What is Hermitian: Definition and 350 Discussions

Homework Statement I need to prove that the hermitian matrix is a vector space over R Homework Equations The Attempt at a Solution I know the following: If a hermitian matrix has aij = conjugate(aji) then its easy to prove that the sum of two hermitian matrices A,B give a hermitian...

Hermitian operator--prove product of operators is Hermitian if they commute Homework Statement If A and B are Hermitian operators, prove that their product AB is Hermitian if and only if A and B commute. Homework Equations 1. A is Hermitian if, for any well-behaved functions f and g...

I was reading Lie Algebras in Physics by Georgi......second edition... Theorem 1.2: He proves that every finite group is completely reducible. He takes PD(g)P=D(g)P ..takes adjoint...and gets.. P{D(g)}{\dagger} P=P {D(g)}{\dagger} So..does this mean that the projection...

A and B are two Hermitian vector operators. K1=AXB, K2=AXB-BXA. Are K1 and K2 hermitian or anti-hermitian?

A and B are two hermitian vector operators. K1=AXB, K2=AXB-BXA. Are K1 and K2 hermitian or anti-hermitian?

1. Let AH be the hermitian matrix of matrix A, and how the eigenvalues of AH be related to eigenvalues of A? [b]3. what I have done is equation no.1: (AH-r1*I) * x1 = 0, And equation no.2: (A-r2*I) * x2 = 0 time no.1 both sides by x2H ((A*x2)H-r1*x2H)* x1 = 0 Then we have...

Consider the Dirac equation in the ordinary form in terms of a and \beta matrices i\frac{{\partial \psi }} {{\partial t}} = - i\vec a \cdot \vec \nabla \psi + m\beta \psi The matrices are hermitian, \vec a^\dag = \vec a,\beta ^\dag = \beta . Daggers denote hermitian...

Hi there, Does anyone know the Latex code for Hermitian conjugate (dagger) on TeXniccenter? Thank you!

Homework Statement Choose \lambda_{1}, \lambda_{2}, \lambda_{3} along with a set of vectors {v_{1},v_{2},v_{3}} and construct an Hermitian matrix H with the eigenpairs (\lambda_{1},v_{1}),(\lambda_{2},v_{2}),(\lambda_{3},v_{3}) Homework Equations The Attempt at a Solution...

Homework Statement Consider the vector space of square-integrable functions \psi(x,y,z) of (real space) position {x,y,z} where \psi vanishes at infinity in all directions. Define the inner product for this space to be <\phi|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}...

Let P and Q be Hermitian positive definite matrices. We prove that x*Px < or eq. x*Qx, for all x in C^n (C : complex numbers) if and only if x*Q^-1 x < or eq. x*P^-1 x for all x in C^n. I guess I should use the definition of a hermitian positive definite matrix being x*Px > 0 , for all x in...

Homework Statement A and B are noncommuting quantum mechanical operators: AB - BA = iC Show that C is Hermitian. Assume all the appropriate boundary conditions are satisfied. I do not understand how to show this. I isolated C as: C = i(BA-AB) ..and I want to show that C is...

If the Hamiltonian is given by H(x,p)=p^2+p then is it Hermitian? I'm guessing it's not, because quantum-mechanically this leads to: H=-h^2 \frac{d^2}{dx^2}-ih\frac{d}{dx} and this operator is not Hermitian (indeed, for the Sturm-Liouville operator O=p(x)\frac{d^2}{dx^2}+k(x)\frac{d}{dx}+q(x)...

What is the relationship btw the Hermitian inner product btw 2 complex vectors & angle btw them. x,y are 2 complex vectors. \theta angle btw them what is the relation btw x^{H}y and cos(\theta)?? Any help will be good?

Hi there, In all text of QM I have, they tells that the density operator is hermitian. But without considering the math, from the physics base, why density operator must be hermitian? What's the physical significane of the eigenvalue of density matrix? Thanks

Homework Statement Using Dirac notation (bra, kets), define the meaning of the term "Hermitian". Homework Equations The Attempt at a Solution From what I understand, a hermitian operator is simply one that has the same effect as its hermitian adjoint. So, I'm assuming it should...

