A conjugate acid, within the Brønsted–Lowry acid–base theory, is a chemical compound formed when an acid donates a proton (H+) to a base—in other words, it is a base with a hydrogen ion added to it, as in the reverse reaction it loses a hydrogen ion. On the other hand, a conjugate base is what is left over after an acid has donated a proton during a chemical reaction. Hence, a conjugate base is a species formed by the removal of a proton from an acid, as in the reverse reaction it is able to gain a hydrogen ion. Because some acids are capable of releasing multiple protons, the conjugate base of an acid may itself be acidic.
In summary, this can be represented as the following chemical reaction:
Acid + Base ⇌ Conjugate Base + Conjugate Acid
Johannes Nicolaus Brønsted and Martin Lowry introduced the Brønsted–Lowry theory,
which proposed that any compound that can transfer a proton to any other compound is an acid, and the compound that accepts the proton is a base. A proton is a nuclear particle with a unit positive electrical charge; it is represented by the symbol H+ because it constitutes the nucleus of a hydrogen atom, that is, a hydrogen cation.
A cation can be a conjugate acid, and an anion can be a conjugate base, depending on which substance is involved and which acid–base theory is the viewpoint. The simplest anion which can be a conjugate base is the solvated electron whose conjugate acid is the atomic hydrogen.
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...
I am trying to understand why the conjugate of a signal in the time domain doesn't produce an exact flip of the frequency domain spectrum. There is always a one-pixel shift in the result.
The MATLAB code is shown below. I use a flip for the conjugate spectrum to show that it doesn't match the...
I am reading a good paper of J. R. Shewchuk, titled "An introduction to the conjugate gradient method without the agonizing pain", however, I do not fully understand the idea of conjugate directions. As shown in Figure 22a, where the vectors d1 and d2 are not orthogonal. These vectors are...
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...
The definition of the hermitian conjugate of an anti-linear operator B in physics QM notation is
\langle \phi | (B^{\dagger} | \psi \rangle ) = \langle \psi | (B | \phi \rangle )
where the operators act to the right, since for anti-linear operators ( \langle \psi |B) | \phi \rangle \neq...
If I understand correctly (a big caveat), one shows that if one can get from one function to the other via a Fourier transform and multiplication by a constant, then the width of the corresponding Gaussian wave of one gets larger as that of the other gets smaller, and vice-versa, and by a bit...
Momentum ##\vec{p}## and position ##\vec{x}## are Fourier conjugates, as are energy ##E## and time ##t##.
What is the Fourier conjugate of spin, i.e., intrinsic angular momentum? Angular position?
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
Is the last inequality correct? Should it not be ##|A|^2 \cdot 2(1+\cos{(ka)})##? How is the time calculated here? Given ##\Delta v > 10^{-34}##... How come ##mv \Delta v = \Delta (\frac{mv^2}{2})##? Where does the ##(1/2)## come from?
I recently had to find what f(7) equals if f(x) = \frac{x^2-11x+28}{x-7}. I first tried \frac{x^2-11x+28}{x-7} \cdot \frac{x+7}{x+7}, and it seemed like a perfect fit since I eventually got to \frac{x^2(x-4)-49(x+4)}{x^2-49}=(x-4)(x+4), but that gave me f(7)=33, instead of the right answer...
Are all complex integers that have the same norm associates of each other?
I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...
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...
Homework Statement
This post contains the answer to my thread of 10th August...
[/B]
in which I asked if anyone could point out how to derive
##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0 \,_{pq} - g_{pq} ^{(4)} \Gamma ^0\, _{rs} g^{rs}) g^{pq} g^{jq}##
from
##\mathfrak {L}## = (4)R...
Homework Statement If you look in Wikipedia for ADM formalism, you are given a Derivation, which starts from the Lagrangian:
##\mathfrak {L}## = (4)R ##\sqrt{^{(4)}g}## and moves rapidly to...
The conjugate momenta can then be computed as
##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0...
As a preface to this theorem stated in my text, it states that:
"If all the coefficients of a polynomial ##P(x)## are real, then ##P## is a function that transforms real numbers into other real numbers, and consequently, ##P## can be graphed in the Cartesian Coordinate Plane."
It then goes on...
Hi.
I have been trying to solve this problem that has been keeping me up at night for a coupe weeks at least. If anyone can help me, I would be greatly appreciated.
Hot air enters a cylindrical duct. The duct has some R-value and radiation and convection is being accounted for on the outside...
Hi. There is a problem that I have been working on and I seem to be getting somewhat unrealistic results. Can anyone critique my modeling method?
Problem: Heated air enters a duct of length L at temp T_h. The outside of the thin walled duct will have convection and radiation both being...
I am continuing to work through Lessons on Particle Physics. The link is
https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf
I am on page 22, equation (1.5.58). The authors are deriving the Hermitian conjugate of the Dirac equation (in order to construct the current). I am able to...
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...
