Hermitian Definition and 347 Threads

Hi, the task is as follows: I now have to show the following $$\begin{align*} \langle f , \Delta g \rangle &= \langle \Delta f , g \rangle\\ \int_{V} dx^3 \overline{f(x)} \cdot \Delta g&= \int_{V} dx^3 \overline{\Delta} \overline{f(x)} \cdot g\\ \int_{V} dx^3 \overline{f(x)} \cdot...

Consider a general ket ##|\psi\rangle##, expressed in the ## |\alpha_i\rangle## basis: $$ |\psi\rangle = \sum \langle \alpha_i | \psi \rangle |\alpha_i\rangle, $$ where## \langle \alpha_i | \alpha_j \rangle = \delta_{ij} ##, with ## \delta_{ij} ## being the Kronecker delta function, and ##...

It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But I haven't seen a satisfying justification. As an example of the precision of our discourse, someone has said that ##-i\hbar\partial_t## satisfies the definition of Hermicity, but it is not an...

The eigenvectors of a hermitian matrix corresponding to unique eigenvalues are orthogonal. This is not too difficult of a statement to prove using mathematical induction. However, this case is seriously bothering me. Why is the dot product of the vectors not rightly zero? Is there something more...

Why are the eigenvectors of this hermitian matrix not checking out as orthogonal? The eigenvalues are certainly distinct. ChatGPT also is miscalculating repeatedly. I have checked my work many times and cannot find the error. Kindly assist.

Consider a generic Hermitian 2x2 matrix ##H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}## where ##a,b,c,d## are real numbers, ##I## is the identity matrix and ##\sigma_{i}## are the 2x2 Pauli Matrices. We know that the eigenvalues for ##H## is ##d\pm\sqrt{a^2+b^2+c^2}## but now suppose I have the...

Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it UAU=D but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.

Hi, unfortunately, I have problems with the following task I tried the fast way, unfortunately I have problems with it I have already proved the following properties, ##\bigl< f,xg \bigr>=\bigl< xf,g \bigr>## and ##\bigl< f, \frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g...

Show that an ##n\times n##-matrix ##H## is hermitian if and only if ##H^2 = H^\dagger H##.

Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##. Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##. However, I don't really know what to do with this expression. ##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...

##H## is an ##n\times n## Hermitian matrix with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It's claimed that ##G = \int_{0}^{\infty} e^{tH} dt## is such that ##G = H^{-1}##. I was looking at\begin{align*} G\mathbf{e}_i &= \int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n}{n!} H^n...

Consider a set of ##n## position operators and ##n## momentum operator such that $$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$ Lets now perform a linear symplectic transformation $$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$ $$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$ such that the canonical commutation...

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...

Momentum operator is ##p=-i\frac{d}{dx}## and its adjoint is ##p^\dagger=i\frac{d}{dx}##. So, ##p^\dagger=-p##. How is the momentum Hermitian?

Hello, I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!

Hi Pf, I am reading this article about generalization of Pauli matrices https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices#Generalized_Gell-Mann_matrices_%28Hermitian%29 When i receive a qubit in a given density matrix , i can measure the mean values of the Pauli matrices by...

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...

The definition of the hermitian conjugate of an anti-linear operator B in physics QM notation is \langle \phi | (B^{\dagger} | \psi \rangle ) = \langle \psi | (B | \phi \rangle ) where the operators act to the right, since for anti-linear operators ( \langle \psi |B) | \phi \rangle \neq...

Product of two Hermitian matrix ##A## and ##B## is Hermitian matrix only if matrices commute ##[A,B]=0##. If that is not a case matrix ##C=AB## could have complex eigenvalues. If A=\sum_k \lambda_k|k \rangle \langle k| B=\sum_l \lambda_l|l \rangle \langle l| AB=\sum_{k,l}\lambda_k\lambda_l|k...

Suppose we have a Hamiltonian containing a term of the form where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use where I have set ħ=1 so that This term is Hermitian because r and p both are.*...

