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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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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Generalized Gaussian integral with imaginary coefficient [migrated]
From this question and answers, one can prove the following identity (the Gaussian integral for the case of purely imaginary coefficients) holds:
$$\int_{-\infty}^{\infty} dx \ \exp\left(iax^2+iJx\...
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Can $\int e^{ix^2 t} dx$ be defined without going to imaginary time? [duplicate]
The motivation is trying to define the path integral, where at some point we get the integral
$$
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}+\...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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-...
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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 ...
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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$. ...
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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 ...
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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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Calculating non-dielectric reflectance without using complex numbers [closed]
I need a to calculate the fresnel reflection ratio of a non dielectric material given the incident angle, the refractive indexes of the incident and interfacing materials and the extinction ...
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Hermitian Conjugate Terms in Lagrangians
In Lagrangians (for example, that of the Standard Model) one sometimes sees $+ \text{H.c.}$ for the hermitian conjugate of a term but I am not sure what happens if there is a covariant derivative.
For ...
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Motivation for splitting the Lorentz Algebra using $J_{\pm i}$
On page 116, Zee (in QFT in a nutshell) introduces the combinations:
$$J_{\pm i}\equiv \frac{1}{2} \left( J_i\pm iK_i \right) \tag{1}$$
Where the $J_i $'s are the 3 generators of the rotation group ...
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What is the physical meaning of the pressure of an acoustic point source being complex?
Context
From various sources of Acoustics (such as "Acoustics -
An Introduction to Its Physical Principles and Applications" by Allan D. Pierce and "Fundamentals of General Linear ...
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How does the wavefunction transform under an arbitrary change of variables?
Suppose we have a variable $x$ and a probability density $\rho(x)$. The pushforward of this density under a bijective function $y = f(x)$ is given by
\begin{equation*}
\rho'(y) = \frac{\rho(f^{-1}(y))}...
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Lagrangian having an $O(N)$ symmetry
A Lagrangian with $SO(N)$ symmetry can be written like:
$${\mathcal{L}} = \frac{1}{2}(\partial_\mu \Phi)^T (\partial^\mu \Phi) - (\frac{1}{2}\mu^2 \Phi^T \Phi + \frac{1}{4}\lambda (\Phi^ T \Phi)^2).$$...
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Which order multiplet of a given $SU(N)$ is real or complex?
I am studying the $SU(2)$ symmetric Lagrangian in particle physics.
$${\mathcal{L}} = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - (\mu^2 \Phi^ \dagger \Phi + \lambda (\Phi^ \dagger \Phi)^2).$$
...
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What does it mean to measure a complex electric field?
One of the Event Horizon Telescope papers says the following:
Every antenna $i$ in an interferometric array records the incoming complex electric field as a function of time, frequency, and ...
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Conditions on EM wave complex exponentials and generalizations
The complex exponential ansatz for electromagnetic waves is utilized for algebraic simplicity. However, we admit that, depending on the setup for the ansatz, only the real or imaginary part is of ...
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Vortex energy calculation in contour integral
I encountered a problem asking me to calculating the contour integral to evaluate the energy of a vortex as shown in the picture. I am asked to compute the energy using the following equation.
$$E=-\...
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Complex conjugate and expectation values in QM
I'm currently trying to understand expectation values within Quantum Mechanics. I have a few questions that I could need a little bit of help with understanding how to interpret and how to further do ...
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What is the physical meaning of the imaginary part in $LRC$ circuits or AC circuits? [duplicate]
When we learned about AC L-R-C circuits, there is a phase difference in the voltage across the inductor and capacitor with the current. We were told that representing these phase differences on the ...
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Complex phase of the path integral in QM?
The square modulus of an amplitude must be real. Given that, I am having some trouble understanding the square modulus of a path integral being absolutely real. Given
\begin{equation}
\int\!Dq(t)\...
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In the real world, what is the stream function $Ψ$?
So I know that complex potential is just $Ω=Φ+iΨ$
$Φ$ is the potential function which associates a scalar value to every point on the field. Lines where $Φ$ is constant are the equipotential lines. In ...
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Reference to understand this branch cut question
I am currently reading a physics paper in which the authors have complexified an ordinary differential equation (ODE). They mention the following statement in the paper:
"These branch points ...
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What are the necessary conditions for this statement?
I’m trying to figure out if $\left< a \middle| b \right> = \left< b \middle| a \right> $ when $\left| a \right>$ and $\left| b \right>$ are eigenfunctions with the same eigenvalue $\...
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What is the missing part of the argument needed to justify the claim of (2.52) in Peskin and Schroeder's QFT textbook?
[This paragraph has been added to make clear that this is not a homework question having been branded as such by a mod of some kind. The question is attempting to the core of a very important question ...
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Why is a wave function $\psi$ needed for QM? Is it possible to make a differential equation involving just the p.d.f. $|\psi|^2$ of a particle? [duplicate]
Why do you need a wave function $\psi$ for quantum mechanics? Can't you just make a differential equation involving just the p.d.f. $|\psi|^2$ of a particle? Since basically with quantum mechanics the ...
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