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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41
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11answers
14k views

About the complex nature of the wave function?

1. Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an ...
34
votes
11answers
8k views

QM without complex numbers

I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?
25
votes
7answers
2k views

Can one do the maths of physics without using $\sqrt{-1}$?

The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation ...
35
votes
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 ...
2
votes
2answers
5k 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?
7
votes
3answers
1k views

Born's Rule, What is the Reason? [duplicate]

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 ...
21
votes
5answers
13k views

The meaning of imaginary time

What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in ...
10
votes
6answers
2k views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
13
votes
5answers
1k 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 ...
19
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6answers
2k 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 ...
10
votes
4answers
2k views

Minkowski spacetime: Is there a signature (+,+,+,+)?

In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if ...
7
votes
2answers
4k views

What is imaginary time? [duplicate]

I am not professional physicist; but I am curious about Stephen Hawking's "imaginary time". It would be better to elaborate exactly what it is. I am not confused because of the word "imaginary" but I ...
12
votes
1answer
3k 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 ...
8
votes
2answers
242 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
votes
1answer
6k views

What does $\Psi^*$ mean in Schrodinger's formulation of Quantum Mechanics?

I am not a physics student. In one of my courses, some fundamental concepts of Quantum mechanics were needed, so I was going through them when I stumbled upon this. It says $$\text{probability} = ...
2
votes
1answer
3k views

Solving the time independent Schrodinger equation: Does a complex solution make sense?

In my notes, I have the Time Independent Schrodinger equation for a free particle $$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$ The solution to this is given, in my notes, ...
12
votes
6answers
1k 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 & ...
9
votes
2answers
513 views

Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
7
votes
2answers
415 views

Complex coordinates in CFT

The Setup: Let's say we want to study a Euclidean $\mathrm{CFT}_2$ on $\mathbb R^2$ with coordinates $\sigma^1$ and $\sigma^2$ and metric $ds^2 = (d\sigma^1)^2+(d\sigma^2)^2$. It seems to me that ...
2
votes
3answers
508 views

What is the rationale behind representing a state function by a complex valued function in QM?

What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a ...
8
votes
3answers
525 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 ...
6
votes
3answers
426 views

Special relativity and imaginary coefficient of the time coordinate

I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary ...
4
votes
2answers
1k 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 ...
2
votes
1answer
82 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 ...
76
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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 $10^{19}$ ...
5
votes
1answer
1k 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 ...
7
votes
3answers
1k 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 ...
4
votes
6answers
2k views

Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
9
votes
3answers
802 views

Is quantum tunneling related to imaginary time?

I was studying for my exam and looking at the chapter which talks about Potential-energy graphs. Let's take this as an example: My book states that: "If the object is in $B$ and has a total energy ...
5
votes
3answers
1k views

What does the Schrodinger Equation really mean?

I understand that the Schrodinger equation is actually a principle that cannot be proven. But can someone give a plausible foundation for it and give it some physical meaning/interpretation. I guess ...
3
votes
1answer
468 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). ...
1
vote
4answers
608 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 ...
14
votes
7answers
2k views

Quaternions and 4-vectors

I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity: $$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$ Is it ...
5
votes
2answers
359 views

QFT Hilbert spaces over other rings than the complex numbers $\mathbb{C}$

I would like some help evaluating a physics theory recently proposed by a physics professor at the College of Dupage. I think the theory is utterly wrong, for very simple reasons. If an amateur ...
8
votes
2answers
620 views

Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
5
votes
5answers
8k views

Confused over complex representation of the wave

My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$ I'm having trouble visualizing ...
4
votes
3answers
1k views

Born Interpretation of Wave Function

I have just started Griffiths Intro to QM. I was studying Born's interpretation of Wave function and it says that the square of the modulus of the wave function is a measure of the probability of ...
3
votes
2answers
13k views

How to get “complex exponential” form of wave equation out of “sinusoidal form”?

I am a novice on QM and until now i have allways been using sinusoidal form of wave equation: $$A = A_0 \sin(kx - \omega t)$$ Well in QM everyone uses complex exponential form of wave equation: $$A ...
1
vote
1answer
266 views

Concrete example of the application of complex analysis in electrostatics [closed]

I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple ...
9
votes
3answers
2k views

Complex numbers in optics

I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers?
3
votes
1answer
319 views

Is it okay to Wick rotate to give the negative of the Euclidean metric? Also, could we make the space-like coordinates imaginary instead?

There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + ...
2
votes
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 ...
2
votes
2answers
378 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 ...
2
votes
0answers
328 views

Probability and probability amplitude [duplicate]

What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ...
1
vote
1answer
302 views

What is the advantage of using exponential function over trigonometric function in analyzing waves?

A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ...
0
votes
2answers
199 views

Properties of Hodge Duality

So we know that Hodge duality works this way $$⋆(dx^i_1 \wedge ... \wedge dx^i_p)= \frac{1}{(n-p)!} \epsilon^{i_1..i_p}_{i_{p+1}..i_n} dx^{i_{p+1} } \wedge dx^{i_n}$$ where $p$ represents the $p$ in ...
0
votes
2answers
153 views

Quantum Mechanical States

What can be the precise answer to the question that Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they ...
8
votes
6answers
3k 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 ...
6
votes
2answers
407 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 ...
4
votes
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 ...