# What is Conjugate: Definition and 259 Discussions

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.

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1. ### I Power of a complex conjugate

Can someone please tell me why this is true? This isn't exactly the De Moivre's theorem. Thank you.
2. ### Complex conjugate of a pole is a pole?

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...
3. ### MATLAB FFT of conjugate doesn't coincide exactly at the negative frequency

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...
4. ### A Understanding of conjugate directions

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...

31. ### I Product of complex conjugate functions with infinite sums

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...
32. ### Conjugate Beam M/EI: Explaining Diagrams & Theorem

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...
33. ### A Conjugate variables in the Fourier and Legendre transforms

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}...
34. ### I Canonically conjugate pairs

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...
35. ### Real Scalar Field, Hamiltonian, Conjugate Momentum

## 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...
36. ### I What is the conjugate of (A+iB)exp(C+iD)?

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...
37. ### Complex Conjugate Inequality Proof

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.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
38. ### I Complex conjugate and time reversal operator

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...
39. ### A Conjugate variable clarification

[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...
40. ### I Schrodinger equation in terms of complex conjugate

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
41. ### I What is the complex conjugate of the momentum operator?

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!
42. ### Two conjugate elements of a group have the same order PROOF

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...
43. ### [Linear Algebra] Conjugate Transpose of a Matrix and vectors in ℂ

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...
44. ### Do we really mean Hermitian conjugate here?

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 ##...
45. ### SU(2) lepton doublet conjugation rules

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...
46. ### Hydrogen atom ground state wave function complex conjugate

For hydrogen atom ground state we know φ=π-1/2a-3/2e-r/1 I want to know the complex conjugate of φ* ?
47. ### Proof: 3 Reversible evolutions -- Hermitian Conjugate

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...
48. ### Why does the complex conjugate of psi pop out?

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...
49. ### Representation of conjugate momentum

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...
50. ### MHB Conjugate Bra Ket properties

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> = -...