Hermitian Definition and 347 Threads

Homework Statement Show that the Hamiltonian operator (\hat{H})=-((\hbar/2m) d2/dx2 + V(x)) is hermitian. Assume V(x) is real Homework Equations A Hermitian operator \hat{O}, satisfies the equation <\hat{O}>=<\hat{O}>* or ∫\Psi*(x,t)\hat{O}\Psi(x,t)dx =...

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...

Homework Statement I don't understand why the Pauli matrix σx is hermitian. Nonetheless, I am able to prove why the σy matrix is hermitian. Homework Equations The Attempt at a Solution Whenever I do the transpose and then the conjugate I get the negative of σx instead. Am I doing...

I have this review question: If operators A and B are hermitian, prove that their commutator is "anti-hermitian", ie) [A,B]†=-[A,B] What has me confused is the placement of the dagger on the commutator. Why [A,B]† and not [A†,B†]? Also, I am using Griffith's Intro to QM as a text. I have...

I have a few operators here, and was wondering how to go about proving whether or not they are Hermitian: a) ix^2 b) e^x c) 3x + P_hat/2 d) x^2*P_hat e) ix*P_hat

Hiya :) the title is meant to be prove it isn't hermitian Homework Statement Prove the operator d/dx is hermitian Homework Equations I know that an operator is hermitian if it satisfies the equation : <m|Ω|n> = <n|Ω|m>* The Attempt at a Solution Forgive the lack of latex , I...

Homework Statement \int d^{3} \vec{r} ψ_{1} \hat{A} ψ_{2} = \int d^{3} \vec{r} ψ_{2} \hat{A}* ψ_{1} Hermitian operator A, show that this condition is equivalent to requiring <v|\hat{A}u> = < \hat{A}v|u> Homework Equations I changed the definitions of ψ into their bra-ket forms...

Hi. I'm just a bit stuck on this question: Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators? Im not quite sure how to start it. Thanks!

Homework Statement Show that (AB)^+ = A^+ B^+ using index notation Homework Equations + is the Hermitian transpose The Attempt at a Solution I know that AB = Ʃa_ik b_kj summed over k so (AB)^+ = (Ʃa_ik b_kj)^+ = Ʃ (a_ik b_kj)^+ = Ʃ (a_ik)^+(b_kj)^+ = A^+ B^+ I am not...

I encountered this part in Griffith's Introduction to Quantum Mechanics that I have been unable to figure out. It is probably obvious, but I am not seeing it. I probably need more practice with operators in order to have it fully understood. Equation 2.64 in the second edition states...

Homework Statement Given a Hermitian operator A = \sum \left|a\right\rangle a \left\langle a\right| and B any operator (in general, not Hermitian) such that \left[A,B\right] = \lambdaB show that B\left|a\right\rangle = const. \left|a\right\rangle Homework Equations Basically those...

Homework Statement I need the steps to follow when finding the polar decomposition of a hermitian matrix If someone could direct me to a website that would help, or put up an example here please. thanks :) Homework Equations The Attempt at a Solution

I'm looking for a proof of the fact that orthogonal eigenfunctions of a Hermitian operator have distinct eigenvalues. I know the proof the converse: that eigenfunctions belonging to distinct eigenvalues are orthogonal. thanks alot!

Hello, I am trying to understand what the differences would be in replacing the symmetry equation: g_mn = g_nm with the Hermitian version: g_mn = (g_nm)* In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about...

When defining the radial momentum operator, we don't use the classical analogue which would be \underline{x}.\underline{p}/r where \underline{x} and \underline{p} are operators. Instead we choose 1/2(\underline{x}.\underline{p}/r+\underline{p}.\underline{x}/r). If it is because the former...

Hi! Q is postive definite A is any matrix. Why Q^{-1}AQ^{-1} is hermitian??

Hi, Suppose that we have a complex matrix \mathbf{H} that is Hermitian. The real part of the matrix will be symmetric, and the imaginary part of the matrix will be anti-symmetric. But what about the diagonal elements in the imaginary part? I mean we deduce that the elements in the diagonal of...

