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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Does an imaginary expectation value correspond to 0?

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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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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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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316 views

Can average momentum be imaginary?

I am new to quantum physics. We just learnt about wave equations, observables and expectation values today. What really caught my attention was the expectation value of average momentum and energy: $$\...
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Improper Integrals and Contour Integration

I am reading a physics paper which employs contour integration to evaluate some integrals and I am a little confused about something. According to the author if we want to integrate some function from ...
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2answers
85 views

Hermicity of Lorentz group generators

In Ashok Das Lectures on QFT book, pg. 135-136, he stated the following hermicity properties for the Lorentz group generators: $$ {J_i}^\dagger=-J_i\,,\quad{K_i}^\dagger=K_i \tag{4.45}\label{4.45} $$ ...
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Complex conjugate of the Dirac equation

(Following the calculations done in 'Quantum Field Theory in a Nutshell' [Second Edition] by Zee, Page 101) The Dirac equation in the presence of an electromagnetic field is given by: $$ [i \gamma^{\...
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Quantum Mechanics without Complex Numbers [duplicate]

I have been studying some Lie theory recently and I came across the idea of representing complex numbers using matrices, e.g. $$1= \begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix} , i= \begin{...
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Why is the $i\epsilon$-prescription necessary in the Klein-Gordon propagator?

When evaluating the Klein-Gordon propagator, in the book by P&S, p. 31, I see that, it is customary to shift the poles and add $i\epsilon$ in the denominator. I don't understand, why this is ...
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Polchinski Eq 3.2.4 and Eq 3.2.5: Deforming contours in path integral

Here is the section of the book I'm talking about. I'm confused about the following two points: (i) Why is the path integral oscillatory? (ii) What does it mean, "we can deform contours just as ...
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Path integral calculation in complex scalar field theory

I have some trouble understanding a particular expansion in my QFT lecture. Consider a complex scalar field $\phi$, with the Lagrangian $$\mathcal{L} = \partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi....
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Derivation of Cayley-Klein parameters in Goldstein

In his derivation for Cayley-Klein parameters, Goldstein introduces the matrix $$\mathbf{Q}=\begin{pmatrix}\alpha & \beta \\ \gamma & \delta \end{pmatrix}$$ and says that the following unitary ...
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How well does the concept or model of imaginary time work?

In order to make the Minkowski metric, in special relativity, equivalent to the Euclidean metric, one idea is to allow time to take imaginary values. As far as I have learned about SR, it does make ...
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Scattering Greens function exactly at energy of bound state

I have a small bit of confusion about the expansion I am seeing in literature for the Greens function in time independent scattering theory. For example here is an excerpt from Scattering Theory of ...
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2answers
37 views

Reflection and transmission coefficient in Fresnel equations

If I have a look at the Fresnel equations on Wikipedia, I get the following expressions for reflection and transmission at perpendicular light incidence $$ R = \left|\frac{\hat n_1 - \hat n_2}{\hat ...
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Gaussian Integral with Complex Parameters — Divergence and Convergence

Although this is more of a mathematical question, I will in what follows refer to an answer of @Qmechanic that has been posted in this forum (I am sorry for creating a new post for this, but I ...
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String theory assumes that strings are complex geometric objects

In string theory, a string is a one-dimensional space that can be closed on itself as a circle (closed string) or open as a linear interval (open string), but does a string have one real dimension or ...
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How to generalize the momentum reversal operator?

In non-relativistic quantum mechanics, in position space, the complex conjugate operator $C$ flips the sign of the momentum operator, $CpC=-p$ (and thus also flips the sign of the orbital angular ...
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How possible is using complex variables to model the electric field system of a dipole? [duplicate]

If one considera a dipole system, is it possible to model the system using complex variables and if ao, how can we use complex analysis to model it? I have the idea that we can model dipole moment as ...
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Do 2d CFTs define healthy 4d QFTs?

When doing 2d CFTs we typically complexify coordinates and formally consider $\mathbb C^2$, with the understanding that, in the end, we are to restrict to the real slice $\bar z=z^*$. If we do not ...
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In path-integral, when do we have to insert fact $i$ in front of the action $S$ in the exponent?

I have got stuck in these concepts for a fews days: Wick rotation, Euclidean spacetime and QED in gravity. Generally, in Minkowski space time, there is a factor $i$ in front of the action $S$, e.g., ...
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Query in inner product axioms in QM

In inner product spaces in $\mathbb R$ we have an axiom stating that: $$ \langle x, x\rangle \geq 0\ \ \text{and} \ \ \langle x, x\rangle = 0 \iff x = 0$$ In Griffiths' textbook for Quantum ...
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Importance of complex functions in quantum mechanics

In quantum mechanics, we work with the space $\mathcal{H} = L^2(\mathbb{R})$ of functions with complex value square integrable. Thus Hermitian operators will play a central role since they have a real ...
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What is the advantage of using imaginary units for time in the Minkowski Space rather than regular euclidian space as Lorentz used? [duplicate]

I do understand that Lorentz transformations became as a rotation of coordinates as of a hyperbolic rotation. But what is its advantage over real vector? What is the new thing that it introduces and ...
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How do we perform 'time' translation in Euclidean QFT?

