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

    Complex conjugate as a Mobius transformation

    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). [...
  2. J

    Proving Complex Conjugate is Real: Euler's Identity

    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...
  3. ShayanJ

    Canonically conjugate operators

    I've searched for this but found nothing,so I ask it here. What are canonically conjugate operators? Is [A,B]=cI a definition for A and B being canonically conjugate? Thanks
  4. N

    How does abstract algebra relate to canonically conjugate in physics?

    Momentum and position are canonically conjugate in physics because they are the Fourier transforms of each other. In the context of abstract algebra what would that mean. More precisely, Let G be the group they both (p and x) belong to and let ψ:G->G/H be the natural homomorphism where H is...
  5. mnb96

    Question on conjugate closure of subgroups

    Hello, Let's have a group G and two subgroups A<G and B<G such that the intersection of A and B is trivial. I consider the subgroup \left\langle A^B \right\rangle which is called conjugate closure of A with respect to B, and it is the subgroup generated by the set: A^B=\{ b^{-1}ab \;|\; a\in...
  6. J

    What is the Conjugate of sin(z)?

    The problem is to show sin\overline{z} = \overline{sinz}. What I need is help to get going.We know that sinz = \frac{e^{iz}-e^{-iz}}{2i}I can't see the first step in this. What I've tried to do is expressing sin\overline{z} and \overline{sinz} in terms of the above equation, but I don't know...
  7. T

    Impedance matching derivation (next best after complex conjugate method)

    (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...
  8. M

    Confirming Conjugate Factor of Cubic Equation

    Homework Statement If a cubic equation, f(x) has a factor of (3+√2), then the conjugate of the factor, (3-√2) is also a factor for f(x). Homework Equations The Attempt at a Solution Just to confirm is that statement correct? I read it else where but i not sure is it correct or...
  9. M

    Dirac notation and conjugate transpose in Sakurai

    In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate...
  10. H

    Integrating a function of the complex conjugate of x with respect to dx

    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...
  11. I

    Conjugate fields and conjugate subgroups of an automorphism group

    Suppose E and D are both finite extensions of F, with K being the Galois closure of \langle D,E \rangle (is this the correct way to say it?) Is it correct that E and D are conjugate fields over F iff G,H are conjugate subgroups, where G,H\leqslant \text{Aut}(K/F) are the subgroups which fix...
  12. S

    How/when can I take a wave function and its complex conjugate as independent?

    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...
  13. B

    Complex conjugate on an inner product

    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} +...
  14. S

    Complex conjugate of a derivative wrt z

    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...
  15. S

    Hermitian conjugate of an operator

    Hey guys, I'm doing a third year course called 'Foundations of Quantum Mechanics' and there's this thing in my notes I don't quite get. I was hoping to get your help on this, if you don't mind. It's about Hermitian conjugate operators. The sentences go (v, Au) = (A†v|u) <v|A|u> = <v|(A|u>)...
  16. C

    Finding the solutions to a quadratic equation with a complex conjugate

    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...
  17. R

    Fixed points of conjugate functions

    Homework Statement suppose f and g are conjugate show that if p is an attractive fixed point of f(x), then h(p) is an attractive fixed point of g(x). Homework Equations f and g being conjugate means there exist continuous bijections h and h^-1 so that h(f(x)) = g(h(x)) a point p...
  18. S

    Proving Zero Result with Complex Conjugates and Dot Product

    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...
  19. L

    Conjugate Addition and Ether Formation

    Homework Statement http://i43.tinypic.com/nxvw41.jpg Ok here it is molecule B. http://i39.tinypic.com/333aaro.jpg Ok and here it is problem nr. 8, second molecule. Homework Equations No equations needed. The Attempt at a Solution 1) So why is it that we do not add any...
  20. C

    Conjugate of a wave function

    if the wave function ψ(x)=R(x)eikx then what is it conjugate, ψ*?
  21. S

    Derivative wrt Complex Conjugate

    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...
  22. M

    Proof that if a polynomial has a complex zero it's conjugate is also a zero

    Homework Statement If P(x) is a polynomial with real coefficients, then if z is a complex zero of P(x), then the complex conjugate \bar{z} is also a zero of P(x). Homework Equations Book provides a hint: assume that z is a zero for P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} and...
  23. Fredrik

    Linear or conjugate operators and automorphisms on the lattice of subspaces

    I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even...
  24. R

    Prove there is no homeomorphism that makes two functions conjugate

    Homework Statement let f(x) = x3 and g(x) = x - 2x3. Show there is no homeomorphism h such that h(g(x)) = f(h(x)) Homework Equations Def let J and K be intervals. the function f:J->K is a homeomorphism of J onto K if it is one to one, onto, and both f and its inverse are...
  25. T

