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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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Orthonormal basis written in Dirac Notation

$\left\{ e _ { i } \right\}$ is an orthonormal basis which has the orthonormal condition as following: $$e _ { i } ^ { T } \cdot e _ { j } = \delta _ { i j }$$ In Dirac Notation where $| i \rangle = | ...
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Contour Integration in Schwartz

In Matthew Schwartz's QFT text, on page 39, he has the following contour integral: $$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$ This can be split into two terms, one ...
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49 views

Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $\psi$ and its "conjugate" $\psi^*$ (quotation marks theirs) has Lagrangian density: $$L=-\...
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What are the applications of phase-integral method in plasma physics? [closed]

Which kinds of equations have to be solved by using phase integral method (Vlasov equation)? Are there any PHYSICAL phenomena explained by using this method? Or it is just a pure mathematical technics ...
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Can we get rid of complex number? [duplicate]

Complex number is very weird to me, especially so when they appear in engineering and physics equations. It is possible to represent complex number as real matrix where $i = \begin{pmatrix} 0&1\\...
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Derivation for angular acceleration from quaternion profile

Given a profile of unit quaternions $q(t)$ that represents the orientation of a body over time, I like to get the angular acceleration $\dot \omega (t)$. I tried to find a formula myself, but I get ...
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50 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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33 views

Confused over the complex term in the simple harmonic wave equation

I am trying to derive the general equation of Lamb wave. My book says that $$y = A\exp(i(kx−\omega t))$$ is the general equation of simple harmonic wave propagating in +ve $x$ direction. but I am ...
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49 views

Is it necessary to learn complex analysis in order to learn classical electrodynamics?

I am reading "Classical electricity and magnetism, chapter 1" by Wolfgang K. H. Panofsky and Melba Phillips. I am having little trouble on page 13 and afterwards. It talks about singularity, poles, $2^...
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Why doesn't Feynman take the real part of a complex EM wave before finding the intensity?

This is from The Feynman Lectures vol 2 chapter ch 33 on reflection of complex EM waves on metal. In the above I would expect $\frac{I_r}{I_i} = \frac{Re(E'_0)^2}{Re(E_0)^2}$, rather then $\frac{...
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Pure imaginary Schroedinger wave function

I know that the solutions to the time-dependent Schrodinger equation are always linear combinations of the form $$ \Psi(x,t)=\sum_n c_n e^{-iE_nt/\hbar} \psi_n(x) $$ If $ \Psi(x,0) $ is PURELY ...
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54 views

Action of complex conjugation on Hamiltonian

Consider a finite-dimensional non-relativistic QM system with hamiltonian $H$. Let $K$ denote the complex conjugation operator. What does $K H K$ simplify to, if the system is: (a) spin-zero; (b) spin-...
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But how exactly do you calculate the Joukowsky Airfoil, within a minimal margin of error?

After reading a fair bit of theory around the uses of conformal mapping to solve for the forces of lift acting on a wing, or a 2D cross section of the wing, in relation to the angle of attack. However ...
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130 views

Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
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47 views

When can viscosity be ignored?

I am working on understanding the motion of a solid ball in an infinite elastic (or possibly viscoelastic) medium when subjected to a sinusoidal driving force. The frequency-dependent complex wave ...
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51 views

Scalar particles are described by a real scalar field or by a complex one?

Well, in the title is already stated my main question. I know you can use a complex scalar field to describe two real scalar fields, by using just one that involves both of them. But, in the modern ...
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Reflectivity with Complex Refractive Index

I would like to ask a followup question on a previous post found here. Starting from the general expression for reflectivity: $$R = \bigl\lvert\frac{n_1-n_2}{n_1+n_2}\bigr\rvert^2$$ and substituting ...
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25 views

Phasor rotation direction in electromagnetism

In electromagnetism, we often define a field propagating in the $z$ direction over time $t$, with wavenumber $k$ and angular frequency $\omega$ as $\text{cos}(-kz+\omega t + \phi)$. Choosing $z = 0$ ...
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How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate

If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean? When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
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Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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Interpretation of the magnetic potential ($A$-field) in the quantum mechanical probability of current

The probability of current in quantum mechanics when the is a magnetic potential, A, is defined as: $$\boldsymbol j=\frac{1}{2m}(\psi^*\hat{\boldsymbol p} \psi-\psi\hat{\boldsymbol p}\psi^* -2q{\...
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How does the Dirac operator change under adjoint transformation?

I'm trying to demonstrate an identity $$\int \overline{\psi}D\phi = \int \overline{D\psi}\phi$$ by substituting in the dirac operator as $D = i\gamma^{a}\partial_{a}$ and $\overline{\psi} = \psi^{\...
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Some basics about Bracket Notation

I'm trying to prove something. Sorry this post is so long but I wanted to keep things as basic as possible so people have an easier time understanding. Let's assume we have a quantum system $\rho$ ...
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44 views

Electrodynamics : Problem with Notations for fields $\vec{(\vec{r},t)}$ and $\vec{B}(\vec{r},t)$(complex and real notations)

I'm sutyding a course on electrodynamics and am stuck on a few lines I can't make sense of. The professor uses $$\vec{E}(\vec{r},t) = \vec{U_0} cos (\vec{k}\cdot \vec{r} - \omega t + \phi)$$ (so far, ...
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Complex solutions to an Underdamped Oscillator

In many of the books talking about damped simple harmonic motion, the underdamped oscillator is treated as follows: Newton's second law says $$m\ddot{x} + r\dot{x} + sx = 0 $$where s is stiffness ...
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54 views

The hermitian conjugate of anti-linear operator

Some quantum mechanics books tell us that the definitions of hermitian are If $\langle\psi|A\phi\rangle=\langle B\psi|\phi\rangle$ for linear operators, then $B=A^\dagger$ If $\langle\psi|C\...
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Solution to Klein-Gordon equation: real field condition and other questions

Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. ...
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Would the time dependent wave equation of a free particle be the same in 4D Euclidean spacetime as it is in 4D Minkowski spacetime?

