Hermitian Definition and 347 Threads

I have been following a series of on-line lectures by Dr Physics A. He clearly describes what Hermitian operators for polarization and spin are and what they do. But when he gets to the position and momentum operators I am rather lost. They are no longer represented by square matrices. The...

The gamma matrices ##\gamma^{\mu}## are defined by $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$ --- There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis. --- Is it possible to prove the relation...

In the dirac equation we have a term which is proportional to \alpha p . In the book they say that \alpha must be an hermitian operator in order for the Hamiltonian to be hermitian. As I understand, we require this because we want (\alpha p)^\dagger = \alpha p. But (\alpha p)^\dagger =...

Is there any way that i can convert a non-hermitian matrix to a hermitian matrix ?

Basically I've seen some expressions involving Hermitian Operators that I can't seem to justify, that others on the internet throw around like axiomatic starting points. (AB+BA)+ = (AB)++(BA)+? Why does this work? Assuming A&B are hermitian, I get why we can assume A+B is hermitian, but does...

Are All symmetric matrices with real number entires Hermitian? What about the other way around-are all Hermitian matrices symmetric?

Homework Statement Not actually a homework question but is an exercise in my lecture notes. Homework Equations I'm following this which demonstrates that the momentum operator is Hermitian: The Attempt at a Solution$$KE_{mn} = (\frac{-\hbar^2}{2m}) \int\Psi_{m}^{*} \Psi_{n}^{''} dx $$ $$...

Hello everyone, There's something I am not understanding in Hermitian operators. Could anyone explain why the momentum operator: px = -iħ∂/∂x is a Hermitian operator? Knowing that Hermitian operators is equal to their adjoints (A = A†), how come the complex conjugate of px (iħ∂/∂x) = px...

Homework Statement I have the matrix form of the Hamiltonian: H = ( 1 2-i 2+i 3) If in the t=0, system is in the state a = (1 0)T, what is Ψ(x,t)? Homework Equations Eigenvalue equation The Attempt at a Solution So, I have diagonalized given matrix and got...

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 If a matrix is both Hermitian and unitary show all its eigen values are ±1 Have no idea how to solve ,Have an idea what's hermitian and unitary matrix I know eigen values of hermitian matrix are real and for a unitary matrix it on a unit circle . Thanks

Homework Statement Be V the set ##\{f \in \mathbb{R}[X]| deg\,f \leq 2 \}##. This becomes to an euclidic vector space through the inner product ##\langle f,g\rangle:=\sum_{i=-1}^1f(i)g(i)## . The same goes for ##\mathbb{R}## with the inner product ##\langle r,s\rangle :=rs\,\,\,##. a) For...

Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?

Hello everyone, I have a question. I'm not sure if it is trivial. Does anyone have ideas of finding a matrix ##A\in M_n(\mathbb{C})##, where ##A## is normal but not self-adjoint, that is, ##A^*A=AA^*## but ##A\neq A^*?##

I understand this question is rather marginal, but still think I might get some help here. I previously asked a question regarding the so-called computable Universe hypothesis which, roughly speaking, states that a universe, such as ours, may be (JUST IN PRINCIPLE) simulated on a large enough...

Homework Statement I am trying to prove the following: if ##A\in C^{m\ \text{x}\ m}## is hermitian with positive definite eigenvalues, then A is positive definite. This was a fairly easy proof. The next part wants me to prove if A is positive definite, then ##\Delta_k##=\begin{bmatrix} a_{11} &...

Homework Statement $$M=C/m(k.k'g^{\mu\nu} - k^{\nu}k'^{\mu})\epsilon ^*_{\mu}(k,\lambda)\epsilon _{\nu}(k',\lambda ')$$ Calculate $$\sum _{\lambda} |M|^2$$ Homework Equations $$\sum _{\lambda}\epsilon ^*_{\mu}\epsilon _{\nu}=-g_{\mu\nu}$$ The Attempt at a Solution Firstly, I find...

Homework Statement Hi, my task is to show that the momentum operator is hermitian. I found a link, which shows how to solve the problem: http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf But there are two steps that I don't understand: 1. Why does the wave function approach...