    The Dirac equation and its conjugate

    This isn't really a homework problem, just a form of writing I don't quite understand. The Dirac equation is: ("natural units") (i\gamma^{\mu}\partial_{mu}-m)\Psi = 0 When I try to build the conjugated equation, where \bar{\Psi} := \Psi^{+}\gamma^{0}, I get...
  26. Z

    Complex & Hermitian Conjugate of Matrix

    Homework Statement 1x2 Matrix A = [(5) (-2i)] What is the complex conjugate and Hermitian conjugate of this matrix? Homework Equations The Attempt at a Solution D^T = 5 -2i D^H = 5 +2i What do you think of my answers?
  27. V

    Complex conjugate of absolute exponential

    Hello all, I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x) Thanks for your help. -Brian
  28. L

    Normal Subgroup Conjugate of H by element

    Homework Statement Let H be a subgroup of group G. Then H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G \Leftrightarrow xH=Hx \forall x\in G \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G...
  29. V

    Finding the Minor and major axis lenths of an ellipse from conjugate diameters

    Hello, I'm having a problem finding the minor and major axsis lengths of an ellipse from three points, the ellipse's center, and two conjugate end point diameters. I have no problem solving the problem when the conjugate diameters align with the minor and major axsis, but when they don't the...
  30. A

    Complex Conjugate: just replace i by -i even in denominator or inside argument?

    to get a Complex Conjugate of a #, is it ok to just replace i by -i even in denominator or inside argument?
  31. D

    Question on derivative of conjugate

    I was trying to find the derivative of \overline{x} for some x \in \mathbb{C} I solved this as \frac{\mbox{d}}{\mbox{d}x} \left(\overline{x}\right) = \lim_{h \rightarrow 0}\frac{\overline{x+h}-\overline{x}}{h} \frac{\mbox{d}}{\mbox{d}x} \left(\overline{x}\right) = \lim_{h...
  32. N

    [complex analysis] differentiation w.r.t. complex conjugate; does it make sense?

    Hello, Differentiability of f : \mathbb C \to \mathbb C is characterized as \frac{\partial f}{\partial z^*} = 0. More exactly: \frac{\partial f(z,z^*)}{\partial z^*} := \frac{\partial f(z[x(z,z^*),y(z,z^*)])}{\partial z^*} = 0 where z(x,y) = x+iy and x(z,z^*) = \frac{z+z^*}{2} and...
  33. M

    Conjugate beam: help with clarification of given diagram, not with my work

    Homework Statement See problem number 2 as attached. The Attempt at a Solution I found the reactions and shear and moment diagrams for the original beam. I also drew the conjugate beam and loaded it with M/EI. My only question for this problem is what the 4EI and EI under the beam...
  34. M

    Slope and deflection diagrams using conjugate beam method

    Homework Statement Construct the slope and deflection diagrams. I've attached the problem with the original diagram (problem #1). The Attempt at a Solution Considering the number of diagrams required, I thought it would be best to attach a photo of my work. I drew the shear and moment...
  35. M

    [Linear Algebra] Finding T*; complex conjugate linear transformation

    [Linear Algebra] Finding T* adjoint of a linear operator Homework Statement Consider P_1{}(R), the vector space of real linear polynomials, with inner product < p(x), q(x) > = \int_0^1 \! p(x)q(x) \, \mathrm{d} x Let T: P_1{}(R) \rightarrow P_1{}(R) be defined by T(p(x)) = p'(x) +...
  36. V

    Hermitian conjugate of operators

    Homework Statement Find the hermitian conjugates, where A and B are operators. a.) AB-BA b.) AB+BA c.) i(AB+BA) d.) A^\dagger A Homework Equations (AB)^\dagger =B^\dagger A^\dagger The Attempt at a Solution Are they correct and can I simplify them more? a.)...
  37. B

    Conjugate Relief in Gears/ Transmission Error

    Folks, 1) Transmission Error: I understand it is the deviation of the driven gear from its theoretical position based on the driving gear moving with uniform angular velocity. Ie it is a measure of the where the driven gear actually is compared to where it should be...but I don't understand...
  38. V

    Solving Hermitian Conjugate Homework

    Homework Statement a.) Show \hat {(Q^\dagger)}^\dagger=\hat Q , where \hat {Q^\dagger} is defined by <\alpha| \hat Q \beta>= <\hat Q^ \dagger \alpha|\beta> . b.) For \hat Q =c_1 \hat A + c_2 \hat B , show its Hermitian conjugate is \hat Q^\dagger =c_1^* \hat A^\dagger + c_2^* \hat...
  39. Q

