What is Hermitian: Definition and 349 Discussions

problem based on hermitian operator Homework Statement A is an hermitian operator and as we know the eigenstates a of A with eigenvalues a satisfy A psi a = a psi a. How do we show that lambda psi a (lambda is a non zero complex number) is an eigen state belonging to the same eigen...

Homework Statement WTS \langle \hat{A} \rangle = \langle \hat{A} \rangle^\ast The Attempt at a Solution \langle \hat{A} \rangle^\ast = \left(\int \phi_l^\ast \hat{A} \phi_m dx\right)^\ast=\left(\int (\hat{A}\phi_l)^\ast \phi_m dx\right)^\ast= \int \phi_m^\ast \hat{A}\phi_l dx. So...

I have some questions about the properties of a Hermitian Operators. 1) Show that the expectaion value of a Hermitian Operator is real. 2) Show that even though \hat{}Q and \hat{}R are Hermitian, \hat{}Q\hat{}R is only hermitian if [\hat{}Q,\hat{}R]=0 Homework Equations The...

Question: Is the Fourier Transform of a Hermitian operator also Hermitian? In the case of the density operator it would seem that it is not the case: \rho(\mathbf{r}) = \sum_{i=1}^N \delta(\mathbf{r}-\mathbf{r}_i) \rho_k = \sum_{i=1}^N e^{-i\mathbf{k} \cdot \mathbf{r}} I have a hard...

i've just started going through QM and I'm having major problems with following the significance of hermitian matrices. the main problem is i can't visualise what's happening to a matrix when you calculate its transpose or adjoint. can anybody give me a useful way of visualising this?

Its quantum computing but related to math: Homework Statement show every hermitian matrix can be diagonalized by unitary matrix. Prove this using. N x N matrix. Homework Equations H= hermitian matrix. U = unitary matrix show U-1(inverse)HU = D (diagonal) using N x N matrix. The...

Homework Statement Show that if \Omega is an hermitian operator, and \varphi and \psi are (acceptable) wavefunctions, then then \int \phi^{*} \Omega \psi dz = \int \psi (\Omega \phi)^{*} dz Homework Equations Consider the wave function \Psi = \phi + \lambda\psi The Attempt at a...

Homework Statement a = x + \frac{d}{dx} Construct the Hermitian conjugate of a. Is a Hermitian? 2. The attempt at a solution <\phi|(x+\frac{d}{dx})\Psi> \int\phi^{*}(x\Psi)dx + <-\frac{d}{dx}\phi|\Psi> I figured out the second term already but need help with first term... am...

Can anybody give me a hint about how can i show that if an operator is linear then it's hermitian conjugate is linear. Thanks for your help from now.

How can i show that (a_{1} A_{1}+a_{2} A_{2})^{\dagger}=a_{1}^{\ast} A_{1}^{\dagger}+a_{2}^{\ast} A_{2}^{\dagger} notice: a_{1},a_{2}\in C and A_{1}^{\dagger},A_{2}^{\dagger} are hermitian conjugate of A_{1},A_{2} operators

Proove that position x and momentum p operators are hermitian. Now, more generaly the proof that operator of some opservable must be hermitian would go something like this: A\psi_{n}=a_{n}\psi_{n} Where A operator of some opservable, \psi_{n} eigenfunction of that operator and a_{n} are the...

Dear experts! I have a small Hermitian matrix (6*6). I need to inverse this matrix. The program memory is bounded. What method is optimal in this case? Can you give any e-links? Thanks In Advance.

Homework Statement How do I show that the annihilation operator \hat{a} is hermitian WITHOUT explicitly using the condition where an operator X is hermitian if its adjoint is also X ie. X=X^+ Homework Equations none. The Attempt at a Solution I could show \hat{a} \hat{x}...

1.What does it mean for an operator to be hermitian? Note: the dagger is represented by a ' 2. How do I show that for any operator ie/ O' that O + O' , i(O-O') and OO' are hermitian? Thanks in advanced

I just have a simple question about hermition conjugates of spinors. Is the hermitian conjugate of: \epsilon \sigma^\mu \psi^\dagger equal to: -\psi \sigma^\mu \epsilon^\dagger where both psi and epsilon are 2-component spinors of grassmann numbers?

