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.

Filter by
Sorted by
Tagged with
2
votes
0answers
68 views

Imaginary time & predictions

Is the imaginary time just a different convention to express the time evolution to make the calculations easier? Hawking said that "It turns out that a mathematical model involving imaginary ...
-1
votes
1answer
43 views

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=...
0
votes
1answer
40 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\...
1
vote
1answer
43 views

Using higher order derivatives of quaternions in equations of motion

It is common to look at the orientation of a rigid body in term of a quaternion which encodes an axis and angle with a vector and scalar. $$ \boldsymbol{q} = \pmatrix{ \boldsymbol{\hat{z}} \sin\left( \...
0
votes
0answers
21 views

Possible to convert accelerometer $x,y,z$ measurements into quaternion?

I have an inertial measurement unit (IMU) with an accelerometer that reports the acceleration (a) along the $x$, $y$ and $z$-axis in milli-$g$. I can get the Euler angles for roll ($\phi$) and pitch ($...
0
votes
1answer
46 views

Does Wick rotation work for time-dependent Hamiltonian?

Consider a quantum system that is governed by a Hamiltonian with explicit time dependence $H(t)$. Is it always legitimate to perform a Wick rotation $t \rightarrow -i\tau$, and calculate the time-...
0
votes
2answers
53 views

Ladder operators vs. conjugate variables

In the book Introduction to Many-Body Physics by Piers Coleman, it states on page 12 that ... the particle field and its complex conjugate are conjugate variables. In other words, the particle ...
-3
votes
0answers
43 views

How is a velocity vector derived from a Quaternion?

Edit: Sorry for poor explanation I will try and improve... Background: I am working with output values from a computer program and trying to translate these given values. Question: How can I get ...
2
votes
1answer
84 views

Independence of $z$ and $z^*$ in coherent states

In the book of Lowell Brown on QFT its mentioned that $$\int_{\mathbb{R}^2} \frac{dq'dp'}{2\pi} e^{(-z^{*}z + z^{*}_1z + z^{*}z_2)} = e^{z^{*}_1z_2}\tag{1.8.12}$$ where $$z=\frac{q'+ip'}{\sqrt{2}} \...
3
votes
2answers
58 views

Drawing the Wavefunction

In lecture, my professor had drawn the wavefunction for a particle which encounters a potential energy barrier, whose energy it fails to exceed. The graph was highly similar to the one which appears ...
0
votes
1answer
49 views

Operators as complex numbers

I recently came across a paper where the following manipulation had been done after writing considering Heisenberg Operators as complex numbers $$\delta a^\dagger*a_s=a_s*\delta a^\dagger$$ (where $a$...
0
votes
2answers
47 views

Is translation of position vectors allowed?

I have read that position vectors represent certain specific points in three dimensional space. So, therefore, translation of position vectors must not be allowed. Since, in any case it's done, it ...
0
votes
1answer
64 views

Does it make a difference if we treat the wavefunction as having two real components instead of one real and one imaginary component? [duplicate]

I understand that the wavefunction in Quantum Mechanics is usually treated as a complex vector with one real and one imaginary component. Does it make an actual difference in terms of the answers we ...
0
votes
1answer
23 views

How do you polar-plot the complex-valued array factor (AF) of a phased antenna array? [closed]

If a normalized array factor for an $N$-element linear antenna array (without per-element phase shifts) is calculated as $$ AF = \frac{1}{N}\sum_{m=0}^{N-1} e^{jmkd\cos\theta} $$ where: $\theta$ is ...
1
vote
1answer
74 views

Continuity equation in quantum mechanics - Verification Sakurai 2.7.30 [closed]

I am trying to verify Equation 2.7.30 from Sakurai's "Modern Quantum Mechanics" 2ed. The bottom line of my question is: $?? \psi^{*}\vec{A}\cdot\nabla\psi+\psi\vec{A}\cdot\nabla\psi^{*}=0 ?? ...
0
votes
2answers
45 views

Complex conjugate of a real wave function [closed]

In the context of applying operators to find expectation values; is the 'complex conjugate' of a wave function, $\psi^*$, where $\psi$ has no complex numbers, just simply itself? For example, given ...
1
vote
2answers
77 views

Lorentz boost expressed as Hyperbolic versors

At this link https://en.wikipedia.org/wiki/Versor#Hyperbolic_versor it is claimed that an hyperbolic versor, defined as: $$ \exp(a \mathbf{r})=\cosh a+\mathbf{r}\sinh a $$ where $||\mathbf{r}||=1$ ...
0
votes
0answers
89 views

Discrepancy in two-point correlator integral of Harmonic Oscillator

The two point correlator of Quantum H.O. of natural frequency $\omega$, calculated using path integrals, is $$C_{2}=D\left(t_{2}-t_{1}\right) \propto \int \frac{d w^{\prime}}{2 \pi} \frac{e^{-i w^{\...
0
votes
1answer
80 views

Can I write a complex field, in some cases, as a real field?