When people want to find a conserved current which is constructed from a Dirac spinor, they consider the Dirac equation and its "Hermitian conjugate". But the equations they consider are ## (i\gamma^\mu \partial_\mu -m)\psi=0 ## and ##\bar{\psi}(i\gamma^\mu \overleftarrow{\partial_\mu}+m)=0 ##...

We know that operators can be represented by matrices. Every operator in finite-dimensional space can be represented by a matrix in a given basis in this space. If the transpose conjugate of the matrix representation of an operator in a given basis is the same of the original matrix...

Exercise 2.24 on page 71 of Nielsen and Chuang's Quantum Computation and Quantum Information asks the reader to show that a positive operator is necessarily Hermitian. There is a hint given; namely, that you first show an arbitrary operator can be written $A=B+iC$, where $B$ and $C$ are...

Homework Statement Is there a non-degenerate 2x2 matrix that has only real eigenvalues but is not Hermitian? (Either find such a matrix, or prove that it doesn't exist) Homework EquationsThe Attempt at a Solution Here's my problem. I'm getting Contradicting results. So, I found this 2x2...

If C NOT Hermitian, show we can decompose C into $\frac{1}{2}\left( C + {C}^{\dagger} \right) +\frac{1}{2i}i\left( C- {C}^{\dagger} \right) $ I've managed to prove C = C a couple of times, EG taking Hermitian or conjugate of both sides, probably there is a bit of info I am not thinking of or...

Homework Statement Consider hermitian matrices M1, M2, M3, M4 that obey the property Mi Mj + Mj Mi = 2δij I where I is the identity matrix and i,j=1,2,3,4 a) Show that the eigenvalues of Mi=+/- 1 (Hint: Go to the eigenbasis of Mi and use the equation for i=j) b) By considering the relation Mi...

Homework Statement Consider a qubit whihc undergoes a sequence of three reversible evolutions of 3 unitary matrices A, B, and C (in that order). Suppose that no matter what the initial state |v> of the qubit is before the three evolutions, it always comes back to the sam state |v> after the...

Homework Statement I'm having some trouble with questions asking me to "show" or "prove" instead of computing an answer so I'm looking for some input if I'm actually doing what I'm supposed to or not (and for the last one I don't know where to get started really.) 1. Show that ##T^*## is...

Its usually said(like https://en.wikipedia.org/wiki/Superselectiond ) that superselection rules imply that not all Hermitian operators can be considered to be physical observables. But I don't understand how that follows. Can someone explain? Thanks

Using the Feshbach-Villars transformation, its possible to write the KG equation as two coupled equations in terms of two fields as below: ## i\partial_t \phi_1=-\frac{1}{2m} \nabla^2(\phi_1+\phi_2)+m\phi_1## ## i\partial_t \phi_2=\frac{1}{2m} \nabla^2(\phi_1+\phi_2)-m\phi_2## Then we can...

Homework Statement Find all 2x2 Matrices which are both hermitian and unitary. Homework Equations Conditions for Matrix A: A=A^† A^†A=I I = the identity matrix † = hermitian conjugateThe Attempt at a Solution 1. We see by the conditions that A^† = A and by the second condition, we see that...

Homework Statement Show that the sum of two nxn Hermitian matrices is Hermitian.Homework Equations Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †. I will denote the complex conjugate with a *. The Attempt at a...

I thought I had these, but then I get to Sturm-Liouville and my confidence wavers ...please confirm / correct / supplement: 1) An Adjoint operator is written $ A^† = (A^T )^*≡(A^* )^T $ We can identify an operator A as adjoint $ (A^† ),iff <ψ_1 |Aψ_2> = <Aψ_1 | ψ_2> $ An adjoint operator...

Asked to determine the eigenvalues and eigenvectors common to both of these matrices of \Omega=\begin{bmatrix}1 &0 &1 \\ 0& 0 &0 \\ 1& 0 & 1\end{bmatrix} and \Lambda=\begin{bmatrix}2 &1 &1 \\ 1& 0 &-1 \\ 1& -1 & 2\end{bmatrix} and then to verify under a unitary transformation that both can...

