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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Quantum vector space - A complex vector space

Why is vector space of states, a complex vector space? and not a real vector space or perhaps a space based on a new field altogether, which we would have to create specifically for quantum mechanics? ...
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29 views

Imaginary number for extinction coefficient in complex refractive index

In complex refractive index on a material, $n=n'+ ik$, the imaginary part $k$ is physical meaning, as it shows absorption in the material but it is an imaginary. How we measure an imaginary values in ...
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2answers
121 views

2D Gauss law vs residue theorem

I used to have a vague feeling that the residue theorem is a close analogy to 2D electrostatics in which the residues themselves play a role of point charges. However, the equations don't seem to add ...
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77 views

“Imaginary” Perfect Time

In the definition (in one spatial dimension) of $\Delta \tau$ there is the relation: $(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2$ which is invariant. If $(\Delta x)^2 > (\Delta t)^2$ then there ...
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32 views

Is polarization of a wave just a description of its motion in three dimensions?

Since a polarization of the wave is described by complex numbers, we can try to give that mathematical formalism geometrical meaning. With having two different axes, one imaginary and other real, it ...
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38 views

Physical meaning of taking twice the real part of a Fourier transform

In my previous question, Calculating the coherence length from a spectrum, it turned out that I can calculate the coherence length of my light source from the autocorrelation function, which can be ...
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2answers
26 views

Calculating the Probability Current of a Travelling Wave

Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : ...
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69 views

From Minkowski to Euclidean Time in Path Integrals

I'm trying to prove the following equality: $$ <x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge ...
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1answer
92 views

Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook

When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value ...
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78 views

Complex variables in classical mechanics [duplicate]

In quantum mechanics complex numbers are absolutely essential because of the relation $$[\hat q_i,\hat p_j]=i\hbar\delta_{ij}.$$ But is complex number also essential anywhere in the formalism of ...
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1answer
243 views

Why do we must initially assume that the wavefunction is complex?

The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial ...
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54 views

Why $A\sin(2\pi ft) =\frac{A}{2j}(e^{j2\pi ft}-e^{-j2\pi ft})$ but not $\frac{A}{2}(e^{j2\pi ft}-e^{-j2\pi ft})$? [migrated]

$$v_{\mathrm{in}}(t)=A\sin(2\pi ft) =\frac{A}{2j}\left(e^{j2\pi ft}-e^{-j2\pi ft}\right) \\ |H(f)|=|H(-f)|;\angle H(f) = -\angle H(f) \\ v_{\mathrm{out}}=H(f)v_{\mathrm{in}}=\frac{A}{2j}H(f)e^{j2\pi ...
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71 views

Could the phase factor $i$ be replaced by “matrix representation” totally in quantum mechanics? [duplicate]

It seems that $i$ plays an important role in quantum mechanics (Q.M.). On the other hand, linear algebra plays such an important role in Q.M. too. So would linear algebra, such as a matrix be able to ...
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1answer
32 views

What is the physical significane of Complex Time Evolution in EM Waves? [duplicate]

So, I have been having a hard time understanding why there is even a complex phase for EM waves: $$\phi=\exp[i\omega t]=\cos(\omega t)+i\sin(\omega t)$$ Don't understand why it is there? Any one ...
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1answer
91 views

Why do we use $\psi$ instead of a straightforward probability?

What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
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5answers
474 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
80 views

Index of Refraction in Metal: Approximating Complex Perturbation

If you consider waves in a metal, you can write the index of refraction for the metal as, $$ n^2 = 1 - \frac{\omega_p^2}{\omega^2} $$ I am interested in what will happen if the index is perturbed by ...
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2answers
228 views

Dimensional aspects of the imaginary unit $i$ in physics [duplicate]

From a real world perspective each dimension in the 3-D Cartesian System can be represented by an axis that is perpendicular to 2 other axes. I read somewhere else that the effect of ${\sqrt {-1}}$ is ...
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1answer
37 views

Square of the momentum operator (issue with taking dot product of complex numbers)

So the momentum operator in coordinate space is: $$ \vec{p} = -i\hbar\vec{\nabla}$$ And the hamiltonian for a free particle is: $$ H = \frac{p^2}{2m}$$ All over the internet I see this written as: $$ ...
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81 views

Transmission + Reflection coefficients >1 For Potencial Barrier with Negative Complex Part Contradicts Paper

I am studying reflection and transmission coefficients for a barrier consisting of a a step potencial defined by: $$V(x):=\begin{cases}0&{\rm if}\,|x|>a/2 \\ V_0+iW_0 & {\rm ...
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1answer
146 views

Phasor form of Maxwell's Equations

I'm interested in the transformation from the standard Maxwell's equations to their phasor equivalents. From the literature, this means injecting: \begin{equation} E = Re(\boldsymbol{E}e^{j\omega ...
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4answers
369 views

Why complex functions for explaining wave particle duality?