I am learning quantum field theory. Now I am considering this case: Suppose a spin-0 particle which obeys the Klein-Gordon field equation and its anti-particle obeying the same equation do not have ...
2
votes
2answers
53 views

Friedmann Equation with imaginary values

Consider the Friedmann equation with no radiation: $$ \frac{H(t)^2}{H_0^2} = \Omega_{m,0} a^{-3} + \Omega_{k,0} a^{-2} + \Omega_{\Lambda,0} $$ We can have values for $a(t)$ and the density parameters ...
3
votes
4answers
978 views

Why do complex number seem to be so helpful in real-world problems? [closed]

Complex numbers are often used in Physics especially in Electrical Circuits to analyze them as they are easy to move around like phasors. They make the processes easy but it seems kind of amusing to ...
1
vote
1answer
73 views

Can I multiply a solution by a complex number to make it real in quantum mechanics?

I am trying to understand the solution to the infinite square well centered at zero in Principles of Quantum Mechanics by Shankar. Here is how it goes: Inside the well (region II - Outside left is I ...
0
votes
0answers
22 views

How do complex conjugate operator act on $k$-space?

I was calculating TR symmetry for a system and I came to realization that I do not know how $K$ (complex conjugate operator) act on a given Hamiltonian in $k$-space. More specifically, let $H(k)$ be a ...
-1
votes
4answers
127 views

Quantum Computing without Complex Numbers

p.s. I am trying to get a handle on what actual computing operations a quantum computer program does. Any information on that would be appreciated [noting the issue that that might count as a ...
1
vote
1answer
45 views

Equation of motion for rigid body dynamics with quaternions

I'm trying to understand the equation of motion for rigid body dynamics in the presence of a quaternion joint for the root of a humanoid robot. But the dimensionality inconsistency issue is confusing ...
2
votes
1answer
57 views

What happens to conservation laws if the spatial variable is complex?

This is more of a conceptual question. Normally a conservation law will look something like $$\frac{\partial j}{\partial t}+\frac{\partial F}{\partial x}=0\tag{1}$$ where $x$ is typically a real-...
0
votes
0answers
41 views

Books for Complex Methods in Sciences

I am looking to study QM but I found out that I don't understand all the complex number representation like plane waves and many more. So what are some good books to study this topic of complex ...
1
vote
3answers
60 views

Electromagnetic waves - complex numbers

The solution for the wave equation for the electric field is generally: $$\vec{E} = E_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)} $$ My question is about the complex part, why do we use complex numbers? ...
0
votes
3answers
25 views

Conditions for various levels of damping in oscillatory motion (concern about negative frequencies)

Consider the following generic equation for a damped oscillator (eq 5.33 in Taylor's CM book) $$ x(t) = e^{-\beta t} \bigg(C_1 e^{\sqrt{\beta^2-\omega_0^2}t} + C_2 e^{-\sqrt{\beta^2-\omega_0^2}t}\...
3
votes
1answer
43 views

Normalization of overcomplete coherent states as basis

In a complete orthonormal basis $|x\rangle$, we often use the completeness relation: $$\sum_{n=0}^\infty | x \rangle \langle x | = \mathbb{I}$$ if the basis is continuous we use the natural extension $...
3
votes
1answer
115 views

What happens if the wave function is multiplied by i?

I want to look at the complex wave function $\psi$ in quantum mechanics. If a complex number $a + bi$ is multiplied by $i$ it is rotated by 90 degree in the complex plane. What does this mean for a ...
1
vote
1answer
38 views

Sign of pair of Dirac spinor bilinear

I don't understand the following statement: Any pair of Dirac spinors verifies $(\bar{\Psi}_1\Psi_2)^\dagger=\bar{\Psi}_2\Psi_1$ and it is valid for both commuting and anti-commuting (Grassmann-valued)...
0
votes
2answers
85 views

Does there exist anything more general (extension) than operator of QM?

