What is Hermitian operator: Definition and 60 Discussions

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product


{\displaystyle \langle \cdot ,\cdot \rangle }
(equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint:




{\displaystyle \langle Av,w\rangle =\langle v,Aw\rangle }
for all vectors v and w. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator


{\displaystyle {\hat {H}}}
defined by







{\displaystyle {\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi ,}
which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.
The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail.

View More On Wikipedia.org
  1. cianfa72

    I Eigenstates of particle with 1/2 spin (qbit)

    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}##...
  2. 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...
  3. K

    Linear algebra inner products, self adjoint operator,unitary operation

    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!
  4. 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...
  5. nomadreid

    I Conjugate variables: two descriptions (Link?)

    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...
  6. 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...
  7. Johny Boy

    A Expectation Value of a Stabilizer

    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...
  8. 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...
  9. 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) |...
  10. 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 > ?
  11. Marrrrrrr

    I QM: I as an Observable & Its Eigenvectors & Eigenvalue

    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...
  12. 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 ##...
  13. 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†...
  14. binbagsss

    QM Bra & Ket Linear Algebra Hermitian operator proof -- quick question

    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##...
  15. L

    Finding Eigenvalues and Wave Function in a Basis of Orthonormalized Vectors

    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...
  16. A

    Proof for the Hermitian operator

    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...
  17. nomadreid

    Hermitian operator represented as a unitary operator

    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...
  18. arcoon

    Is (i/x^2 d/dx) a Hermitian Operator?

    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...
  19. A

    Eigenvalues and Eigenvectors of a Hermitian operator

    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...
  20. carllacan

    Conmutative Hermitian operator in degenerate perturbation theory

    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 =...
  21. D

    Expectation value of a hermitian operator prepared in an eigenstate

    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...
  22. R

    How Can Hermitian Operators Prove Key Quantum Mechanics Equations?

    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...
  23. S

    Proof that HK is hermitian operator only if HK=KH

    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...
  24. Ikaros

    Show a real, smooth function of Hermitian operator is Hermitian

    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...
  25. S

    Quick Matrix Element Question using Hermitian Operator

    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!
  26. S

    Hermitian Operator Expectation Values

    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...
  27. L

    Normalized Eigenvectors of a Hermitian operator

    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 +...
  28. A

    Hermitian operator <=> observable?

    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...
  29. P

    Understanding Hermitian Operators

    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...
  30. A

    Square of a hermitian operator in matrix form

    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...
  31. G

    Eigenvectors of a 2D hermitian operator (general form)

    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...
  32. D

    Determine whether or not is a Hermitian operator

    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...
  33. U

    Hermitian Operator in Inner Product

    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...
  34. F

    Derive RMS around average of a hermitian operator

    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...
  35. L

    Is the Operator O Hermitian if (a, Oa) Equals (Oa, a) for Any Vector a?

    Homework Statement A Hermitian operator is such that, for arbitrary vectors ai and aj in a vector space,we have (ai,Oaj) = (Oai, aj). Prove that if for an arbitrary vector a in the vector space, the operator O satisfies (a,Oa) = (Oa, a), then O is Hermitian Homework Equations...
  36. L

    Show that the eigenvalues of a hermitian operator are real.

    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 =...
  37. F

    Prove that eigenstates of hermitian operator form a complete set

    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).
  38. Shackleford

    Show eigenvalues of hermitian operator are real

    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.
  39. V

    Is Hamiltonian operator a Hermitian operator?

    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} =...
  40. T

    Definition of Hermitian Operator in Dirac Notation

    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...
  41. L

    Show \hat{O}^2 is Hermitian Given \hat{O} is Hermitian

    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...
  42. P

    Commutator and hermitian operator problem

    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...
  43. P

    Prove that any Hermitian operator is linear

    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...
  44. S

    Eigenkets of a function of a hermitian operator

    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...
  45. K

    Check If Operator Is Hermitian: Real Eigenvalue Test

    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?
  46. D

    When are the solutions for \hat{R} being Hermitian?

    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...
  47. D

    Is the Operator C Hermitian and What are the Eigenfunctions and Eigenvalues?

    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...
  48. D

    Show How to Write A as B + iC: Hermitian Operators

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

    Spectrum of Hermitian operator

    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...