I recently thought of this, please excuse me if it is way off the mark! If I act on a state with a hermitian operator, am I able to find the psi(p) (momentum), where I had psi(x) (position) before (and wise versa)? Or does the operator do what it appears to do, and that is find the derivative...

A Hermitian matrix is a square matrix that is equal to it's conjugate transpose. Now let's say I have a Hermitian operator and a function f: [ H.f ] The stuff in the square is the complex conjugate as the functions are in general complex. If I do not consider the matrix representation of...

We know in general a Hermitian operator is not guaranteed to have eigenvalues, but self-adjoint operator is(if I remember correctly). Then why we still claim all observables are hermitian instead of claiming them to be self-adjoint?

Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...

Prove that the sum of two hermitian matrices A and B gives us a hermitian matrix. I'm not sure if this is a legit proof: A+B=A*+B* =(conjugate of A)T+(conjugate of B)T =(conjugate(A+B))T =(A+B)T

A,B and C are three hermitian operators such that [A,B]=0, [B,C]=0. Does A necessarily commutes with C?

Homework Statement Suppose that the commutator between two Hermitian operators â and \hat{}b is [â,\hat{}b]=λ, where λ is a complex number. Show that the real part of λ must vanish. Homework Equations Let A=â B=\hat{}b The Attempt at a Solution AΨ=aΨ BΨ=bΨ...

I'm in my second year of a physics degree and my QM lecturer showed us how to calculate the RMS around the expectation of an operator by considering the E of a system in equal superposition of two energy eigenstates u_1 and u_2. He then says "This gives some measure of how far off we would be...

Homework Statement Let A, B, C, D be nxn complex matrices such that AB and CD are Hermitian, i.e., (AB)*=AB and (CD)*=CD. Show that AD-B*C*=I implies that DA-BC=I The symbol * indicates the conjugate transpose of a matrix, i.e., M* is the conjugate transpose of M. I refers to the identity...

Homework Statement Show that the operator O = i \frac{d2}{ dx2 (please not 2 a squared term, Latex not working. So i (d2/dx2)) is not hermitian operator for a particle in 1D with periodic boundary conditions. Homework Equations The Attempt at a Solution I know to prove an...

Homework Statement Show that the eigenvalues of a hermitian operator are real. Show the expectation value of the hamiltonian is real. Homework Equations The Attempt at a Solution How do i approach this question? I can show that the operator is hermitian by showing that Tmn =...

Hi, In the definition of the Hermitian product is says that <v.v> >= 0. But <v.v> is a complex number, how is "larger then" defined for a complex number? Does the definition refer to the length of the complex number? Thanks

Not really sure how to go about this. Our lecture said "it can be shown" but didn't go into any detail as apparently the proof is quite long. I'd really appreciate it if someone could show me how this is done. Thanks. (Not sure if this is relevant but I have not yet studied Hilbert spaces).

Homework Statement 1)Prove that the characteristic roots of a hermitian matrix are real. 2)prove that the characteristic roots of a skew hermitian matrix are either pure imaginary or equal to zero. Homework Equations The Attempt at a Solution

Hi, this is actually more a math-problem than a physics-problem, but I thought I'd post my question here and see if anyone can help me. So I'm writing an assignment in which I have to define, what is understood by a hermitian operator. My teacher has told me to definere it as: <ϕm|A|ϕn> =...

I decided to go over the mathematical introductions of QM again.The text I use is Shankar quantum, and I came across this theorem: "If \Omega and \Lambda are two commuting hermitain operators, there exists (at least) a basis of common eigenvectors that diagonalizes them both." in the proof...

Homework Statement Prove: (QR)*=R*Q*, where Q and R are operators. (Bij * I mean the hermitian conjugate! I didn't know how to produce that weird hermitian cross) The Attempt at a Solution I have to prove this for a quantum physics course, so I use Dirac's notation with two random functions f...

If we're attempting to solve H\psi = E\psi where H is real and Hermitian, are we allowed to assume \psi is real? Why or why not? My gut tells me the answer is "yes," since we know E is real, but I can't make my idea rigorous.

