What is Hermitian: Definition and 349 Discussions

Homework Statement Prove that the eigenvalues of a Hermitian matrix is real. http://www.proofwiki.org/wiki/Hermitian_Matrix_has_Real_Eigenvalues The website says that "By Product with Conjugate Transpose Matrix is Hermitian, v*v is Hermitian. " where v* is the conjugate transpose of v...

Description 1. Prove that operators i(d/dx) and d^2/dx^2 are Hermitian. 2. Operators A and B are defined by: A\psi(x)=\psi(x)+x B\psi(x)=d\psi/dx+2\psi/dx(x) Check if they are linear. The attempt at a solution I noted the proof of the momentum operator '-ih/dx'...

Homework Statement Prove that i d/dx and d^2/dx^2 are Hermitian operators Homework Equations I have been using page three of this document http://www.phys.spbu.ru/content/File/Library/studentlectures/schlippe/qm07-03.pdf and the formula there. The Attempt at a Solution I have...

Why all operators in QM have a Hermitian Matrices ?

Homework Statement If M is a square matrix, prove: (A, MB) = (adj(M)A, B) where (A, MB) denotes the scalar product of the matrices and adj() is the adjoint (hermitian adjoint, transpose of complex conjugate, M-dagger, whatever you want to call it!) Homework Equations adj(M)=M(transpose of...

Homework Statement Consider the set of functions {f(x)} of the real variable x de fined on the interval -\infty< x < \infty that go to zero faster than 1/x for x\rightarrow ±\infty , i.e., \lim_{n\rightarrow ±\infty} {xf(x)}=0 For unit weight function, determine which of the...

1. The problem statement, all variables and given known data Compute: ∫x2(Hn(x))2e-x2dx The boundaries of the integral are -∞ to +∞ Homework Equations By Rodrigues' formula: Hn(x) = (-1)nex2dn/dxn(e-x2) The Attempt at a Solution I proceed to plug in my expression for H into the integral...

Hi all Homework Statement Given is a Hermitian Operator H H= \begin{pmatrix} a & b \\ b & -a \end{pmatrix} where as a=rcos \phi , b=rsin \phi I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are: \mid +...

bra - ket?? Hi, maybe a stupid question, but i would like to know if, if We have a real number, but we are i a vector space, and the operator is hermitian, is |a> is equal to < a |*? i assume this, because if a is the vector (1,0) (spin up), and only real entries. im trying to make...

Hi, Homework Statement Let A and B be hermitian operators. Show that C=i[A,B] is hermitian aswell. Homework Equations - The Attempt at a Solution Well, I tried just to use the definition but I'm not sure if that's enough (my guess would be no lol)...

Homework Statement Show that the unitary time evolution time operator requires that the Hamiltonian be hermitian. And then it tells us to use the infinitesimal time evolution operator. The Attempt at a Solution U(dt)=1-\frac{iHdt}{\hbar} so now we take...

Homework Statement If <h|Qh> = <Qh|h> for all functions h, show that <f|Qg> = <Qf|g> for all f and g. f,g, and h are functions of x Q is a hermitian operator Hints: First let h=f+g, then let h=f+ig Homework Equations <Q>=<Q>* Q(f+g)= Qf+Qg The Attempt at a Solution...

Homework Statement a)For a general operator A, show that and i(A-A+) are hermitian? b) If operators A and B are hermitian, show that the operator (A+B)^n is Hermitian. Homework EquationsThe Attempt at a Solution The first part I did, (A+A+)+=(A++A)=(A+A+)...

If A and B are hermitia operators , then prove (A+B)^n is also hermitian. Justw ondering if this would suffice ? ∫ ψ^*(A+B) ∅ dt= ∫((A+B) ψ)^* ∅ dt assuming (A+B) is hermitian I can do that again ∫ ψ^*(A+B) ∅ dt= ∫((A+B) ψ)^* ∅ dt multiply them together ∫((A+B) ψ)^(2*) ∅^2...

My question is about both sides of the same coin. First, does a hermitian operator always represent a measurable quantity? Meaning, (or conversely) could you cook up an operator which was hermitian but had no physical significance? Second, are all observables always represented by a...

If H is a Hermitian operator, then its eigenvalues are real. Is the converse true?

Hi. In a question I needed to figure out whether -\frac{i\hbar}{m} \hat{p} is hermitian or not. Since the constant doesn't matter this is similar to whether i \hat{p} is hermitian or not. I thought that since \hat{p} is hermitian, then i times it would not be, since it would not...

Homework Statement I can find my eigenvalues just fine, and they're both real, as expected. My first eigenvalue is -3, which I know is correct. I have the equations 5x+(3-i)y=0, (3+i)x+2y=0 Both of the equations come from my hermitian matrix, after I substituted λ=-3. Homework...

Which is the role of hermitian and unitary operators in quantum mechanics and which operator is neither hermitian nor unitary

I am running a program that has to diagonalize large, complex Hermitian matrices (the largest they get is about 1000x1000). To diagonalize the matrix once isn't too bad, but I need to diagonalize thousands to millions of different Hermitian matrices each time I run a simulation. If I only need...

Homework Statement Show that linear combinations A-iB and A+iB are not hermitian if A and B (B≠0) are Hermitian operators Homework Equations Hermitian if: A*=A Hermitian if: < A l C l B > = < B l C l A > The Attempt at a Solution So I've seen this question everywhere but not...