If we have an operator in a $1+1$ dimension QFT then we get the Hamiltonian, which comes from and generates translations in the $t$ direction and a momentum operator which comes from and generates ...
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Meaning of complex conjugate in $T$-symmetry

I have a question about the meaning of complex conjugation in time reversal symmetry in quantum mechanics. $T$-symmetry in classical mechanics is defined simply by the substitution $t \to -t$. If I ...
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1answer
66 views

Why the imaginary unit in time axis? [duplicate]

Why time is not like other dimensions is a real amount? In relativity time axis is $i*c*t$, where $i$ is the imaginary unit and $c$ is light speed in free space. Did science or philosophy reached to ...
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Why we need Hermitian conjugate amplitude rather than complex conjugate amplitude for calculating cross section?

I am trying to understand some concepts related to scattering in QED, so I would phrase my question in similar context. After calculating the scattering amplitude $\mathcal{M}$ for a process, we take ...
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Can we handle the wave function as if it was a real valued function? [duplicate]

I am trying to analyze in general simple one dimensional QM problems. To be more specific let's consider this kind of Hamiltonian: $$H=\frac{\hat{p}^2}{2m}+V(\hat{x})$$ From this one we can derive the ...
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Solving a classical damped free particle via the Fourier transform (and residue calculus)

Deriving the transfer function Suppose we have a free particle in one dimension with position $x$ and momentum $p$, and some damping $\Gamma$: \begin{equation} \begin{aligned} \dot{x} &= p/m, \\ \...
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Why multiplying complex current $\hat{I}(\omega)$ with $e^{-i\omega t}$ and taking the real part gives actual current?

In modern electrodynamics by Andrew Zangwill chapter 14, section 14.13.2 an analysis of RLC circuit is shown where Fourier transform of current, EMF, and impedance is used. And equation is $\hat{E}(\...
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What will happen if I multiply a ket vector by a complex number?

I was reading Zettili’s Quantum Mechanics book. There I have seen when a ket (or bra) multiplied by complex number, we also get a ket (or bra) But how do we infer this by mathematics?
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Complex Nature of the Wavefunction

I am doing this problem and I realized the wavefunction is real. But we also already showed that the wavefunction needs to be complex. Why is it that the wavefunction given here is real? I first ...
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Green function for Laplace's equation in complex three dimensional space

The Green function for Laplace's equation in three dimensions for a source at the origin is $$ \nabla^2 G(\mathbf{r})= \delta(\mathbf{r}) =\delta(x)\delta(y)\delta(z) $$ where is $\mathbf{r}=\mathbf{...
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Do quantum wave functions rotate through imaginary space?

Watching a visualization of Schrödinger’s equation, I noticed that the wave function for a 2-dimensional particle was placed in a 3-dimensional graph consisting of 2 Real axes and an Imaginary axis. ...
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Complex Lie algebra vs Real Lie algebra in Physics

A Lie algebra is a vector space $\mathfrak{g}$ over some field $F$ together with a binary operation $$\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following ...
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1answer
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Why don't the specificities of quantum mechanics (like the necessity of complex number) appear in classical mechanics?

It's well known that classical mechanics is a crude approximation of reality, and that it can be derived from quantum mechanics. But if this is so, why is it not a linear theory, like quantum ...
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Contradiction? Getting a complex Fourier variable when solving the Helmholtz equation in lossy media

I want to find the general solution of the Helmholtz equation in the context of electromagentism $$(\nabla^2+n(\omega)^2\omega^2)\hat{E}(\omega, \vec{r})=0$$ For this I tried Fourier transforming the ...
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Analytic Continuation: Replacement of $t \rightarrow - i \tau$ Mathematical Justification [duplicate]

It's commonly used in imaginary-time path integral that "analytic continuation" means replacing $t \rightarrow - i \tau$ or reparametrizing the theory in terms of imaginary time $\tau = i t$....
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2answers
79 views

Normalising a wave-function

So I have a small confusion when normalising an infinite well wave-function. The wave-function for my problem is $$Ψ(x) = Ae^{i(kx-wt)}+Be^{-i(kx-wt)}+Ce^{i(kx-wt)}+De^{-i(kx-wt)}.\tag{1}$$ Applying ...
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1answer
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Why is it necessary to wrap our contour around the branch cut at $+ im$ in the spacelike Klein-Gordon propagator? (P&S)

This question is in reference to eq. (2.52) on the bottom of page 27 in Peskin and Schroeder. To evaluate the Klein-Gordon field propagator along a spacelike interval we wrap the contour around the ...
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2answers
253 views

The Functional Determinants in Peskin and Schroeder (Eq.9.77)

I'm working on the Eq.9.77 in Peskin (page 304): To demonstrate this, we need only apply standard identities from linear algebra. First notice that, if a matrix $B$ has eigenvalues $b_i ,$ we can ...
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1answer
94 views

Why do people use real values for the Wick-rotated time $\tau$?

In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in ...
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1answer
157 views

Imaginary time & predictions

Is the imaginary time just a different convention to express the time evolution to make the calculations easier? Hawking also said that "It turns out that a mathematical model involving ...
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1answer
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What is the use of $i=\sqrt{-1}$ in plane wave equation? [duplicate]

I mean we can represent plane waves using just sine and cosine functions. Why do we need to use Euler's formula to represents plane waves as complex exponentials? What is the intuition behind using $i=...
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1answer
54 views

Can we apply Euler's formula on plain waves?

I don't speak English so edit my question if it is not accurate. Euler's formula for a complex number is: $$e^{i\theta}=\cos\theta + i \sin\theta$$ But when I write a plain wave as $$e^{i\vec{k}\cdot\...

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