# 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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### Complex physical quantities

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, ...
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### Why not just complex conjugate bras and kets instead of Hermitian conjugate?

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate ...
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### Does swapping “$i$” with “$-i$” change a physical theory?

Mathematically speaking, "$i$" and "$-i$" are the two roots of the equation $x^2+1=0$ and it seems to me at least that there is no obvious way of distinguishing between them. Thus, ...
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### Do we necessarily need real number for quantum? [closed]

In the quantum mechanics, one asked if the complex number was necessary? A typical answer was that it was not, or that it's simple direct product of real numbers. However, consider rational number to ...
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### Complex exponential method of solving differential equations

In the twenty third Feynman lecture, the solution of the following differential equation is discussed: $$\frac{d^2 x}{dt^2} + \frac{kx}{m} = \frac{F}{m}$$ AFter 'complexifying' this differential ...
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### How to find the input impedance of a woodwind instrument? (playing frequency of a woodwind instrument)

I'm trying to reproduce the model described in this paper https://hal.archives-ouvertes.fr/file/index/docid/683477/filename/clarinette-logique-8.pdf. The logical clarinet is a succession of 18 ...
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### The use of complex fields in electromagnetism

Virtually all treatments of electromagnetic wave propagation, and in particular of monochromatic plane waves, use basic complex analysis to simplify calculations. I am comfortable with these ...
1answer
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### Derivative of a complex potential for the $\lambda \Phi^{4}$-model

A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given ...
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### Complex notation in harmonic oscillator

For a simple harmonic oscillator, $$x(t) = A \cos(\omega t).$$ We can also write $x(t)$ as: $$x(t) = C_1 e^{i\omega t} + C_2 e^{-i\omega t}.$$ Why is it necessary that the coefficients $C_1$ and $C_2$ ...
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### Question regarding step potential

We are learning about step potential in class. I have completely understood that the behavior of the wave function representing the particle, can have different responses depending on the energy of ...
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### Why could $i\hat i$ in complex quaternions be identified as $\sigma_x$?

In the complex quaternions algebra $\mathbb{C}\otimes\mathbb{H}$, there're 8 elements: 1, $i$, $\hat i$, $\hat j$, $\hat k$ (quateronic ijk), $i\hat i$, $i\hat j$, $i\hat k$. The last three objects ...
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### Conjugate complex of linear operators in quantum mechanics

I'm pretty new to quantum mechanics (I would like to understand it broadly as an hobbyist). I'm trying to reading Principles of Quantum Mechanics by Dirac. I've found difficult to understand a ...