If \hat{O} is hermitian, show that \hat{O}^2 is hermitian. we have <\psi|\hat{O}^2|\psi>^* = <\psi|\hat{O}\hat{O}|\phi>^*=<\phi|\hat{O}^{\dagger} \hat{O}^{\dagger}|\psi>=<\phi|\hat{O}\hat{O}|\psi>=<\phi|\hat{O}^2|\psi> which works (hopefully)! to do this in integral notation is the...

Homework Statement Homework Equations The Attempt at a Solution I've gone round in circles doing this! I started of by writing it as an integral of (psi* x A_hat2 x psi) w.r.t dx, then using the equation above but I keep coming back at my original equation after flipping it...

pretty simple question. have to prove \hat{O} \hat{O}\dagger is a Hermitian operator. i found that \left( \int \int \int \psi^{\star}(\vec{r}) \hat{O} \hat{O}^{\dagger} \phi(\vec{r}) d \tau \right)^{\star} = \int \int \int \phi^{\star}(\vec{r}) \hat{O}^{\dagger} \hat{O} \phi(\vec{r}) d...

Homework Statement C is an operator that changes a function to its complex conjugate a) Determine whether C is hermitian or not b) Find the eigenvalues of C c) Determine if eigenfunctions form a complete set and have orthogonality. d) Why is the expected value of a squared hermitian...

Homework Statement Im am considering a covariant differential: D_\mu H = ( partial_\mu + \frac{1}{2} i g \tau_j W_{j\mu} + ig B_\mu ) H H is an isospiner, \tau_j are the pauli spin matrices, \partial_\mu is the four-gradient \frac{\partial}{\partial x^\mu} and W_{j \mu} and B_\mu are...

Homework Statement Let (u,v)1 be a second Hermitian scalar product on a vector space V. Claim: There exists a positive transformation T with respect to the given scalar product (u,v) such that (u,v)1 = (Tu,v) for all u,v in V. Homework Equations A transformation T is positive if...

Hi all, i cannot find where's the trick in this little problem: Homework Statement We have an hermitian operator A and another operator B, and let's say they don't commute, i.e. [A,B] = cI (I is identity). So, if we take a normalized wavefunction |a> that is eigenfunction of the operator A...

is the set of nxn traceless hermitian matrices under addition a group? is the set of nxn traceless hermitian matrices under multiplication a group? is the set of nxn traceless non-hermitian matrices under addition a group? question 1-I thought that traceless means trace=0 is this right...

I have a question..I am trying to solve a differential equation that arises in my research problem. Because the differential equation has no solution in terms of well known functions, I had to construct a series solution for the differential equation which is physical and agrees with the...

If you had an operator A-hat whose eigenvectors form a complete basis for the Hilbert space has only real eigenvalue how would you prove that is was Hermitian?

What is the Hermitian conjugate of a complex #, say, 5+6i??

Homework Statement If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well? Homework Equations The Attempt at a Solution

Homework Statement Simply--Prove that any Hermitian operator is linearHomework Equations Hermitian operator defined by: int(f(x)*A*g(x)dx)=int(g(x)*A*f(x)dx) Linear operator defined by: A[f(x)+g(x)]=Af(x)+Ag(x) Where A is an operatorThe Attempt at a Solution I am at a complete loss of how to...

Homework Statement Hi all. Let's say that I have a Hermitian 2x2 matrix A with two distinct eigenvalues, and thus two eigenvectors. Question 1: What space is it they span? Is it R2? Now let us say I have another Hermitian 2x2 matrix B with two distinct eigenvalues, and thus two...

Hello, I'm a little bit confused regarding Hermitian Adjoint. I want to show that <x,y> = A(x,y) is an inner product where the function A : V*V --> V be defined as A(x,y) = x^T*A_hat*y. A_hat = [2 1 0; 1 4 1; 0 1 4]. How would i go about showing that <x,y> = A(x,y) is an inner product...

Homework Statement Let V be the \mathbb{R}-vector space \mbox{Herm}_n( \mathbb{C} ). Find \dim_{\mathbb{R}} V. The Attempt at a Solution I'd say the dimension is 2n(n-1)+n=2n^2-n, because all entries not on the main diagonal are complex, so you have n(n-1) entries which you have to...