I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as ...
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37 views

Complex Fourier Particular Solution [closed]

I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous ...
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3answers
242 views

Applications of analytic continuation to physics

I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the ...
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1answer
342 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 ...
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6answers
1k views

Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
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1answer
221 views

Why is the wave function complex? [duplicate]

Why should an equation (TDSE) in which first time derivative is related to second space derivative have a solution that contains $i$?The wave function is supposed to be complex, but I am unable to ...
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2answers
257 views

Real Part of the Wave Function

In Quantum Mechanics the square of the wave function is compared to a probability density. Is there no similar relation to waves in the sense that something meaningful can be ascribed to the real part ...
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763 views

What does the complex electric field show?

We have a complex electric field. Is there any definition for absolute and imaginary part of a complex electric field? What do they stand for?
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2answers
303 views

Global phase symmetry for complex scalar field theory

I have started to study QFT. And I have some difficulties in such classical situation. Suppose i want to calculate $\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\phi$ for lagrangian ...
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2answers
132 views

“Complex Variables Method” in Diff. Eq. - Justification and physical meaning?

A common method of simplifying calculations that involve differential equations - particularly involving oscillation - is to replace $\cos(\theta)$ with $e^{i \omega t}$, evaluate, and then take the ...
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5answers
436 views

What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & ...
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6answers
630 views

What does imaginary number maps to physically?

I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but ...
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1answer
282 views

Finding Stagnation Points from the complex potential

I am trying to find the stagnation point of a fluid flow from a complex potential. The complex potential is given by $$\Omega(z) = Uz + \cfrac{m}{2\pi}\ln z.$$ From this I found the streamfunction to ...
5
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1answer
185 views

Are electrodynamics problems in the complex plane relevant to real life?

This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of that question. I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of ...
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101 views

Some complex-number manipulation when calculating coefficients

I am going through Sakurai's quantum mechanics, and at one point the solution to a problem says: $(\sin(\beta)\cos(\alpha)-i\sin(\beta)\sin(\alpha))b+\cos(\beta)a=a$ ...
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1answer
63 views

Nonequilibrium themal QFT

Wick rotation to thermal of QFT in Minkowski space to thermal QFT, which is after this transformation analogue to statistical mechanics, does only describe equilibrium statistical mechanics. On page ...
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3answers
565 views

Born's Rule, What is the Reason?

As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives ...
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3answers
210 views

States and observables in quantum mechanics

I'm beginning learn quantum mechanics. As I understand, state is a map $\phi$ from $L^2(\mathbb R)$ such that $|\phi|^2$ describes probability density of a particle's position. By integrating ...
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1k views

Variational Derivation of Schrodinger Equation

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ...
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1answer
96 views

Using angular momentum in complex coordinates

So given the angular momentum operator: $$L_{z} = - ih\biggl(x \frac{\mathrm{d}}{\mathrm{d}y} - y \frac{\mathrm{d}}{\mathrm{d}x}\biggr)$$ I know how to write these in terms of polar coordinates ...
4
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1answer
229 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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0answers
58 views

Any problem at assuming the wave function as a real (rather than complex) function? [duplicate]

I'm a beginner at quantum mechanics but whenever I check the wave functions, they always have a complex factor. I can't clearly understand because complex number is an imaginary number so it can't ...
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1answer
229 views

Formulas for kinetic energy

I was reading ABC of relativity from Bertrand Russell and some formulas about kinetic energy caused me some problems. Here is the extract : The kinetic energy is, in the usual form ...
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0answers
52 views

What is the relationship between complex time singularities and UV fixed points?

In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
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1answer
57 views

Phasor representation of voltage in frequency domain

In a text on application of electromagnetism in transmission line, there introduces a phasor for the voltage (in frequency domain) $$\tilde{V}(x) = V^+e^{-i\beta x} + V^-e^{i\beta x.}$$ Here $V^+$ ...
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466 views

What is a physical example of a Saddle-Node Bifurcation?

I am doing a presentation on bifurcations and would like physical examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node ...
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1answer
59 views

Work done by complex field on complex plane

A force field is given by $F = 3z+5$. Find the work done in moving an object in this force field along the parabola $z = t^2 + it$ from $z = 0$ to $z = 4+2i$. I don't understand why conjugate ...
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2answers
201 views

Is there a physical motivation to study finite fields?

Clearly finite groups are of immense value in physics and these are also substructures of fields. However I never came across any computations involving finite fields at university and so I never ...
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229 views

Usage of Schrödinger equation vs Madelung equations

It is well known that Madelung formulation is alternative to the Schrödinger Formulation, cf. this previous Madelung transformation Phys.SE post. I wanted to know what makes Schrödinger's formulation ...