What is Complex conjugate: Definition and 79 Discussions
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of
a
+
b
i
{\displaystyle a+bi}
is equal to
a
−
b
i
.
{\displaystyle a-bi.}
The complex conjugate of
z
{\displaystyle z}
is often denoted as
z
¯
{\displaystyle {\overline {z}}}
.
In polar form, the conjugate of
r
e
i
φ
{\displaystyle re^{i\varphi }}
is
r
e
−
i
φ
{\displaystyle re^{-i\varphi }}
. This can be shown using Euler's formula.
The product of a complex number and its conjugate is a real number:
a
2
+
b
2
{\displaystyle a^{2}+b^{2}}
(or
r
2
{\displaystyle r^{2}}
in polar coordinates).
If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.
I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$...
This isn't a homework problem, but a more general question.
Let ##f## be a function with two singular points ##r## and its complex conjugate ##r^*##.
let
$$f=\frac{g}{z-r} \quad \text{and assume} \quad g(r)\neq 0$$
so ##r## is a simple pole of ##f##.
we have conjugates that are singular...
It is a rather simple question:
In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$
$$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...
I've been studying quantum mechanics this semester in school and have ran into an issue I can't find an answer for. I understand why we take the complex conjugate of the wave function, such as when calculating expectation values. I'm a little confused though as to why we take the complex...
I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0.
Thank you
Hi everyone.
Yesterday I had an exam, and I spent half the exam trying to solve this question.
Show that ##\left\langle\Psi\left(\vec{r}\right)\right|\hat{p_{y}^{2}}\left|\phi\left(\vec{r}\right)\right\rangle =\left\langle...
When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...
I would like to ask about unitary transformation.
UA(IV)
UB*UA(IV)
UAT(UB*UA(IV))=UB(IV)
UB(IV)*(X)
IVT(UB(IV)*(X))=UB(X)
UBT*UB(X)=X
From the information above, UAT,IVT and UBT are the transpose of the complex conjugate. The aim of this code is to get the value of X in the step 4. This is...
Boyd - Nonlinear Optics page 5, there says 'Here a laser beam whose electric field strength is represented as $$\widetilde{E}(t) = Ee^{-iwt} + c.c$$But why is it written like this? Is it because the strength is the real part of the complex electric field? Then why doesn't he divide it by 2 after...
Hello,
I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 .
Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...
An exercise asks me to determine whether the following operator is Hermitian:
{\left( {\frac{d}{{dx}}} \right)^ * }.
I don't even know what that expression means.
a) Differentiate with respect to x, then take the complex conjugate of the result?
b) Take the complex conjugate, then...
1. Given a Markov state density function:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ##
##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...
Homework Statement
$$
\left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right |
$$
Where z and w are complex numbers not equal to zero.
2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
Homework Statement
Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms.
Degree 4: No real zeros, complex zeros of 1+i and 2-3i
Homework Equations
(-b±√b^2-4ac)/2a
The Attempt at a Solution
I want...
Hi. I'm confused about the action of the complex conjugate operator and time reversal operator on kets.
I know K(a |α > ) = a* K | α > but what is the action of K on | α > where K is the complex conjugation operator ? What is the action of the time reversal operator Θ on a ket , ie. what is Θ...
I know there's a similar post, but i didn't understand it. Why the derivative respect to t in terms of the complex conjugate of ψ is:
instead of being the original S.E in terms of ψ*
or the equation in terms of ψ with the signs swapped
Hello, i am kind of confused about something.
What is the complex conjugate of the momentum operator? I don't mean the Hermitian adjoint, because i know that the Hermitian adjoint of the momentum operator is the momentum operator.
Thanks!
I am rather new to the whole idea of complex conjugates and especially operators.
I was trying to understand the solution to a problem I was doing, but the math is confusing me rather than the physics. In the last row of calculations, why does the sin change to a cos, and the d/dx change to...
I just started teaching myself multivariable calculus and I know what the modulus of a complex number is but what is the complex conjugate and why does it pop out when we take the mod square of psi?
Like the first minute or two of video...
What are complex conjugates, how does one find them...
Hi, an exercise asks to show that $ \int_{0,0}^{1,1} {z}^{*}\,dz $ depends on the path, using the 2 obvious rectangular paths. So I did:
$ \int_{c} {z}^{*}\,dz = \int_{c}(x-iy) \,(dx+idy) = \int_{c}(xdx + ydy) + i\int_{c}(xdy - ydx) = \frac{1}{2}({x}^{2} + {y}^{2}) |_{c} + i(xy - yx)|_{c}...
Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation.
This is incorrect, but I think it is close:
X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2]
I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?
I'm just starting this, but what would the complex conjugate of Ψ(x,t) in the equation :
|Ψ(x,t)|^2= Ψ(x,t)* Ψ(x,t)
be.. Let's just say, for example, that x is 4 and t is 9... Please help if you can..
Could you please help me out with the steps to completing this, because I really want to...
Homework Statement
prove that sqrt2|z| greater than or equal to |Rez| + |Imz|
Homework Equations
|z|^2 = x^2 + y^2
Rez=x, Imz=yThe Attempt at a Solution
so far I've worked it down to this.
2(x^2 + y^2) greater than or equal to x^2 + 2xy + y^2
I've used a few different values for x and y and...
Homework Statement
Find U(x,y) and V(x,y) for f(z) = -(1-z)/(1+z)
Find Ux, Vy, Vx, Uy (partial derivatives)
Homework Equations
z = (x+iy)
The Attempt at a Solution
I found U(x,y) and V(x,y), and I used the quotient rule to find the partial derivatives Ux, Vy.
They should be...
