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.

learn more… | top users | synonyms

0
votes
0answers
46 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
4
votes
1answer
46 views

Should the complex conjugate of a derivative of a Grassmann number include a sign?

Take a real Grassmann variable, by which I mean $\theta=\theta^*$. We have $$\int d\theta~ \theta =1,\qquad \frac{\partial}{\partial\theta}\theta=1$$ If I define the conjugation of Grassmann ...
6
votes
1answer
68 views

What is the mathematical motivation for complexifying momenta in BCFW?

One of the first steps in obtaining the on-shell BCFW recursion relations is complexifying the momenta of the external particles. Now complexifying things is not unprecedented (the dispersion program ...
2
votes
2answers
99 views

Why does a electric Potential have to be real, but not a Potential in quantum mechanics?

So I had this Problem when I had to learn about classical electromagnetism: Why is it, that we use complex numbers when calculating stuff, but in the end only the real part is important (for example ...
0
votes
1answer
45 views

Why Does there Have to be Linearity in Ket and Skew Symmetry?

I'm reading Shankar's "Principles of Quantum Mechanics," and on page 8 he states that one axiom in Dirac notation is linearity in ket, and because they are also skew symmetric there is anti-linearity ...
2
votes
2answers
62 views

A general complex electric field

When dealing with a plane wave solution to the electric field such as $$\vec{E}(r,t)=E_{0}\cos(kz-\omega t+\phi)$$ we usually introduce a complex electric field $\tilde{E}(r,t)$ such that $\vec{E}(r,t)...
0
votes
1answer
53 views

Question about Eigenvalues of Hermetian Operators Being Real Numbers

I'm still slogging through Quantum Mechanics: The Theoretical Minimum and I've reached another area that baffles me. Susskind uses the following to show that the eigenvalues of Hermitian operators ...
0
votes
1answer
27 views

Does the sign of imaginary part of complex permittivity have any physical meaning?

I have noticed some papers having written complex permittivity as $e' + je'' $ and others as $e' - je''$. The data in literature does not specify the sign. What should I use and does the sign of $e''$ ...
1
vote
2answers
173 views

Is the $i$ in QM a time component in disguise?

In SR, it is possible to replace the Minkowski metric $\eta_{\mu\nu}$ with a (pseudo) euclidean metric $\delta_{\mu\nu}$ provided that time is measured in imaginary units. I was wondering if the same ...
0
votes
0answers
31 views

Variables in the Dirac Equation Lagrangian [duplicate]

(Warning: I'm a student of mathematics with no training in physics.) In derivations of the Dirac equation from an action principle, one encounters the action $$S= \displaystyle\int\,d^4x \,\bar\psi(x)...
0
votes
0answers
64 views

Usage of Complex Numbers in Quantum Mechanics [duplicate]

In Griffiths 2nd Edition Quantum Mechanics page 148, it says when describing the eigenfunction to a part of the central potential problem as $$\mathrm e^{i m \phi}$$ "In electrodynamics we would ...
80
votes
1answer
5k views

Is there such thing as imaginary time dilation?

When I was doing research on General Relativity, I found Einstein's equation for Gravitational Time Dilation. I discovered that when you plugged in a large enough value for $M$ (around $10^{19}$ ...
0
votes
0answers
30 views

Applications of octonions in special relativity?

According to the Wikipedia article on octonions: Octonions [...] have applications in fields such as string theory, special relativity, and quantum logic. However, I couldn't find any ...
1
vote
2answers
40 views

Steady state RLC circuit analysis

In an RLC series circuit let applied EMF be given $V=V_0\sin\omega t$, $$Z=Z_C+Z_R+Z_L=R+i\left(\frac{1}{\omega C}-\omega L\right)$$ $$|Z|=\sqrt{R^2+\left(\frac{1}{\omega C}-\omega L\right)^2}$$ Then ...
0
votes
2answers
59 views

Confusion of Schrödinger equation and complex conjugates

I have a similar question that was asked in the following link: (Schrödinger's Equation and its complex conjugate). But I find both the question and answers not specific enough. So let me ...
1
vote
1answer
44 views

Kirchhoff's laws in phasor domain

While analysing AC circuits, we write voltage, current etc all with complex numbers namely "phasors". While studying the same, I wondered if Kirchhoff's laws held good with current and voltage in ...
0
votes
0answers
32 views

Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same Schroedinger equation?

