What is Hermitian: Definition and 349 Discussions

Numerous things are named after the French mathematician Charles Hermite (1822–1901):

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  1. L

    A Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?

    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.
  2. L

    Show that an operator is Hermitian

    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...
  3. Euge

    POTW Prove Hermitian Matrices Satisfy ##H^2 = H^\dagger H##

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

    Is an operator (integral) Hermitian?

    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##...
  5. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

    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...
  6. ergospherical

    I Hermitian Inverse: Exploring Eigenvalues of ##H^{-1}##

    ##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...
  7. A

    A Finding the Hermitian generator of a Symplectic transformation

    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...
  8. S

    I A strange definition for Hermitian operator

    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...
  9. H

    B Is the Momentum Operator Hermitian? A Proof

    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?
  10. A

    I General worked out solution for diagonalizing a 4x4 Hermitian matrix

    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!
  11. H

    A How can we measure these Hermitian operators?

    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...
  12. LCSphysicist

    I Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

    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...
  13. pellman

    I Anti-unitary operators and the Hermitian conjugate

    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...
  14. L

    Understanding Commutativity and Eigenvalues in the Product of Hermitian Matrices

    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...
  15. S

    A Converting between field operators and harmonic oscillators

    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.*...
  16. dykuma

    Hermitian Matrix and Commutation relations

    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...
  17. JD_PM

    Show that the Hamiltonian operator is Hermitian

    $$<f|\hat H g> = \int_{-\infty}^{\infty} f^*\Big(-\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x) \Big) g dx$$ 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} -...
  18. M

    A Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix

    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...
  19. M

    Show that the Hamiltonian is Hermitian for a particle in 1D

    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...
  20. Haynes Kwon

    Hermiticity of AB where A and B are Hermitian operator?

    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...
  21. H

    I Is the product of two hermitian matrices always hermitian?

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

    I Hermitian operators in QM and QFT

    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...
  23. J

    I Pauli exclusion principle and Hermitian operators

    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...
  24. J

    I The Multiplication Table is a Hermitian Matrix

    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...
  25. C

    Hermitian operators in quantum gravity

    Are there new hermitian operators in quantum gravity? Background: In many worlds interpretation (MWI). We have the preferred basis problem and the basis are for example position, momentum, spin. Each of those bases come from a hermitian operator: they are the eigenbasis of the (for example)...
  26. S

    If A and B are Hermitian operators is (i A + B ) Hermitian?

    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 -...
  27. learn.steadfast

    I Hermitian and expectation values.... imaginary?

    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...
  28. S

    Prove that the exchange operator is Hermitian

    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) |...
  29. Jd_duarte

    I Hermitian Operator Proof - Question

    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 > ?
  30. Luck0

    A Diagonalizing Hermitian matrices with adjoint representation

    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...
  31. Gene Naden

    A Hermitian conjugate of the derivative of a wave function

    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...
  32. J

    Skew-Hermitian or Hermitian Matrix?

    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?
  33. SemM

    A Hilbert-adjoint operator vs self-adjoint operator

    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...
  34. S

    A How does a pseudo-Hermitian model differ from a Hermitian?

    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...
  35. S

    I Can a Hermitian matrix have complex eigenvalues?

    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...
  36. W

    I Hermitian Operators: Referencing Griffiths

    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...
  37. Dyatlov

    Hermitian conjugation identity

    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...
  38. binbagsss

    Quantum mechanics Hermitian operator

    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 ##...
  39. B

    Diagonalization of Gigantic Dense Hermitian Matrices

    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...
  40. Vitani11

    Show that if H is a hermitian operator, U is unitary

    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†...
  41. T

    I Hermitian operators, matrices and basis

    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...
  42. D

    I Proof that parity operator is Hermitian in 3-D

    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...
  43. Z

    Exponential of hermitian matrix

    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...
  44. N

    I Symmetric, self-adjoint operators and the spectral theorem

    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...
  45. A

    Find the Hermitian conjugates: ##x##, ##i##,##\frac{d}{dx}##

    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...
  46. J

    I Hermitian Operators in QM

    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...
  47. S

    A Hermitian properties of the gamma matrices

    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...
  48. H

    I Hermitian Operators in Dirac Equation

    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 =...
  49. I

    I From Non Hermitian to Hermitian Matrix

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

    Understanding Hermitian Operators: Exploring Their Properties and Applications

    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...
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