Take a wavefunction ##\psi## and let this wavefunction be a solution of Schroedinger equation,such that:
##i \hbar \frac{\partial \psi}{\partial t}=H\psi##
The complex conjugate of this wavefunction will satisfy the "wrong-sign Schrodinger equation" and not the schrodinger equation,such that ##i...
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...
Homework Statement
Prove that the conjugate of ##g(x) = f(Ax + b)## is ## g^*(y) = f^*(A^{-T}y) - b^TA^{-T}y ## where A is nonsingular nXm matrix in R, and b is in ##R^n##.
Homework Equations
This is from chapter 3 of Boyd's Convex Optimization.
1. The conjugate function is defined as ##...
If w[n] are samples of the white gaussian noise process, I know that
E[w[n1] w[n2]] = 0 for a WGN process.
what would the following expression lead to:
E[w[n1] w*[n2]] = ?
Would it also be zero?
Thanks a lot!
Homework Statement
Given that a complex number z and its conjugate z¯ satisfy the equation z¯z¯ + zi = -i +1. Find the values of z.
Homework EquationsThe Attempt at a Solution
In a lecture on introductory quantum mechanics the teacher said that Heisenberg uncertainty principle is applicable only to canonically conjugate physical quantities. What are these quantities?
Homework Statement
Consider a charge ##q##, with mass ##m##, moving in the ##x-y## plane under the influence of a uniform magnetic field ##\vec{B}=B\hat{z}##. Show that the Hamiltonian $$ H = \frac{(\vec{p}-q\vec{A})^2}{2m}$$ with $$\vec{A} = \frac{1}{2}(\vec{B}\times\vec{r})$$ reduces to $$...
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
I don't understand how the diagram of M/EI of conjugate beam drawn , can someone explain about it ? According to conjugate beam theorem ,
Homework EquationsThe Attempt at a Solution
i know that the M/EI represent the w(x) , which is force per unit length of the beam . I...
In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
Hi PF
I read a paper in which Lewandowski writes:
the Gauss law has the form
##\partial E^a / \partial x^a + c_{jk}E^{aj}\gamma ^k_a = 0##
wherec are the structure constants
he then writes that if we are in a semisimple algebra they are skew symmetric in the indices and it can be rewritten as...
## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z##
In order to compute the Hamiltonian I first of all need to compute...
I came across this in QM course while trying to work out the time evolution equation of a wave-packet.
##(A+iB)^{1/2}*e^{C+iD}##
*Thank you I got it: I converted A+iB to exponential form and used De Moivre theorem to find the sqrt of A+iB, and finally combined the two exponentials and worked...
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.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
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...
[Mentor's note: forked from https://www.physicsforums.com/threads/conjugate-variable-clarification.878112/] [Broken]
I think the list is very interesting. From https://en.wikipedia.org/wiki/Conjugate_variables
The energy of a particle at a certain event is the negative of the derivative of...
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!
Homework Statement
Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order.
Homework Equations
Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html
The Attempt at a Solution
Since y is a conjugate of x...
Homework Statement
Let A be an n x n matrix, and let v, w ∈ ℂn.
Prove that Av ⋅ w = v ⋅ A†w
Homework Equations
† = conjugate transpose
⋅ = dot product
* = conjugate
T = transpose
(AB)-1 = B-1A-1
(AB)-1 = BTAT
(AB)* = A*B*
A† = (AT)*
Definitions of Unitary and Hermitian Matrices
Complex Mod...
When people want to find a conserved current which is constructed from a Dirac spinor, they consider the Dirac equation and its "Hermitian conjugate". But the equations they consider are ## (i\gamma^\mu \partial_\mu -m)\psi=0 ## and ##\bar{\psi}(i\gamma^\mu \overleftarrow{\partial_\mu}+m)=0 ##...
I have a left-handed ##SU(2)## lepton doublet:
##
\ell_L = \begin{pmatrix} \psi_{\nu,L} \\ \psi_{e,L} \end{pmatrix}.
##
I want to know its transformation properties under conjugation and similar 'basic' transformations: ##\ell^{\dagger}_L, \bar{\ell}_L, \ell^c_L, \bar{\ell}^c_L## and the general...
Homework Statement
Consider a qubit whihc undergoes a sequence of three reversible evolutions of 3 unitary matrices A, B, and C (in that order). Suppose that no matter what the initial state |v> of the qubit is before the three evolutions, it always comes back to the sam state |v> after the...
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...
We know that in Cartesian position basis the representation of momentum is -ihbar (d/dx)
Consider a cylindrical/spherical/whatever curvilinear coordinates. To make life simple, consider a particle constrained to move on a circle so that its position can described by θ only. Suppose we express...
Just checking (while trying to prove the Schwarz inequality for $<f|H|g>$, I know $ <f|g>=<g|f>^* $ please confirm/correct :
If $ \psi=f+\lambda g, \:then\: \psi^*=f^*+\lambda^* g^* $
Is $ <f^*|g>=<g^*|f>^* $ and $ <f^*|H|g>=<g^*|H|f>^* $ (H hermitian)?
Is $ <f^*|H|g><g^*|H|f> = -...