Hermitian operator Definition and 60 Threads

QM uses separable Hilbert spaces as model to represent quantum system's states. Take for instance a 1/2-spin particle: its quantum pure state is represented by a ray in the abstract Hilbert space ##\mathcal H_2## of dimension 2. Take an observable represented by an Hermitian unitary operator...

A very basic doubt about a QM system (particle) with spin 1/2 (qbit). From the Bloch sphere representation we know that a qbit's pure state is represented by a point on the surface of the sphere. Picking a base, for instance the pair of vector/states ##\ket{\uparrow}## and ##\ket{\downarrow}##...

In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given: An operator is Hermitian if and only if it has real eigenvalues. I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We...

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!

The basis he is talking about: {1,x,x²,x³,...} I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the...

If I understand correctly (a big caveat), one shows that if one can get from one function to the other via a Fourier transform and multiplication by a constant, then the width of the corresponding Gaussian wave of one gets larger as that of the other gets smaller, and vice-versa, and by a bit...

I need help with part d of this problem. I believe I completed the rest correctly, but am including them for context (a)Show that the hermitian conjugate of the hermitian conjugate of any operator ##\hat A## is itself, i.e. ##(\hat A^\dagger)^\dagger## (b)Consider an arbitrary operator ##\hat...

Given that operator ##S_M##, which consists entirely of ##Y## and ##Z## Pauli operators, is a stabilizer of some graph state ##G## i.e. the eigenvalue equation is given as ##S_MG = G## (eigenvalue ##1##). In the paper 'Graph States as a Resource for Quantum Metrology' (page 3) it states that...

Trying to prove Hermiticity of the operator AB is not guaranteed with Hermitian operators A and B and this is what I got: $$<\Psi|AB|\Phi> = <\Psi|AB\Phi> = ab<\Psi|\Phi>=<B^+A^+\Psi|\Phi>=<BA\Psi|\Phi>=b^*a^*<\Psi|\Phi>$$ but since A and B are Hermitian eigenvalues a and b are real, Therefore...

Homework Statement [/B] Let P be the exchange operator: Pψ(1,2) = ψ(2,1) How can I prove that the exchange operator is hermitian? I want to prove that <φ|Pψ> = <Pφ|ψ>Homework Equations [/B] <φ|Pψ> = <Pφ|ψ> must be true if the operator is hermitian. The Attempt at a Solution [/B] <φ(1,2) |...

Hi, I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html. Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?

So, hermitian linear operators represent observables in QM. I (a matrix whose elements are all 1) is certainly a hermitian linear operator. Does this mean that I represent a measurable property? If so, what do we call that property? Identity? Moreover, for any state-vector A, A would be an...

Homework Statement I have the criteria: ## <p'| L_{n} |p>=0 ##,for all ##n \in Z ## ##L## some operator and ## |p> ##, ## |p'> ##some different physical states I want to show that given ## L^{+}=L_{-n} ## this criteria reduces to only needing to show that: ##L_n |p>=0 ## for ##n>0 ##...

Homework Statement Show that if H is a hermitian operator, then U = eiH is unitary. Homework Equations UU† = I for a unitary matrix A†=A for hermitian operator I = identity matrix The Attempt at a Solution Here is what I have. U = eiH multiplying both by U† gives UU† = eiHU† then replacing U†...

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...

Homework Statement Eigenvalues of the Hamiltonian with corresponding energies are: Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β Iv2>=(I1>-I3>) /21/2 E2=α-β Iv3>= (2I2> - I1> I3>)/61/2 E3=α-β Write the matrix of the Hamiltonian in the basis of...

1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...

Homework Statement I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary...

Homework Statement Hi, I'm doing a Quantum chemistry and one of my question is to determine if is hermitian or not? I am learning and new to this subject... Cant figure out how to do this question at all. Please helppp! ^Q= i/x^2 d/dx is hermitian or not? Homework Equations The Attempt at a...

Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...

Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...

Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...

Prove the equation A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle where A is a Hermitian operator and \left\langle\psi |\psi\bot\right\rangle = 0 \left\langle A\right\rangle = The expectation value of A. \Delta A...

Let H and K be hermitian operators on vector space U. Show that operator HK is hermitian if and only if HK=KH. I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here...

Homework Statement If B is Hermitian, show that BN and the real, smooth function f(B) is as well. Homework Equations The operator B is Hermitian if \int { { f }^{ * }(x)Bg(x)dx= } { \left[ \int { { g }^{ * }(x)Bf(x) } \right] }^{ * } The Attempt at a Solution Below is my...

Hi there, This should be very simple... If I have a state <1|AB|2> where A and B are Hermitian operators, can I rewrite this as <2|BA|1> ? That would be, taking the complex conjugate of the matrix element and saying that A*=A and B*=B. Thank you!

Hey, I have the following question on Hermitian operators Initially I thought this expectation value would have to be zero as the eigenvectors are mutually orthogonal due to Hermitian Operator and so provided the eigenvectors are distinct then the expectation would be zero... Though...

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 +...

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...

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...

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...

what is it?

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...

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 \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...

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 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 =...

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).

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.

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 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...

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...

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...

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...

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?

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...

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

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...