In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there ...
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Value of complex angle in Snell's law

According to Snell's law: $$\sin \theta_t = \displaystyle \frac{n_1}{n_2} \sin \theta_i$$ where $\theta_i$ is the incidence angle in medium 1 and (after the ray crossed the interface between medium ...
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36 views

Forced Oscillations and Complex Representation

An oscillating force $F \cos \omega t = \Re\{Fe^{i\omega t}\}$, where $F$ is real, is applied to a mass $m$ on the end of a spring with spring constant $k$. The displacement, $x$, of the particle can ...
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139 views

Lagrangian density for a complex scalar field

I am taking a course on classical field theory, and am not entirely sure as to what motivates the for of the Lagrangian density for a complex scalar field. In my lecture notes, this is first ...
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Complex permittivity of mixture

I have been reading a few papers regarding the estimation of the permittivity $\epsilon_{mix}$ of a mixture of materials of permittivities $\epsilon_1$ and $\epsilon_2$ (I am not a phisicist or a ...
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Must a classical Lagrangian or a Hamiltonian be a real function?

$\bullet$ Is it fair to assume that the classical Hamiltonian or Lagrangian of a system (a particle or a field) is always a real-valued function? $\bullet$ If not, can you provide counter-examples? ...
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Finding measurements in non-Hermitian operators

I know how the measurement postulate in quantum mechanics works, in regard to hermitian operators, but what if an operator is non-hermitian? Can i apply the following reasoning? If an operator is ...
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Significance of the complex component in the underdamped harmonic motion equation [closed]

The following differential equation represents the motion of a body of mass $m$ and displacement $x$ from the mean position, that is attached to a spring of force constant $a$ and viscous damping ...
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Momentum Operator for Schrödinger's equation notation

I'm studying Chemistry but we are studying Quantum Physics as a separate module and I had a question regarding the momentum operator. In the lecture, we were given the following information: The ...
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Is the mass of Tachyons real or imaginary?

I have always considered quantities like mass, charge, momentum etc to be completely real quantities as them being imaginary doesn't make much sense to me. But for tachyons to exist, they should have ...
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Why is there a physical preference to real numbers?

In quantum mechanics, operators can only be observables if the eigenfunctions they operate on have real eigenvalues. If they are complex, I am told that, surely, some observable of a physical system ...
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How is the component of a wave vector parallel to an interface made to always be real?

This question stems from what I've seen as the definition of a wave vector and from not being able to reconcile how, when the incident medium is lossy, the parallel component can always be real. Let's ...
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2answers
59 views

What is the most useful to learn out of complex analysis and differential equations for undergraduate studies in physics? [closed]

Next year I'm planning to start on my bachelor's in physics, however, I have already started taking some undergraduate courses in mathematics and next semester I will have to choose between complex ...
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96 views

How is the complexification of the Lorentz Lie algebra related to the need for Dirac's 4-component spinor in QFT?

There have been several questions with good answers in physics.stackexchange about the motivation of the complexification of the Lorentz Lie algebra, basically as a mathematically nice way to deal ...
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Why is $\langle c \cdot f|g\rangle=c^*\langle f|g\rangle$?

Why is $\langle c \cdot f|g\rangle=c^*\langle f|g\rangle$? $c$ is a complex number and $c^*$ is the conjugate. I think that $\langle c \cdot f|g\rangle=c\langle f|g\rangle$ because that's how scalar ...
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Is plane wave equation $\Psi(\mathbf r,t)=\Psi_0e^{i(\mathbf k \cdot \mathbf r-\omega t)}$ for quantum-mechanical wavefunctions really complex? [duplicate]

The equation given for plane sine wave (for instance used to "derive" the Schrödinger equation) is $$ \Psi(\mathbf r,t)=\Psi_0e^{i(\mathbf k \cdot \mathbf r-\omega t)} $$ I would have assumed that ...
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What is the difference between solutions to 2nd order homogeneous ODE?

I’m studying Vibrations, and we have two forms to the 2nd order homogeneous ODE: $$mx ̈+kx ̇=0$$ $$x(t)=C_1 e^{iw_n t}+C_2 e^{-iw_n t}$$ and $$x(t)=A\cos(w_n t)+B\sin(w_n t)$$ Even though I can use ...
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558 views

Nonexistence of a Probability for Real Wave Equations

David Bohm in Section (4.5) of his wonderful monograph Quantum Theory gives an argument to show that in order to build a physically meaningful theory of quantum phenomena, the wave function $\psi$ ...
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57 views

Why consider only real part when summing several simple harmonic motions?

I have been studying vibrations and I stumbled upon the overlapping of simple harmonic motions. Consider the case where the number of oscillators $n$ is $n \gg 1$, all of them have the same angular ...
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86 views

Wick rotation of the propagator in quantum mechanics

I am told that making the substitution $t\to-i\tau$, or a 'Wick rotation', can be used to study the propagator in imaginary time, making some problems easier. For example, this source proposes that we ...
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Why is the $\mathrm{SU}(2)$ algebra taken over the complex field?

The lie algebra of $\mathrm{SU}(n)$ is composed by the $n \times n$ antihermitian matrix with null trace over the real field, but physicists prefer to use hermitian matrix. Does this mean taking the ...
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Vanishing Skew-Hermitian Inner Product

Context In section 4.3 of "Statistical Mechanics of Nonequilibrium Liquids" by Evans and Morriss, the following identity is noted: $$ \langle \dot{A} A^* \rangle = 0,$$ with $$ \langle A B^* \...