Homework Statement Within the framework of quantum mechanics, show that the following are Hermitian operators: a) p=-i\hbar\bigtriangledown b) L=-i\hbar r\times\bigtriangledown Hint: In Cartesian form L is a linear combination of noncommuting Hermitian operators. Homework Equations...

show that if A and B are both Hermitian, AB is Hermitian only if [A,B]=0. where or how do io start?

Let \mathcal{L} = \frac{d}{dx} p(x) \frac{d}{dx} + q(x) be a self-adjoint operator on functions f : [a,b] \rightarrow \mathbb{C}. Under what circumstances is the operator Hermitian with <u|v> = \int_a^b u^*(x) v(x) dx ? Can someone give me a hint on this one? I know that hermitian operators...

Hi again, Question: \hat{A} is an Hermitian Operator. If \hat{A}^{2}=2, find the eigenvalues of \hat{A} So We have: \hat{A}\left|\Psi\right\rangle=a\left|\Psi\right\rangle But I actually don't know how to even begin. \hat{A} is a general Hermitian operator, and I don't know where...

Hi, I'm doing a Quantum mechanics and one of my question is to determine if \frac{d^2}{dx^2} (a second derivative wrt to x) is a Hermitian Operator or not. An operator is Hermitian if it satisfies the following: \int_{-\infty}^{\infty}\Psi^{*}A\Psi =...

asking here because i originally asked in the wrong place :) this question is two parts, both dealing with telling if combinations of hermitian operators are hermitian. the first combination is PX + XP, where P stands for the momentum operator, (h bar /i)(d/x), and X is the "x operator"...

Find the diagonal form of the Hermitian matrix A=\left( \begin{array}{cc} 2 & 3i\\ -3i & 2 \end{array} \right) The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix. I put the columns of P as the eigenvectors (with unit length) of A...

Please help! -Dirac delta potential-, Hermitian Conjugate Im trying to solve problem 2.26 from Griffiths (1st. ed, Intro to Q.M.). Its about the allowed energy to double dirac potential. I came up with a final equation that is trancedental. (After I separate the even and odd solution of psi.)...

The question was: If B is Hermitian show that A=B^2 is positive semidefinite. The answer was: B^2 has eigenvalues \lambda_1 ^2, ... \lambda_n^2 (the square of B's eigenvalues) all non negative. My question is: Why do we know that B^2 has eigenvalues \lambda^2 ?

would anyone mind showing me, for example, how to prove that d^2/dx^2 is a hermitian operator? I've tried to work it out from two different books; they both prove that the momentum operator is hermitian, but when i try to apply the same thing to the operator d^2/dx^2 i get lost pretty quick...

I'm getting some confusing information from different sources. If an inner product satisfies conjugate symmetry, it is called Hermitian. But the definition of a hermitian inner product says it must be antilinear in the second slot only. Doesn't conjugate symmetry imply that it's antilinear in...

hey, it's good to be back at pf. :cool: anyway, today i had an exam in my honors modern course, and one of the questions was a proof that the parity operator is hermitian. i don't think i got it right. :/ here's what i did: 1: \int(P_(op) \psi_2(x))^* \psi_1(x) dx = \int \psi_2^*(-x)...

My memory is fading. Can somebody please remind me how I would go about determining in each of the following cases whether the operator A is Hermitian or not? Case 1. A\psi(x) = \psi(x+a) Case 2. A\psi(x) = \psi^*(x) where the star indicates complex conjugation.

This is the problem: Let T be a complex linear space with a complex inner product <.,.>. Define T in L(V,V) to be Hermitian if <Tv,v> = <v,Tv> for all v in V. Show that T is Hermitian iff <Tv,w> = <v,Tw> for all v,w in V [Hint: apply the definition to v+w and to v+iw]. So this was my...

At the risk of arrousing the ire of the moderaters for posting the same topic in two forums, I again ask this question as no one in the quantum forum seems to be able to help. So... Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian...

I'm not sure if this is the appropriate section, perhaps my question is better suited for Linear Algebra. At any rate, here goes. Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian operator A: Given A|\phi_1\rangle = a_1|\phi_1\rangle...