I have read in different places something like the following: Hermitian operators have real eigenvalues Hermitian operators/their eigenvalues are the observables in Quantum Mechanics e.g energy I am not sure what this means physically. Let us say I have a Hermitian operator operating on a...

If we have a hermitian operator Q and we know it's matrix representation [Q], does that mean that [Q2] = [Q]2? For example, I'm pretty sure that's the case for p2 for a harmonic oscillator. We have p=ic(a+-a-) and so p2=c2(a+-a-)(-a++a-)*=c2(a+-a-)(a+-a-)=p p Which tells us that [p2]=[p]2...

Theorem: For every Hermitian operator, there exists at least one basis consisting of its orthonormal eigen vectors. It is diagonal in this basis and has its eigenvalues as its diagonal entries. The theory is apparently making an assumption that every Hermitian operator must have eigen...

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>)...

Homework Statement Let ψ(r)= c_n ϕ_n (r) + c_m ϕ_m (r) where ϕ_n(r) and ϕ_m (r) are independent functions. Show that the condition that  is Hermitian leads to ∫ψ_m (r)^* Âψ_n (r)dr = ∫Â^* ψ_m (r)^* ψ_n (r)dr Homework Equations ∫ψ(r)^*  ψ(r)dr = ∫Â^* ψ(r)^* ψ(r)dr The Attempt...

Homework Statement Find the allowed energies for a spin-3/2 particle with the given Hamiltonian: \hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z} The Attempt at a Solution The final matrix I get is: \begin{pmatrix} \frac{3}{2} & 0 &...

what is it?

Can Someone Please Explain Hermitian Conjugates To Me!? I'm working on some problems about the Hermitian of a Harmonic Oscillator - I keep coming across the Hermitian written in a form with A[dagger]A and similar things - when I've looked in textbooks and online I find it explained using...

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?

Homework Statement Calculate the eigenvectors and eigenvalues of the two-dimensional matrix representation of the Hermitean operator \hat{O} given by |v_k'>\left(O|v_k>= {{O_11,O_12},{O_21,O_22}} where all Oij are real. What does Hermiticity imply for the o- diagonal elements O12...

I tried to prove that a hermitian matrix remains hermitian under a unitary similarity transformation.I just could do it to he point shown below.Any ideas? [ ( U A U ^ {\dagger}) B ] ^ {\dagger} = B ^ {\dagger} (U A U ^ {\dagger}) ^ {\dagger} = B (U A^ {\dagger} U ^ {\dagger}) thanks

hi, i don't have the expression, but my problem is this: in the article of Velo-Zwanzinger appears a step... passing from a equation to other which they call the hermitian form. i going to explain it... this form contains the original form...but appear an extra term..i suppose that it's the...

Homework Statement Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...Homework Equations U=exp(iC) C=C* U*U=I U=A+iB exp(M) = sum over n...

Homework Statement This is something I've been trying to prove for a bit today. My quantum mechanics book claims that the following two definitions about hermitian operators are completely equivalent my operator here is Q (with a hat) and we have functions f,g \langle f \mid \hat Q f...

Sorry if this question has been asked a million times. Either way, I'm working my way through Griffiths. It's a fantastic book--he doesn't try to slip anything past the reader. He is completely honest, and he doesn't abuse mathematics the way most authors do (screwing around with the Dirac...

Its true that one can say a unitary matrix takes the form U=e^{iH} where H is a Hermitian operator. Thats great, and it makes sense, but how can you compute the matrix form of H if you know the form of the unitary matrix U. For example, suppose you wanted to find H given that the...

Homework Statement The operator F is defined by Fψ(x)=ψ(x+a) + ψ(x-a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator. Homework Equations ∫(x+a)d/dx + (x-a)d/dxψ The Attempt at a Solution f = (1=ax) + (1-ax)ψ What are the steps I need...

Homework Statement The hermitian conjugate of an operator, \hat{F}, written \hat{F}^{\tau} satisfies the condition: ∫∅*(r)\hat{F}^{\tau}ψ(r)dr=(∫ψ*(r)\hat{F}∅(r)dr)* for any choice of wavefunctions ψ and ∅. Show that: (\hat{F}+i\hat{G})^{\tau}=\hat{F}^{\tau} -i\hat{G}^{\tau} (10 marks)...

Homework Statement Given the matrix H= \begin{array}{cc} 4 & 2+2i & 1-i \\ 2-2i & 6 & -2i \\ 1+i & 2i & 3 \\ \end{array} Find a unitary matrix U such that U*HU is diagonal (U* is the conjugate transpose of U, and U* = U-1) The Attempt at a Solution I find the eigenvalues λ1 = 9 λ2 =...

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.)...

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...

Hi guys, I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars) However, then I was asked...

How can I formally demonstrate this relations with hermitian operators?(A^{\dagger})^{\dagger}=A (AB)^{\dagger}=B^{\dagger}A^{\dagger} \langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^* If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian I've tried to prove them...

Hello everybody, long time reader, first time poster. I've searched the forums extensively (and what seems like 60% of the entire internet) for anything relevant and haven't found anything, please point me in the right direction if you've seen this before! Homework Statement Show that even...

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