I'm confused about some of the notation in Hoffman & Kunze Linear Algebra.
Let V be the set of all complex valued functions f on the real line such that (for all t in R)
f(-t) = \overline{f(t)} where the bar denotes complex conjugation.
Show that V with the operations (f+g)(t) = f(t) +...
Let ψ be a wavefunction describes the quantum state of a particle at any (x,t), What does ψ* i.e, the complex conjugate of a wavefunction means? I only know probability of finding a particle is given by ∫|ψ|^2 dx= ∫ ψ*ψ dx But what does ψ*ψ really means? I started learning QM with Griffiths...
Hi,
I know if we have a complex number z written as z = x +iy , with a and real, the complex conjugate is z* = x - iy. Also if we write a complex function f(z) = u(x,y) + iv(x,y), with u and v real valued, then similarly the complex conjugate of this function is f(z)* = u(x,y) - iv(x,y)...
Hi all!
From Wirtinger derivatives, given z=x+iy and indicating as \overline{z} the complex conjugate, I get:
\frac{\partial\overline{z}}{\partial z}=\frac{1}{2}\left(\frac{\partial (x-iy)}{\partial x}-i\frac{\partial (x-iy)}{\partial y}\right)=0
This puzzles me, because I cannot see why a...
Homework Statement
This is not exactly a HW problem but related to my thesis work where I am deriving an expression for the intensity of light after a particular spatial filtering. I have:
I(x) = \left[ comb(2x) \ast e^{i\Phi(x)} \right] \left[ comb^*(2x) \ast e^{-i\Phi(x)} \right]
Where...
Good Day,
I would like to know how to find the complex conjugate of the complex number 1/(1+e^(ix)).
I got (1+e^(-(ix)))/(2+2 cos x) but the solution is 0.5 sec (x/2) e^(i(x/2)).
Any help will be greatly appreciated.
Thanks & Regards
P.S. Apologies for not using LATEX as it was formatting...
Hi,
I need to understand the proof about complex conjugate of a function.
g(z) = g*(z*)
I don't know what it it called in English and can't search for it.
If anyone knows where can I get the proof, please let me know.
Thanks for help.
Hi Guys,
I have two questions which kind of relate. The first relates to the complex conjugate of a function. Specifically, When a function is multiplied by its complex conjugate, what does that mean physically?
For instance, I am reading a book on electromagnetic wave scattering, and often...
Homework Statement
Hi
I have a complex function of the form
\frac{1}{1-Ae^{i(a+b)}}
I want to take the complex conjugate of this: The parameters a and b are complex functions themselves, but A is real. Am I allowed to simply say
\frac{1}{1-Ae^{-i(a^*+b^*)}}
where * denotes the c.c.? I...
Hi guys,
I am having a very stupid problem. I can't figure out what Mobius transformation represents T(z)=z*, where z* is the complex conjugate of z.
In my book we are learning about Mobius transformations and how they represent the group of automorphisms of the extended complex plane (Ʃ). [...
Homework Statement
So we are given \alphaexp(i\varpit) +\alpha*exp(-i\varpit) and are asked to prove the resulting equation is real.
Homework Equations
\alpha + \alpha* = 2Re(\alpha) and Euler's Identity
The Attempt at a Solution
I tried expanding out the exp's to cosines and isines but...
(This is not a question I was given to solve, it is a question about the course notes.)
Homework Statement
In impedance matching, what is the next best method after the complex conjugate method?
If the source has V_s and Z_s, what should Z_L be?
V_s, Z_s, and Z_L are in series.
Homework...
The reason I ask the aforementioned question is because I came across the expectation values of operators in Quantum Mechanics. And part of the computation involves integrating a function of the complex conjugate of x with respect to dx.
So as an example let's say I have:
∫ sin (x*) dx where...
For the last step in the derivation of the Gross-Pitaevskii equation, we have the following equation
0=\int \eta^*(gNh\phi+gN^2\phi^*\phi^2-N\mu\phi)\ dV+\int (N\phi^*h+gN^2(\phi^2)^*\phi-N\mu\phi^*)\eta\ dV,
where \eta is an arbitrary function, g,N,\mu are constants, h is the hamiltonian for...
Homework Statement
Consider the set ##C^2= {x=(x_1,x_2):x_1,x_2 \in C}##.
Prove that ##<x,y>=x_1 \overline{y_1}+x_2 \overline{y_2}## defines an inner product on ##C^2##
Homework Equations
The Attempt at a Solution
##<,y>=\overline {<y,x>}##
##= \overline {y_1x_1} +...
First post!
Is it true that for a complex function f({z},\overline{z})
\overline{\frac{∂f}{∂z}} =\frac{∂\overline{f}}{∂\overline{z}}
I think I proved this while trying to solve a problem. If it turns out it's not true and I've made a mistake, I'll upload my 'proof' and have the mistakes...
Homework Statement
Find all solutions to z^2 + 4conjugate[z] + 4 = 0 where z is a complex number.
Homework Equations
Alternate form: 4conjugate[z] + z^2 = -4
The Attempt at a Solution
I have tried solving this solution using the quadratic formula.
However, √b^2 - 4ac = √16 - 4x1x4 = 0...
Homework Statement
Show that the following = 0:
\int_{-\infty}^{+\infty} \! i*(\overline{d/dx(sin(x)du/dx})*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x where \overline{u} = complex conjugate of u and * is the dot product.
2. Work so far...
I am not sure what the derivative with respect to a complex conjugate is and I have not been able to find it in any books.
I assume I should use the chain rule somehow to figure this out:
\frac{\partial z}{\partial z^*}, \quad z=x+iy
Maybe you can do like this?
\frac{\partial...