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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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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Are the intensities of waves equal when they are represented in real vs complex notation?
The intensities should be equal no matter how a wave is represented. So clearly I think i'm making some elementary mistake, it seems they are not same :
$$
\Phi(x,t) = A_0 \cos{(kx-wt)} \\ \Phi(x,t) = ...
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Complex Analysis books for Physics
I am now in my 6th semester of my physics bachelor and now I'm searching for a complex analysis book.
It shouldn't be too long and deep and not too "mathematical" (I don't need every proof). ...
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Why is the formula for expected value of nonobservables in quantum mechanics different then in regular statistics? [duplicate]
Specifically, why is the operator “sandwiched in” between $\Psi^*$ and $\Psi$? i.e. Why isn’t the formula just $$\langle \hat{Q} \rangle = \int \hat{Q}\cdot|\Psi|^2 dx = \int \hat{Q}\cdot\Psi \cdot \...
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Why does Wick rotation appear like an ordinary substitution in this example?
I've seen across several posts, that Wick rotation is not an ordinary substitution. Instead we're rotating the contour of integral and analytically continuing time $t$ to include imaginary time $-i\...
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Why is time harmonic follow the form of $e^{-i\omega t}$, not $e^{i\omega t}$? [closed]
In physics, when we solve an PDE or ODE, the solution usually has the form of
\begin{equation}
f=C_+e^{i\lambda x}+C_-e^{-i\lambda x}
\end{equation}
and the "causility" will eliminate one ...
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How can the mass of an unstable composite particle become complex?
To show where the resonances in cross sections come from, one usually considers the exact propagator in the interacting theory, which for a scalar is
$$iG(p^2)=\frac{i}{p^2-m_R^2+\Sigma(p^2)+i\epsilon}...
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Relationship between complex refractive index and complex conductivity in condensed matter physics
In my field (time-resolved spectroscopes of semiconductors), people use this equation like it was trivial and never cited a source or provide derivation:
$$\tilde\sigma = i\omega\varepsilon_0 (1-\...
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Planck constant imaginary instead of imaginary PDE coefficients in the Schrödinger equation
Trying to get a first understanding of QM. The Schrödinger equation in standard form for $\Psi$
$$ i \hbar\frac{\partial }{\partial t} \Psi(x,t)
=\left[-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial t^...
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Connection between the Beta Function and Residue Theorem?
When we define the bare coupling in Minimal Subtraction we write it as a Laurent series where the analytic part is identified with the finite, renormalized coupling and the nonanalytic part is ...
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Is Dirac theory just a real Clifford algebra?
The gamma matrices $\gamma^\mu$ appearing in the Dirac equation span the Clifford algebra ${\cal Cl}_{1,3}$ over real numbers. They are generators of Clifford algebra in that sense that their products:...
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Why is $\bar\phi(-k) = \bar\phi ^*(k)$?
In Peskin and Schroeder chapter 2 p. 20, they claim that for a real field $\phi(x)$, its Fourier transform $\bar{\phi}(k)$ obey
$$\bar{\phi}(-k) = \bar{\phi}^*(k)$$
I am confused as to why this is ...
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Time Reversal symmetry, Quaternions, and spin-1/2 systems
When one has a system with no spin and time reversal symmetry, one can conclude that the Hamiltonian entries (in a particular basis, of course) must all be real. Can something be said about the ...
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Complex Scalar Field propagator from equations of motion
I was going through Chapter 9 of Schwartz's QFT book and one of the results bothers me.
Suppose we have a complex scalar field theory, and we want to find the propagator associated to the complex ...
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How to interpret $\int\mathrm{d}^2z$? [duplicate]
In chapter 6 of Tong's lecture notes on string theory when calculating the Virasoro-Shapiro/4-point Tachyon amplitude he arrives at the integral
\begin{align*}
C(a, b) = \int\mathrm{d}^2z\ |z|^{2a-2}|...
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Condition to the holomorphy of a complex function
In Witten's note https://arxiv.org/abs/1803.04993, during the proof of Reeh-Schlieder theorem, he made an arguement that considering a function
$$g(u)=\langle\chi|\phi(x_1)\dots e^{\mathrm{i}Hu}\phi(...
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