Homework Statement If \vec{x} is an eigenvector of a Hermitian matrix H, let V be the set of vectors orthogonal to \vec{x} . Show that V is a subspace, and that it is an invariant subspace of H. The Attempt at a Solution The Hermitian H must act on some linear space, call it K and of...

For a hermitian operator A, does the function f(A) have the same eigenkets as A? This has been bothering me as I try to solve Sakurai question (1.27, part a). Some of my class fellows decided that it was so and it greatly simplifies the equations and it helps in the next part too but I don't...

Hello, what's the difference between Hermitian and self-adjoint operators? Our professor in Group Theory made a comment once that the two are very similar, but with a subtle distinction (which, of course, he failed to mention :smile: ) Thanks!

How to check if an operator is hermitian? I mean what is the condition Actualy, i am using the principe that say that the eigenvalue associated with the operator must be a REAL NUMBER.That is to say that i work out to that eigenvalue and see if it is a real number. Am i right?

Homework Statement Ok, here is another little pickle. I am trying to determine what the eigenfunctions and eigenvalues are for the operator C that is defined such that C phi(x) = phi*(x). Part a wants to know if this is a Hermitian operator. Parts b,c want eigenfunctions and eigenvalues...

Hi. I'm not too good at maths and I'm having some trouble figuring out the basics of what to do with complex conjugates of functions. Our lecturer has set a couple problems requiring us to test if a few operators are Hermitian. Before I can get to those I thought I'd test the basic momentum...

I read from a book and claim that for any hermitian matrix can be diagonalized by a unitary matrix whose columns represent a complete set of its normalized eigenvectors. It then given an equation...

Let us define \hat{R} = |\psi_m\rangle \langle \psi_n| where \psi_n denotes the nth eigenstate of some Hermitian operator. When is \hat{R} Hermitian? Solution? Well, let us just call |psi_m> = |m> and |psi_n> = |n>. Now, we need |m><n| = |n><m| If we left multiply by <m| then we find...

Problem Consider the operator \hat{C} which satisfies the property that \hat{C} \phi (x) = \phi ^ * (x). Is \hat{C} Hermitian? What are the eigenfunctions and eigenvalues of \hat{C}? Solution We have \hat{C} \phi = \phi ^ * \iff \phi^* \hat{C}^\dagger = \phi Substituting back into...

Does anyone has proof of radial momentum operator as an Hermitian operator? Thanks.

How do I show that an arbitrary operator A can be writte as A = B + iC where B and C are hermitian?

So I had a QM test today and I needed to show that the energy operator is hermitian. This was easy to show provided that the the adjoint of d/dt is -d/dt. I know this is the case for the spatial derivative but is it the case with the time derivative? The bra-ket is an integral over x not time...

Homework Statement Show that the spectrum \sigma of a linear continuous Hermitian operator A on a Hilbert space H consist of real numbers, ie \sigma(A)\subset \mathbb{R} . Homework Equations Well the spectrum of A are the elements \lambda\in\mathbb{C} such \lambda I - A is NOT...

I've been struggling for awhile, I've been trying to use CLAPACK to avoid learning Fortan. I think I've just a linking problem, since I've been testing code that's supposed to work. in the VC command prompt i type cl dgesv.c and I get the error LNK2019: unresolved external symbol...

Simple question, and pretty sure I already know the answer - I just wanted confirmation, Considering the Hermitian Conjugate of a matrix, I understand that A^{+} = A where A^{+} = (A^{T})^{*} Explicitly, (A_{nm})^{*} = A_{mn} Would this mean that for a matrix of A, where A is a...

Is there any other difference between self-adjoint operators, and Hermitian operators, than that mathematicians seem to talk mostly about self-adjoint operators, and physicists seem to talk mostly about Hermitian operators?

Homework Statement Since the momentum operator is Hermitian why is this wrong: <psi| (p-hat)^2 |psi> = <psi| p-hat p-hat |psi> = <p-hat psi| p-hat |psi> = (p)^2 where p is the expectation value of the momentum. Homework Equations The Attempt at a Solution

Homework Statement I have to show that in 3-d, Lx (angular momentum) is Hermitian. Homework Equations In order to be Hermitian: Integral (f Lx g) = Integral (g Lx* f) Where Lx=(hbar)/i (y d/dz - z d/dy) and f and g are both well behaved functions: f(x,y,z) and g(x,y,z) The Attempt...