# 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.

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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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? ...
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### 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?
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