Folks, What is the idea or physical significance of simultaneous diagonalisation? I cannot think of anything other than playing a role in efficient computation algorithms? Thanks

Hi Folks, I am looking at Shankars Principles of Quantum Mechanics. For Hermitian Matrices M^1, M^2, M^3, M^4 that obey M^iM^j+M^jM^i=2 \delta^{ij}I, i,j=1...4 Show that eigenvalues of M^i are \pm1 Hint: Go to eigenbasis of M^i and use equation i=j. Not sure how to start this? Should I...

I calculate 1) ##\Omega= \begin{bmatrix} 1 & 3 &1 \\ 0 & 2 &0 \\ 0& 1 & 4 \end{bmatrix}## as not Hermtian since ##\Omega\ne\Omega^{\dagger}## where##\Omega^{\dagger}=(\Omega^T)^*## 2) ##\Omega\Omega^{T}\ne I## implies eigenvectors are not orthogonal. Is this correct?

Homework Statement Are the following matrices hermitian, anti-hermitian or neither a) x^2 b) x p = x (hbar/i) (d/dx) Homework EquationsThe Attempt at a Solution For a) I assume it is hermitian because it is just x^2 and you can just move it to get from <f|x^2 g> to <f x^2|g> but I am not...

Hello, Here's a textbook question and my solution, please check if it is correct, I'm slightly doubtful about the second part. Consider Hermitian matrices M_1, M_2, M_3,\ and\ M_4 that obey: M_i M_j+M_j M_i = 2 \delta_{ij} I \hspace{10mm} for\ i,\ j\ = 1,\ ... ,4 (1) Show that the eigenvalues...

Are all the hermitian and lorentz invariant lagrangians, invariant under the combination of CPT? If yes, how can it be proved?

Homework Statement An operator T(t+ε,t) describes the change in the wave function from to to t+ε. For ε real and small enough so that ε2 may be neglected, considering the eqtn below: (a)If T is unitary, show H is hermitian (b)if H hermitian, show T is unitary. Homework Equations $$ T(t+ε,t)=...

(First of all I never saw Hilbert spaces in a mathematical class, only used it in intro QM so far, so please don't assume I know that much when answering.) Let's consider the Hilbert space on the interval [a,b] and the operator ##\textbf{L} = \frac{d^{2}}{dx^{2}} ##. Then ##\textbf{L}## is...

Hello; I'm reading "principles of quantum mechanics" by R.Shankar. I reached a theorem talking about Hermitian operators. The theorem says: " To every Hermetian operator Ω,there exist( at least) a basis consisting of its orthonormal eigenvectors.Its diagonal in this eigenbasis and has its...

I know that (A\mp )\mp =A . Where A is an Hermitian operator How does one go about proving this through the standard integral to find Hermitian adjoint operators? I should mention, I don't want anyone to just flat out show me step by step how to do it. I'd just like a solid starting place...

Given that any Hermitian matrix M can be transformed into a unitary matrix K = U†MU, for some unitary U, where U† is the adjoint of U, what is the relationship (if any) between the eigenvectors and eigenvalues (if any) of the Hermitian matrix M and the eigenvectors and eigenvalues (if any) of...

Hi. I'm trying to study QM from Shankar on my own. Asking this here because I don't really have a teacher to help me with this: Homework Statement I'm trying to solve problem 1.8.9 -part 3 of "The Principles of Quantum Mechanics" by R Shankar. Here's the problem: Given the values of Mij (see...

Homework Statement In Sakurai's Modern Physics, the author says, "... consider an outer product acting on a ket: (1.2.32). Because of the associative axiom, we can regard this equally well as as (1.2.33), where \left<\alpha|\gamma\right> is just a number. Thus the outer product acting on a ket...

[mentor's note: This thread was forked from https://www.physicsforums.com/threads/is-the-time-derivative-hermitian.791879/ when it looked to to be raising issues beyond the original question. Refer back to that thread for any missing context] Can you elaborate on this statement? For position...

I want to know why the time-derivative acts as though it's Hermitian under conjugation. I have read elsewhere that the time-derivative isn't a true operator in the quantum mechanical sense but I don't understand why that's the case, and if that's the case I still don't understand why...

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...