1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is a Hermitian

  1. Jul 24, 2014 #1
    Definition/Summary

    The Hermitian transpose or Hermitian conjugate (or conjugate transpose) [itex]M^{\dagger}[/itex] of a matrix [itex]M[/itex] is the complex conjugate of its transpose [itex]M^T[/itex].

    A matrix is Hermitian if it is its own Hermitian transpose: [itex]M^{\dagger}\ =\ M[/itex].

    An operator [itex]A[/itex] is Hermitian (or self-adjoint) if it is its own adjoint: [itex]\langle Ax|y\rangle\ =\ \langle x|Ay\rangle[/itex] (in the finite-dimensional case, that means that its matrix is Hermitian).

    In quantum theory, an observable must be represented by a Hermitian operator (on a Hilbert space).

    For other uses of the adjective "Hermitian", see http://en.wikipedia.org/wiki/Hermitian.

    Equations

    [tex] \int \psi _1 ^* (\hat{O} \psi _2 ) \, dx = \int (\hat{O}\psi _1 )^* \psi _2 \, dx [/tex]

    [tex] \langle b |\hat{O} |c \rangle = \langle c |\hat{O} |b \rangle ^*[/tex]

    Extended explanation

    A matrix [itex] M [/itex] is hermitian if:
    [tex] M^{\dagger} = (M^T)^* = M , [/tex]
    where [itex] \dagger [/itex] is called the hermitian conjugate, and is thus a combination of matrix transpose and complex conjugation of each entry in the matrix.

    In quantum mechanics, observable quantities are assigned by hermitian operators. Examples of those are:

    (with continuous spectrum)
    position operator
    [tex] \hat{x}, [/tex]

    momentum operator
    [tex] -i\hbar \dfrac{\partial}{\partial x}, [/tex]

    (with discrete spectrum)
    z-component of angular momentum operator
    [tex] \hat{L}_z . [/tex]

    In terms of wave functions, an operator [itex] \hat{O} [/itex] is hermitian if:
    [tex] \int \psi _1 ^* (\hat{O} \psi _2 ) \, dx = \int (\hat{O}\psi _1 )^* \psi _2 \, dx [/tex]

    In terms of bra-ket:
    [tex] \langle b |\hat{O} |c \rangle = \langle c |\hat{O} |b \rangle [/tex]

    Now, using the wave function formalism, some valuable identities will be presented:

    Let us consider two hermitian operators [itex] \hat{A} [/itex] and [itex] \hat{B} [/itex].

    The expectation value:
    [tex] <\hat{A}> = \int \psi ^* (\hat{A} \psi ) dx = \int (\hat{A}\psi )^* \psi dx,[/tex]
    is real, proof:
    [tex] <\hat{A}>^* = \int ((\hat{A}\psi)^*\psi)^* dx = \int (\hat{A}\psi)\psi^* dx = \int \psi^* (\hat{A}\psi) dx = <\hat{A}> [/tex]
    Since [itex] \hat{A} [/itex] was said to be hermitian, and [itex] \psi _1 = \psi _2 [/itex] when we do expectation values.

    Expectation value of [itex] \hat{A}^2 [/itex]:
    [tex] <\hat{A}^2> = \int \psi ^* (\hat{A}(\hat{A}\psi)) dx = \int \psi^* (\hat{A}\tilde{\psi})dx = [/tex]
    [tex] (\tilde{\psi} = \hat{A}\psi \: \text{ is a new wavefunction} )[/tex]
    [tex] \int (\hat{A}\psi)^*\tilde{\psi}dx = \int (\hat{A}\psi)^*(\hat{A}\psi) dx [/tex]

    Now we can show another useful result:
    [tex] \int \psi^* (\hat{A}(\hat{B}\psi))dx = \int(\psi^*(\hat{B}(\hat{A}\psi)))^*dx ,[/tex]
    prove this as an exercise.

    Two more useful things:
    [tex] I = \int \psi^*(\hat{A}\hat{B}+\hat{B}\hat{A})\psi = I^* [/tex]
    is real, show this as an exercise.

    The operators always to the right if not indicated otherwise. Thus:
    [tex] I = \int \psi^*(\hat{A}(\hat{B}\psi))dx + \int \psi^* (\hat{B}(\hat{A}\psi)) dx [/tex]

    [tex] J = \int \psi^*(\hat{A}\hat{B}-\hat{B}\hat{A})\psi = -J^* [/tex]
    is imaginary, show this as an exercise.

    These identities are needed to prove the uncertainty relations of quantum mechanics.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: What is a Hermitian
  1. What if ? (Replies: 31)

  2. What if (Replies: 2)

  3. What is . (Replies: 3)

  4. What is this (Replies: 2)

  5. What is this? (Replies: 6)

Loading...