I have this question: Let $\psi(x,t)$ solution of the Schroedinger equation for a particle under a potential V(x) independent of time. Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same ...
0
votes
1answer
30 views

Imaginary component in capacitive reactance

This is a trivial question in first year but, unfortunately, this popped up in an elementary yet compulsory lab experiment. The capacitive reactance is defined as $$X_{c}=\frac{1}{\omega c}$$ The ...
5
votes
2answers
134 views

Is the Noether charge always a Hermitian operator?

Noether's theorem tells us that to every continuous symmetry of the Lagrangian there corresponds a conserved current $j^\mu$. From the time component of this current, we can then define the Noetherian ...
0
votes
1answer
44 views

2D standing wave

When we have 1D standing waves, we can write them as the sum of two propagating wave in opposite directions that give the formula $\sin(kx)\cos(wt)$. When I try to do this for 2D waves (I mean 2D by ...
1
vote
0answers
41 views

Uncertainties propagation with complex numbers [closed]

How would one go by to estimate the uncertainties on the result of a calculation when it is done with complex values ? For example I am trying to calculate the impedance of a quadrupole and the ...
2
votes
2answers
153 views

Complex conjugate of the Schrödinger equation?

This might be a very simple question but I don't understand how to compute the complex conjugate of the Schrödinger equation: $$ i\partial_t \psi = H\psi $$ where $H$ is an hermitian operator. How to ...
2
votes
1answer
90 views

Is there a way to prove that a bound state wavefunction can always be chosen real for an arbitrary potential in Quantum Mechanics?

As we can prove many things that always (at least in introductory quantum mechanical problems) apply using an arbitrary potential (like that $E>V_{\rm min}$ or else the solutions are non-...
1
vote
1answer
74 views

Can the quantum mechanical current density be imaginary?

I am dealing with a situation where I get an imaginary transmission current density. Is this possible? Does it imply a zero transmission probability?
4
votes
1answer
122 views

Significance of $i$ in the Schrödinger equation [duplicate]

There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance ...
1
vote
0answers
64 views

Complex tetrad vs. Real metric

I asked this question almost a month ago on mathoverflow (http://mathoverflow.net/q/228138/) but received no response. I thought I could have better luck here: I have a question on the relationship ...
1
vote
1answer
71 views

Can a qubit have an imaginary component?

My knowledge of linear algebra is limited and my physics knowledge mostly comes from high school and Youtube so please bear with me. In the equation $$|x\rangle = a|0\rangle+b|1\rangle,$$ I read that ...
4
votes
0answers
56 views

Non-equivalence between $\omega \to \omega \pm i\varepsilon$ and Cauchy principle value

I am looking to gain a more rigorous and deeper understanding as to how an $i\varepsilon$ prescription actually changes the end result of a divergent integral, specifically in regards to Green's ...
3
votes
0answers
95 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
4
votes
1answer
93 views

Why can the bra and ket be varied independently?

Given a functional which depends on a function (ket), and its complex conjugate (bra), e.g. $$F[\varphi] = \langle \varphi|\hat{F}|\varphi\rangle = \int \varphi^{*}(\mathbf{r}) \hat{F} \varphi(\...
3
votes
3answers
152 views

What is the point of complex fields in classical field theory?

I see a lot of books/lectures about classical field theory making use of complex scalar fields. However why complex fields are used in the first place is often not really motivated. Sometimes one can ...
0
votes
1answer
40 views

Complex Coordinate change

I have a simple question where I must change the coordinates of a system however I am unsure whether I am correct. I am changing from Cartisian to complex coordinates. Let's say I only have $x$ and $...
1
vote
2answers
35 views

Notation of complex valued atomic orbitals

Atomic orbitals are usually labeled $1s$, $2p_x$, $2p_x$, $2p_z$ and so on. These wave functions are defined to be real valued. The original wave functions are complex valued. The $2p_x$ orbital is ...
2
votes
0answers
67 views

Wronskian of complex second order linear differential equation

While studying certain analogue gravity models I came across a differential equation of the form: \begin{align} \frac{d^2y}{dz^2} + \omega^2 (z)~ y(z) = 0 \end{align} where $z$ is a complex variable ...
7
votes
3answers
255 views

Why is the Fourier transform more useful than the Hartley transform in physics?