In classical mechanics we simply have quantities which are simply scalar like momentum, energy of system but when we transit to QM it's an ad-hoc principle, at least to me, that we'll be dealing with ...
0
votes
1answer
39 views

The probability current of the complex scalar field, using the Noether theorem

According to wikipedia, then the Noether current for $$ L= \eta^{\mu\nu} \partial_\mu \phi^* \partial_\nu \phi - m^2\phi^*\phi $$ with invariant transformations: $$ \phi\to e^{i \theta}\phi\\ \phi^*\...
0
votes
2answers
62 views

Impedance RLC circuit

Why is the impedance of the inductor defined as $i\omega L$, and of the capacitor $\frac{1}{i \omega c}$ ? More generally, why are they complex numbers? Is impedance a mere mathematical tool?
0
votes
0answers
59 views

The Lagrangian density of electromagnetism, expressed in geometric algebra such that $\nabla \mathbf{F}=0$ is the equation of motion?

Geometric algebra admits a very short and sweet definition of Maxwell's laws of electromagnetism: $$ \nabla \mathbf{F}=0 $$ where $$ \mathbf{F}=\mathbf{E}+i\mathbf{B} $$ and where $$ \nabla \mathbf{F}=...
1
vote
4answers
112 views

How are complex number being treated in physics?

In various places in physics, EM for example, complex numbers are used to describe things that are physically real. I will point a simple case - solving an ODE for resistance/charge/voltage. We get a ...
0
votes
4answers
70 views

Why exponential terms like $\exp (i\omega t)$ make no contribution when averaged over a long time?

In the physics of waves, I often see expressions like $$A\exp(i\omega t) + f(t)$$ where $A$ is a constant, $w$ is the angular frequency and $f(t)$ is an arbitrary function that depends on time. It ...
0
votes
2answers
121 views

Why do we care about real form spherical harmonics?

I'm studying atomic orbitals and the shape is usually represented with real form spherical harmonics, taken as an appropriate linear combination of the complex ones. If, however, the physical quantity ...
3
votes
1answer
75 views

When does a Hermitian operator have real matrix elements?

I will use braket-notation, but my question is not specific to quantum mechanics. Instead, I would be interested in a general answer for operators in some Hilbert space. Let $H$ be a Hermitian ...
0
votes
0answers
19 views

Orientation Quaternion

I am looking to work out the orientation between two quaternions to establish if they are parallel face-face orientation, side-side orientation or perpendicular orientation. At the moment I am taking ...
1
vote
2answers
82 views

Why do we describe probability amplitude rather than probability itself in quantum mechanics?

In the quantum mechanics, the dynamics of quantum system are described in terms of probability amplitude. However, we want to calculate the probability in the end which can be measured. Why don't we ...
2
votes
0answers
85 views

Why did cross product only exist for three and seven dimensions? [closed]

According to this wikipage, The cross product only exists in three and seven dimensions as one can always define a multiplication on a space of one higher dimension as above, and this space can be ...
1
vote
1answer
40 views

How does one calculate the absolute value of a Feynman diagram's amplitude?

How do I obtain the absolute value of a Feynman diagram's amplitude if I do not have values for the components of this amplitude? If the amplitude of a process such as $e^+(p_1) + e^- (p_2) \to \phi (...
2
votes
1answer
69 views

Why do we use the imaginary time evolution in simulations of some quantum system?

I realize that the imaginary evolution could help us to find the ground state for a system. However, I very puzzled why it works, and what the principle is back up there? I have done some searching on ...
1
vote
0answers
39 views

Singular Integrals in Scattering Theory [closed]

This might seem a bit off-topic question but it comes into different physical theories. While learning quantum scattering I came across certain singular integrals but could not compute them. I am ...
0
votes
3answers
88 views

Why are wavefunctions in Quantum Mechanics shown as complex Circular waves instead of real Planar waves?

I'm currently learning Quantum Mechanics from online video lectures and resources. In most of the web articles and videos, the wave functions are shown as circular waves $e^{i\omega t}$ instead of ...
1
vote
1answer
52 views

Seeking for help with complex conjugate of bra|ket

I have 2 vectors: $$|a\rangle=\begin{pmatrix} 2+i & 2+2i\end{pmatrix}$$ and $$ |b\rangle=\begin{pmatrix} 1+i & 1+2i\end{pmatrix}$$ So basically I need to show that $$\langle a| b\rangle=\...
1
vote
1answer
52 views

Why is (anti)-holomorphicity considered in physics?

As a mathematician, holomorphicity is an extremely good property that provides rigidity, finite dimensionality, algebraicity. etc to whatever theory that's considered. I'm curious about why (anti-)...
2
votes
1answer
88 views

Can the time-reversal operator of a two-level system be represented by a $2\times2$ matrix?

I am studying the time-reversal symmetry in the context of topological insulators. As usual, the minimal non-trivial model to be considered is a two-level system with Hilbert space $\newcommand{\ket}[...

1
2 3 4 5
13