    QM - Spin operator conjugate question

    Homework Statement Okay so I've got a question I really need answered first up! If I have a 2x1 matrix for Psi, is Psi* a 1x2 matrix with all the 'i's turned to '-i's? Now onto the actual question - http://imgur.com/3ucb4" - part b only Homework Equations http://imgur.com/bcEm3"...
  40. B

    Hermitian conjugate of plane wave spinors for Dirac equation

    I need to show that u^{+}_{r}(p)u_{s}(p)=\frac{\omega_{p}}{m}\delta_{rs} where \omega_{p}=\sqrt{\vec{p}^2+m^{2}} [itex]u_{r}(p)=\frac{\gamma^{\mu}p_{\mu}+m}{\sqrt{2m(m+\omega_{p})}}u_{r}(m{,}\vec{0})[\itex] is the plane-wave spinor for the positive-energy solution of the Dirac equation...
  41. V

    Solving z^2 = z Conjugate Complex Equation

    Homework Statement Find all complex solutions to z^2 = z conjugate i.e. (a+bi)^2 = a-bi The attempt at a solution First attempt: factoring out (a+bi)^2 = a-bi leads nowhere. Second attempt: r^2 (cos2v + isin2v) = r (cos-v + isin-v) r must be 1. 2v = -v + 2∏n 3v = 2∏n v= 2∏n/3...
  42. A

    Why is the function f(z) = ¯z not differentiable for any z ∈ C?

    Homework Statement Show that f(z) = ¯z is not differentiable for any z ∈ C. Homework Equations The Attempt at a Solution Is it because the Cauchy-Reimann Equations don't hold? Z (conjugate) = x-iy u(x,y)=x v(x,y=-iy du/dx=1≠dv/dy=-1 du/dy=0≠-dv/dx=0 Edit: Is there another approach? Because...
  43. R

    A conjugate of two permutations question

    Homework Statement Suppose x_{1} = \begin{pmatrix} 2 & 9 & 6 \\ \end{pmatrix}\begin{pmatrix} 3 & 5 & 8 \\ \end{pmatrix}\begin{pmatrix} 4 & 7 \\ \end{pmatrix} and x_{2} = \begin{pmatrix} 1 & 5 & 9 \\ \end{pmatrix}\begin{pmatrix} 2 & 7 & 6 \\ \end{pmatrix}\begin{pmatrix} 3 & 4 \\...
  44. J

    Complex conjugate operator is linear

    My textbook claims that the complex conjugate operator is linear. I can't see how this could be. Could someone give me an example of how it is not linear?
  45. F

    Proving Conjugate Subgroups K=H with Prime Number of Elements

    Homework Statement How would I go about proving that if K = gHg-1, for some g \inG, where K and H are both subgroups of G with a prime number of elements, then K = H? Homework Equations I've tried to prove it by saying that if K = gHg-1 then Kg = gH, and since H = gHg-1, then Hg = gH...
  46. T

    Harmonic conjugate of potential function

    Let V(x,y) be the electric potential associated to an electric field E. Suppose that V is harmonic everywhere on R^2. Let F(x,y) be some function such that V and F satisfy Cauchy-Riemann equations (so that F is the harmonic conjugate of V). Let f : C -> C such that f(x,y) = V(x,y) + iF(x,y)...
  47. P

    Conjugate of amplitude with imag. # in exponent and out of exponent?

    When you have i*e^(i@) as an amplitude, when you conjugate, do both i terms switch signs? I tried this and keep getting wrong answer. Thanks in advance. BTW this has to do with spin half particles.
  48. T

    Curl and divergence of the conjugate of an holomorphic function

    I noted that if [itex]f : C \to C[\itex] is holomorphic in a subset [itex]D \in C[\itex], then [itex]\nabla \by \hat{f} = 0, \nabla \dot \hat{f} = 0[\itex]. Moreover, those two expressions are equivalent to the Cauchy-Riemann equations. I'm rewriting this in plaintext, in case latex doesn't...
  49. L

    Working complex fractions without conjugate method

    Homework Statement Ive been given a question that requires an answer in polar form but the method I must use is the normal addition/subtraction of fractions. This throws me because I'm sure there is a simple method for the reciprocal of a complex number. Q. 1/(13- 5i) - 1/(2-3i)...
  50. A

    How do you take this complex conjugate?

    What's the complex conjugate of \frac{1}{\sqrt{1+it}}, \quad t \geq 0.
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