methods lecture regarding analyticity of functions with two complex variables. My question is, are there some interesting/special functions in $\mathbb C^2$...
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Why can't a superpotential depend on a the hermitian conjugate of a superfield?

I am working through Srednicki's "Quantum Field Theory" and am at the chapter on the Minimal Supersymmetric Standard Model (MSSM). In answer to why two higgs superfields are needed in the ...
Cory's user avatar
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How to calculate relative permittivity from the following chart

So basically, I was using HFSS to run some simulation on a MnZn absorber https://www.researchgate.net/figure/a-Real-and-b-imaginary-dielectric-permittivity-curves-for-RAM-based-on-MnZn-...
Aircraft101's user avatar
1 vote
1 answer
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Meaning of complex-number representation of circular polarization

I am reading the Sakurai's book "Modern Quantum Mechanics". It starts from analogy between electron spin and classical light polarization. So far so good, but I have no idea how and why ...
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Is it possible to determine a final orientation from an initial angular velocity and constant angular acceleration analytically?

I am looking to model the rotation of a ball over time. I have the following information: an initial orientation, as a quaternion an initial angular velocity, as X/Y/Z components, fixed to the global ...
John Doe's user avatar
1 vote
1 answer
90 views

Real and imaginary part of the solution to the Laplace Equation violates uniqueness? [closed]

I am trying to solve for the magnetic vector potential on $\mathbb{R}^2$. I have used the phasor formulation of Maxwell's equations and therefore I believe I am solving the equation on $\mathbb{C}^2$. ...
user911fas's user avatar
1 vote
2 answers
202 views

Analyticity in the upper half plane and causality

Can you, please, help me to understand the following How is the analyticity of a complex-valued function in the upper half plane related to causality and Kramers-Kronig relations? Namely, why is it ...
freude's user avatar
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Complex Hamiltonian formalism [duplicate]

If $x_k=\frac{1}{\sqrt{2}}(q_k+ip_k)$ and $\bar{x_k}=\frac{1}{\sqrt{2}}(q_k-ip_k)$ Show that the Hamilton's equation of motion can be expressed in the form: $\frac{dx_k}{dt}+i\frac{\partial H}{\...
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