The Hartley transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega t)$...
1
vote
1answer
71 views

What is the complex dipole moment?

I have some problems with getting the complex (time dependent) dipole moments of some dipoles in a configuration. I eventually want to get the electric and magnetic fields of the configuration, but my ...
3
votes
3answers
133 views

How can $F_0\cos\omega t$ change to $F_0e^{i\omega t}$ in driven oscillator equation?

I have one thing that confuses me on deriving the solution for the Linear Forced Oscillator. Suppose we have the equation as $$ma + rv + kx = F_0 \cos \omega t$$ What confuses me is when the driving ...
1
vote
1answer
53 views

Why do $\psi_a$ and $\bar{\psi}_{\dot{\alpha}}$ represent two different degrees of freedom?

I am taking a course in QFT and I've been introduced to the concept of left-handed (undotted) and right-handed spinors (dotted). I know that left-handed spinors are associated with the irreducible ...
3
votes
1answer
102 views

Variational proof of the Hellmann-Feynman theorem

I use the following notation and definition for the (first) variation of some functional $E[\psi]$: \begin{equation} \delta E[\psi][\delta\psi] := \lim_{\varepsilon \rightarrow 0} \frac{E[\psi + \...
-1
votes
3answers
88 views

Complex conjugate of hydrogen ground state wave function [closed]

For hydrogen atom ground state we know . I want to know the complex conjugate of .
3
votes
1answer
106 views

Grassmann numbers in the dual space

I'm reading the section on Grassmann numbers in QFT for the Gifted Amateur and I'm confused by something said therein: First, they define a coherent state for fermions $\rvert \eta \rangle$ as \begin{...
1
vote
4answers
138 views

Is there a reason why probability density is written as $\psi^*\psi$ instead of $\psi\psi^*$?

As the title states, I see $|\psi|^2$ written as $\psi^*\psi$ instead of $\psi\psi^*$. Are both correct or is there a reason behind it? As far as I'm aware, the only time I see this sort of ordering ...
2
votes
0answers
42 views

Two-Band k.p Model is not Hermitian for imaginary wavevectors

In E. O. Kane's original work on Zener Tunneling, he uses a two-band $k\cdot p$ model for the semiconductor bandstructure: $$H=\begin{pmatrix}E_g+\frac{\hbar^2k^2}{2m_0}&(\hbar/m)kp\\(\hbar/m)kp&...
-2
votes
1answer
111 views

Schrodinger equation violates mathematics?

By the Hamiltonian formalism of quantum mechanics, given a quantum system in a state $\Psi$ in a Hilbert space $\mathcal H$, the state will instantaneously evolve in time according to $$\dot{\Psi}=\...
3
votes
1answer
221 views

Real versus complex Hamiltonian

While a Hamiltonian must be a Hermitian matrix, it can either be real or complex. Is there a significance for having a real Hamiltonian? Does it have any additional physical symmetries? For example,...
2
votes
2answers
70 views

Expectation value of an imaginary operator acting on a real function

In a video (http://youtu.be/r_gBQ_qhg8U?t=9m58s) it's stated that a matrix element of an imaginary operator acting on a real wave function is zero, i.e. $$\langle\text{real}|\text{imaginary}|\text{...
2
votes
1answer
133 views

Is a non-degenerate wavefunction real or complex?

In this video it is stated that: It can easily be verified that the wavefunction of a non-degenerate quantum mechanical system will be real. However the presenter does not explain why this ...
2
votes
3answers
123 views

Why does Griffiths define the complex inner product differently? [closed]

I have just now noticed that Griffiths (in his book Introduction to Quantum Mechanics) defines the complex inner product as $\big<z,w\big>=\sum_{i=1}^n\overline{z}_iw_i$. In all mathematics ...
1
vote
0answers
53 views

Why Hamiltonian is Hermitian? [duplicate]

Everyone knows that this is needed to make eigenvalues real, but still why we enforcing such a structure at first place? An arbitrary operator can have as complex as real eigenvalues, we can simply ...
1
vote
1answer
190 views

How to interpret the complex index of refraction?

The index of refraction which represents how much light gets refracted when entering a medium is defined as $$n = \frac{c}{v}$$ I have seen it stated in several places, such as here, that we can ...