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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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 ...
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2answers
54 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 ...
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1answer
37 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 ...
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2answers
2k views

Schrödinger's Equation and its complex conjugate

I would like to know why there is a minus sign on the right-hand side of the Schrödinger's complex conjugate equation, whereas in the Schrödinger's equation there isn't. I know it is a simple ...
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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 ...
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1answer
26 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 ...
3
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1answer
276 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
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3answers
3k views

Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
3
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1answer
451 views

Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$ \int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz). ...
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2answers
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Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and ...
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0answers
35 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 ...
5
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2answers
121 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 ...
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1answer
127 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
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1answer
42 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 ...
4
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1answer
284 views

Imaginary masses

While watching this video, at around 5:00, the man mentions a certain type of particle having imaginary mass. He also says that these kind of particles can go faster then light. But how it is possible ...
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4answers
2k views

Relativistic mass and imaginary mass

The (relativistic) mass of an object measured by an observer in the $xyz$-frame is given by $$m = \frac{m_{rest}}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}.$$ Mathematically $v$ could be greater than the ...
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3answers
519 views

What is the kinematics of a particle with complex mass?

particles with real-mass have time-like kinematics ($ds^2 > 0$). particles with zero-mass have light-like kinematics ($ds^2 = 0$). particles with imaginary-mass have space-like kinematics ($ds^2 ...
2
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2answers
149 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
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1answer
73 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 ...
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5answers
999 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 ...
16
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5answers
3k views

What are functions of a complex variable used for in physics?

Whenever someone asks "Why are complex numbers important?" the answer, at least in the context of physics, usually includes things like quantum mechanics, oscillators and AC circuits. This is all very ...
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1answer
68 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?
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1answer
117 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 ...
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0answers
57 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 ...
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1answer
68 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 ...
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3answers
131 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 ...
3
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1answer
170 views

The derivation of the irreducible representations of the Lorentz group

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following: In ...
4
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0answers
55 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
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0answers
94 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 $ψ(x) → 0$ ...
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1answer
171 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 ...
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5answers
4k views

Complex Conjugate of Wave Function

I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what $\Psi^* $ should be for any given function. I know it's the complex conjugate at the same ...
4
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1answer
83 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} ...
3
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3answers
139 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 ...
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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
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2answers
34 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 ...
7
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3answers
249 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 ...
2
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0answers
64 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 ...
2
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3answers
137 views

How is complex permittivity measured?

Complex relative permittivity is defined as $$\epsilon_r = \frac{\epsilon(\omega)}{\epsilon_0}=\epsilon_r^{\prime}(\omega) + i\epsilon_r^{\prime\prime}(\omega) = \epsilon_r^{\prime}(\omega) + ...
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1answer
63 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 ...
1
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1answer
47 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 ...
7
votes
2answers
233 views

Can we treat $\psi^{c}$ as a field independent from $\psi$?

When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and ...
3
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1answer
80 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 + ...
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3answers
80 views
3
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1answer
98 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 ...
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4answers
135 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
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0answers
39 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: ...
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1answer
106 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 ...
3
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1answer
199 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 ...
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2answers
66 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. ...
2
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1answer
119 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 ...