First post so please go easy on me, here goes: I have looked over the basic definition of what is a Hermitian operator such as: <f|Qf> = <Qf|f> but I still am unclear what to do with this definition if I am asked prove whether or not i(d/dx) or (d^2)/(dx^2) for example are Hermitian...

Hi there, Was wondering if anyone could point me in the right direction for this one? Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal? Thanks

Reading back in my book, Greiner's "QM :an introduction" I found a formula I don't understand. Let \alpha be a real number, \Delta \hat{A}, \Delta \hat{B} be Hermitian operators. Now I have \int (\alpha \Delta \hat{A} - i \Delta \hat{B})^* \psi^* (\alpha \Delta \hat{A} - i \Delta \hat{B})...

I haven't read this yet, but I'm putting it up here for discussion as it seems so fascinating: PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics Carl M. Bender, Jun-Hua Chen, Kimball A. Milton A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by...

How do you find the hermitian conjugate of x, i, d()/d(x), a+ 'the harmonic oscilator raising operator'?

Hi everyone! How can I find the Hermitian conjugate of the differential operator D, with D psi = 1/i dpart/dpart(x) psi? I know you can do this with partial integration starting from <phi|D|psi>* = <phi|D+|psi> but how exactly does it work? I'm sorry for using such an ugly...

Sorry if this is in the wrong section. I just want to check my answer since I've been going through the exam. Given that A is an n × n matrix and I is the n × n identity matrix, select all the correct responses below. A Every diagonalisable matrix is normal. B If A is Hermitian, then A^TA...

Let's have hermitian matrix A. Then these three conditions are equivalent: 1) A is positively definite \forall x \in \mathbb{C}^{n} \ {0} : x^{H}Ax > 0 2) All eigenvalues of A are positive 3) There exists regular matrix U such that A = U^{H}U Proof: 2) \Rightarrow 3)...

Hi all, I don't understand to one part of proof of this theorem: All eigenvalues of each hermitian matrix A are real numbers and, moreover, there exists unitary matrix R such, that R^{-1}AR is diagonal Proof: By induction with respect to n (order of matrix A) For n = 1...

i searched the forum, but nothing came up. My question, how do you prove that [A,B] = iC if A and B are hermitian operators? I understand how C is hermitian as well, but i can't figure out how to prove the equation.

Is the second derivative with respect to position a hermitian operator? (i.e. d^2/dx^2)? Can anyone prove it? I don't think it is. Thanks

Is the second derivative with respect to position a hermitian operator? (i.e. d^2/dx^2)? Can anyone prove it? I don't think it is. Thanks

Im having difficulty computing large Hermitian polynomials in C++. I fear I may have to steer away from a recursive formula. Any help would be greatly appreciated. John

In this question I let "x1t , x2t, x3t " be the conjugate of x1, x2, x3 The hermitian form Hc(x) = c*x1t*x1 + 2*x2t*x2 - i*x1t*x2 + i*x2t*x1 + x1t*x3 + x3t*x1 +i*x2t*x3 - i*x3t*x2 (sorry, it`s a bit messy) For which value of c is Hc ositive definite? I have tried to find the...

This didn't seem appropriate for College level so I thought I'd post it here. I'm struggling to find a way to prove that the product of two operators P and Q written PQ have the hermitian conjugate Q*P* where the star denotes hermitian conjugate. Really just can't get off the first line with...

So I understand what a hermitian operator is and how if A and B are hermitian operators, then the product of AB is not necessarily Hermitian since *Note here + is dagger (AB)+=B+A+=BA I also recognize that (AB-BA) is not Hermitian since (AB-BA)+=B+A+-A+B+ In addition, I know that...

Let's say you're wondering around P(oo) (which I'll use to represent the space of polymomials of any degree on the interval [-1,1]) and you decide to calculate the matrix X representing the position operator x. Let's say you do this in the basis: 1, t, t^2, ..., t^n,... you'll find that the...

-hey everyone, this one might be a little too math based for this forum, but I ran across it studying for one of my quantum exams and it seemed like an interesting problem. Haven't figured it out completely. We all know hermitian operators play a central role in quantum and so being able...