# 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.

973 questions
Filter by
Sorted by
Tagged with
1 vote
63 views

### Wick rotation and Exponential mapping of an "imaginary" differential operator acting on a real-valued wavefunction

The position shift operator $T^{a}$ (where $a \in \mathbb{R}$ ) takes a real valued wavefunction $\psi$ on $\mathbb{R}$ to its translation $\psi_{a}$, $T^{a} \psi(x)=\psi_{a}(x)=\psi(x+a)$. A ...
1 vote
106 views

### Euler-Lagrange for Dirac Lagrangian - is $\bar \psi$ independent of $\psi$?

In Peskin and Schroeder (3.34) they write the Dirac Lagrangian: $$L_{Dirac} = \bar \psi (i \gamma^\mu \partial_\mu - m ) \psi$$ where $\bar \psi = \psi^\dagger \gamma^0$. Then, they write: "The ...
54 views

### Fourier transform of Green function using residue theroem

I want to compute the Fourier transform of a Green function in $k$-space : $$G^R_{n,m}(\omega)=\int_0^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(n-m)}}{\omega+i\eta-\epsilon_k}$$ By substituting $\omega$ and ...
45 views

66 views

### How do I show that a transformation is a symmetry of the Lagrangian? [closed]

Hello All I have the following question see part (b). I have already done part a and my answer is $L=\frac{m}{2}\dot{z}\bar{\dot{z}}-\frac{k}{2}z\bar{z}$. I have no idea how to go about part b, I ...
41 views

### What is meant by dimension of the defining representation (and adjoint representation)?

A linear representation of a classical Lie group $G$ is defined by $\rho:g \to GL(V)$ where $g \in G$ is a group element and $V$ is the representation space. The dimension of the representation space(...
1 vote
60 views

### How do I write angular displacment in quaternion form? [closed]

Work done is given by $W = F_ids_i$ and in angular displacement moment form it is $W = \tau_i d\theta_i$. I want to get the work done in using quaternion instead. Like $W = \tau_i q_i$ I cannot figure ...
57 views

### Absolute values when normalising the wavefunction

My question concerns Problem 1.4 from Griffiths. I understand the general working of the problem (given below), and derived the correct result for the normalisation constant $A$, but I am troubled by ...
57 views

### Correct frame for angular velocity in quaternion's kinematic

I am reading a paper where the quaternion's kinematic is used, unfortunately the description of the angular velocity does not match with how it's computed, so I have a doubt on which frame $\omega$ is ...
15 views

### Complex solution in equation of motion [duplicate]

When I tried to solve some motion questions, I got complex numbers for time, displacement, etc. And my teacher said my answer was correct. Is it possible to have a complex number solution in the ...
1 vote
39 views

### Cosmology: Covariance between Gaussian distributions for complex spherical harmonics coeficients

In the context of the computation of a variance about $a_{\ell m}$ spherical coefficients of Legendre, I am faced to an issue : There is a term $\langle \text{Re}(a)\text{Im}(a)\rangle$ that appears ...
151 views

### Quantum Mechanics without Complex Numbers in a multipartite setting

I was fairly convinced that usual QM formalism didn't necessitate the use of complex numbers and that ultimately they're just a matter of convenience and utility rather than anything fundamental. This ...
42 views

### Why is a wave function $\psi$ needed for QM? Is it possible to make a differential equation involving just the p.d.f. $|\psi|^2$ of a particle? [duplicate]
Why do you need a wave function $\psi$ for quantum mechanics? Can't you just make a differential equation involving just the p.d.f. $|\psi|^2$ of a particle? Since basically with quantum mechanics the ...