I think I roughly see what's happening here. > First, I will assume that AB - BA = C, without the complex number. >Matrix AB equals the transpose of BA. (AB = (BA)t) >Because AB = (BA)t, or because of the cyclic property of matrix multiplication, the diagonals of AB equals the diagonals of...

$$<f|\hat H g> = \int_{-\infty}^{\infty} f^*\Big(-\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x) \Big) g dx$$ Integrating (twice) by parts and assuming the potential term is real (AKA ##V(x) = V^*(x)##) we get $$<f|\hat H g> = -\frac{\hbar}{2m} \Big( f^* \frac{dg}{dx}|_{-\infty}^{\infty} -...

Motivation: Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...

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...

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...

Why is ##p^4## not hermitian for hydrogen states with ##l=0## when ##p^2## is? Doesn't this contradict the following theorem?

I have always learned that a Hermitian operator in non-relativistic QM can be treated as an "experimental apparatus" ie unitary transformation, measurement, etc. However this makes less sense to me in QFT. A second-quantised EM field for instance, has field operators associated with each...

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html "Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. " "Postulate 6. The total wavefunction must be antisymmetric with respect to the interchange of all...

I was drawing out the multiplication table in "matrix" form (a 12 by 12 matrix) for a friend trying to pass the GED (yes, sad, I know) and noticed for the first time that the entries on the diagonal are real, i.e. the squares (1, 4, 9, 16, ...), and the off diagonal elements are real and complex...

If A and B are Hermitian operators is (i A + B ) a Hermitian operator? (Hint: use the definition of hermiticity used in the vector space where the elements are quadratic integrable functions) I know an operator is Hermitian if: - the eigenvalues are real - the eigenfunction is orthonormal -...

I've been studying quantum mechanics, and working problems to get a feel for expectation values and what causes them to be real. I was working the problem of finite 1D wells, when I came across a situation I did not understand. A stationary state solution is made up of a forward and reverse...

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 > ?

Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...

I am continuing to work through Lessons on Particle Physics. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am on page 22, equation (1.5.58). The authors are deriving the Hermitian conjugate of the Dirac equation (in order to construct the current). I am able to...

Homework Statement Homework Equations For Hermition: A = transpose of conjugate of A For Skew Hermition A = minus of transpose of conjugate of AThe Attempt at a Solution I think this answer is C. As Tranpose of conjugate of matrix is this matrix. Book answer is D. Am I wrong or is book wrong?

Hi, while reading a comment by Dr Du, I looked up the definition of Hilbert adjoint operator, and it appears as the same as Hermitian operator: https://en.wikipedia.org/wiki/Hermitian_adjoint This is ok, as it implies that ##T^{*}T=TT^{*}##, however, it appears that self-adjointness is...

Hi, I have not been able to learn how a pseudo-Hermitian differs from a Hermitian model. If one has a hermitian model that satisfies all the fundamental prescriptions of quantum mechanics, a non-Hermitian would not, as it yields averages with complex values. How does a pseudo-Hermitian differ...

Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...

I have a few issues with understanding a section of Griffiths QM regarding Hermitian Operators and would greatly appreciate some help. It was first stated that, ##\langle Q \rangle = \int \Psi ^* \hat{Q} \Psi dx = \langle \Psi | \hat{Q} \Psi \rangle## and because expectation values are real...

Homework Statement ##(\hat A \times \hat B)^*=-\hat B^* \times \hat A^*## Note that ##*## signifies the dagger symbol. Homework Equations ##(\hat A \times \hat B)=-(\hat B \times \hat A)+ \epsilon_{ijk} [a_j,b_k]## The Attempt at a Solution Using as example ##R## and ##P## operators: ##(\hat...

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 ##...

Hi there, This is a question about numerical analysis used particularly in the computational condensed matter or anywhere where one needs to DIAGONALIZE GIGANTIC DENSE HERMITIAN MATRICES. In order to diagonalize dense Hermitian matrices size of 25k-by-25k and more (e.g. 1e6-by-1e6) it is not...

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†...

Hello, I would just like some help clearing up some pretty basic things about hermitian operators and matricies. I am aware that operators can be represented by matricies. And I think I am right in saying that depending on the basis used the matrices will look different, but all our valid...

Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...

Homework Statement Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential. a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are...

Hi Guys, at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...

Just doing some studying before my final exam later today. I think I've got this question right but wanted to make sure since the problem is from the international edition of my textbook, so I can't find the solutions for that edition online. Homework Statement The Hermitian conjugate (or...