1. a) The action of the parity operator, \Pi(hat), is defined as follows: \Pi(hat) f(x) = f(-x) i) Show that the set of all even functions, {en(x)}, are degenerate eigenfunctions of the parity operator. What is their degenerate eigenvalue? The same is true for the set of all odd functions...

Hi, I totally understand why \chi\psi=\chi^{a}\psi_{a}=-\psi_{a}\chi^{a}=\psi^{a}\chi_{a}=\psi\chi. Where the first equality is just convention, the second is anticommutation of the fields, the third is due to \chi^{a}\psi_{a}=-\chi_{a}\psi^{a} because of the \epsilon^{ab} . But now if...

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165642.jpg?t=1287612122 http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165727.jpg?t=1287612136 Thanks.

1. Let G be an operator on H (Hilbert Space). Show that: (a) H = 1/2 (G + G^{\dagger}) is Hermitian. (b) K = -1/2 (G - G^{\dagger}) is Hermitian. (c) G = H + iK. Homework Equations ... 3. The Attempt at a Solution : (a) Since the adjoint of the sum of two operators does not change...

Hi, If we start with \psi^{\dag}\bar{\sigma}^{\mu}\chi and take its Hermitian conjugate: \left[\psi^{\dag}\bar{\sigma}^{\mu}\chi\right]^{\dag}=\left[\psi^{\dag}_{\dot{a}}\bar{\sigma}^{\mu\dot{a}c}\chi_{c}\right]^{\dag} I'm basing this on Srednicki ch35 (p219 in my edition). His next line...

Homework Statement I have some operators, and need to figure out which ones are Hermitian (or not). For example: 1. \hat{A} \psi(x) \equiv exp(ix) \psi(x) Homework Equations I have defined the Hermitian Operator: A_{ab} \equiv A_{ba}^{*} The Attempt at a Solution I just don't know where...

Homework Statement Show that the operator x^kp_x^m is not hermitian, whereas \frac{x^kp_x^m+p_x^mx^k}{2} is, where k and m are positive integers. The Attempt at a Solution Is this valid? <x^kp_x^m>^*=\left(\int_{-\infty}^\infty\Psi^*x^k(-i\hbar)^m\frac{\partial^m\Psi}{\partial...

(a) I'm not sure what else to do. I don't think I'm properly treating the i. http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-11192023.jpg?t=1286843024 http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-11192034.jpg?t=1286843025

If anyone has time could they please answer this question. I was looking and concept of the the http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" , I was wonder is their a difference between the two terms? If so how are Hermitian and the Hamiltonian different? Can anyone give...

I find the Lagrangian associated with the Dirac equation given in texts as \mathcal{L}=\bar{\psi}\left(i\gamma^\mu \partial_\mu - m\right)\psi or \mathcal{L}=i\bar{\psi}\gamma^\mu \partial_\mu \psi- m\bar{\psi}\psi \mathcal{L}=i \psi^{\dagger}\gamma^0\gamma^\mu \partial_\mu \psi-...

is it possible on a station? how long?

Homework Statement I attached the problem in a picture so its easier to see. Homework Equations The Attempt at a Solution These are the values i got \lambda_ 1 = 1 \lambda_ 2 = -1 x_1 = [-i; 1] (x_1)^H = [i 1] x_2 = [ i; 1] (x_2)^H = [-i 1] * where x_1 and x_2 are...

Hi, there. It should be yes, but I'm very confused now. Consider a simple one-dimensional system with only one particle with mass of m. Let the potential field be 0, that's V(r) = 0. So the Hamiltonian operator of this system is: H = -hbar^2/(2m) * d^2/dx^2 \hat{H} =...

Homework Statement Hello, the problem is asking me to find a unitary matrix U such that (U bar)^T(H)(U) is diagonal. And we have H = [{7,2,0},{2,4,-2},{0,-2,5}] The Attempt at a Solution I don't know where to start. I tried getting the eigenvalues of the matrix A but that lead to...

Homework Statement Hello, I have the following problem: Suppose A is a hermitian matrix and it has eigenvalue \lambda <=0. Show that A is not positive definite i.e there exists vector v such that (v^T)(A)(v bar) <=0 The Attempt at a Solution Let w be an eigenvetor we have the following...