# 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.

924 questions
Filter by
Sorted by
Tagged with
47 views

• 73
46 views

• 113
115 views

• 127
68 views

### Are the equations of the Poynting vector and energy density of an electromagnetic wave only for the real waves?

So, my book says that the Poynting Vector associated to an electromagnetic wave in matter with permeability $\mu$ is $\mathbf{S} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B}$. The thing is, I am ...
• 31
1 vote
51 views

### Rotation by 360°, spin-1/2 fermions and quaternions

Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1. Representing a continuous 360° rotation as a quaternion is also a multiplication by -1. Is there a relationship between these ...
• 190
247 views

### Do commutation relations stay the same when we use different definitions to quantize a free scalar field?

I saw different ways to write the scalar field. For example (Tong p.23): $$\phi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} \left[a_pe^{ipx}+a_p^\dagger e^{-ipx}\right].\tag{2.18}$$ And we ...
• 1,305
1 vote
105 views

### Quaternions as rotation generators

The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8. The unit quaternions $i, j, k$ are generators of rotations about their ...
• 1,945
68 views

• 124
33 views

### Can anyone tell How to write the Real and Imaginary parts of Circular polarisation in terms of Unit vector components?

In my computational simulations, i want to mention the 'real' and 'imaginary' parts of 'Circular' and 'Elliptical' polarised light as an input. Now the syntax of that input is such that i have to give ...
• 1
1 vote
28 views

### How do I find the BH curve from the frequency-complex permeability relationship?

I come here after a lot (I mean a lot) of research about complex permeability. I have a ferrite datasheet which give the complex permeability ($\mu'$ and $\mu''$) over frequency. I don't know how to ...
1 vote
30 views

### Prove: Conformal map $w(z)$ transforms a trajectory in potential $U(z)=|dw/dz|^2$ to one in potential $V(z)=-|dz/dw|^2$

This is a conclusion given without proof in the Chinese version of Arnold's Methematical Methods in Classical Mechancs. Contents related to this conclusion are missing from the English version (I ...
1 vote
74 views

During a QM lecture, we went over Hermitian adjoints. While I understand that it is the Hermitian conjugate of an operator, I do not understand what this represents, besides its definition. Also, I do ...
39 views

### What does the imaginary part of the exponential of an EM wave represent? [duplicate]

I've read about a bit and some people are saying that either the imaginary part is just forgotten, or that is the orthogonal polarisation. These don't make sense to me as I thought the imaginary part ...
• 118
123 views

### Without resorting to QM how do I understand the real world uses of imaginary numbers

I am trying to understand the real world uses of imaginary numbers. I have been told many times that it is not just a convenient tool mathematicians invented but has its places in some fundamental ...
• 129
1 vote
73 views

• 6,487
59 views

• 31
257 views

### Eigenfunctions of $C\psi(x) = \psi^*(x)$

I am considering the complex conjugation operator $C\psi(x) = \psi^*(x)$. As $C^2$ is the identity operator, I've deduced that the possible eigenvalues of the $C$ are $\pm 1.$ To determine the ...
1 vote