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Hermitian

 Definition/Summary The Hermitian transpose or Hermitian conjugate (or conjugate transpose) $M^{\dagger}$ of a matrix $M$ is the complex conjugate of its transpose $M^T$. A matrix is Hermitian if it is its own Hermitian transpose: $M^{\dagger}\ =\ M$. An operator $A$ is Hermitian (or self-adjoint) if it is its own adjoint: $\langle Ax|y\rangle\ =\ \langle x|Ay\rangle$ (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 [url]http://en.wikipedia.org/wiki/Hermitian[/url].

 Equations $$\int \psi _1 ^* (\hat{O} \psi _2 ) \, dx = \int (\hat{O}\psi _1 )^* \psi _2 \, dx$$ $$\langle b |\hat{O} |c \rangle = \langle c |\hat{O} |b \rangle ^*$$

 Scientists Charles Hermite (1822-1901) (pronounced " 'ermeet ")

 Breakdown Physics > Quantum >> Foundations

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 Extended explanation A matrix $M$ is hermitian if: $$M^{\dagger} = (M^T)^* = M ,$$ where $\dagger$ 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 $$\hat{x},$$ momentum operator $$-i\hbar \dfrac{\partial}{\partial x},$$ (with discrete spectrum) z-component of angular momentum operator $$\hat{L}_z .$$ In terms of wave functions, an operator $\hat{O}$ is hermitian if: $$\int \psi _1 ^* (\hat{O} \psi _2 ) \, dx = \int (\hat{O}\psi _1 )^* \psi _2 \, dx$$ In terms of bra-ket: $$\langle b |\hat{O} |c \rangle = \langle c |\hat{O} |b \rangle$$ Now, using the wave function formalism, some valuable identities will be presented: Let us consider two hermitian operators $\hat{A}$ and $\hat{B}$. The expectation value: $$<\hat{A}> = \int \psi ^* (\hat{A} \psi ) dx = \int (\hat{A}\psi )^* \psi dx,$$ is real, proof: $$<\hat{A}>^* = \int ((\hat{A}\psi)^*\psi)^* dx = \int (\hat{A}\psi)\psi^* dx = \int \psi^* (\hat{A}\psi) dx = <\hat{A}>$$ Since $\hat{A}$ was said to be hermitian, and $\psi _1 = \psi _2$ when we do expectation values. Expectation value of $\hat{A}^2$: $$<\hat{A}^2> = \int \psi ^* (\hat{A}(\hat{A}\psi)) dx = \int \psi^* (\hat{A}\tilde{\psi})dx =$$ $$(\tilde{\psi} = \hat{A}\psi \: \text{ is a new wavefunction} )$$ $$\int (\hat{A}\psi)^*\tilde{\psi}dx = \int (\hat{A}\psi)^*(\hat{A}\psi) dx$$ Now we can show another useful result: $$\int \psi^* (\hat{A}(\hat{B}\psi))dx = \int(\psi^*(\hat{B}(\hat{A}\psi)))^*dx ,$$ prove this as an exercise. Two more useful things: $$I = \int \psi^*(\hat{A}\hat{B}+\hat{B}\hat{A})\psi = I^*$$ is real, show this as an exercise. The operators always to the right if not indicated otherwise. Thus: $$I = \int \psi^*(\hat{A}(\hat{B}\psi))dx + \int \psi^* (\hat{B}(\hat{A}\psi)) dx$$ $$J = \int \psi^*(\hat{A}\hat{B}-\hat{B}\hat{A})\psi = -J^*$$ is imaginary, show this as an exercise. These identities are needed to prove the uncertainty relations of quantum mechanics.

Commentary

 tiny-tim @ 05:16 AM Dec10-10 Fixed missing LaTeX. No other changes.

 tiny-tim @ 12:04 PM Feb13-09 Surely "Hermitian" should always be spelled with a capital "H"? It's an adjective derived from a proper name, like "Newtonian". Nouns which are units of measurement are by convention spelled with a small letter, but even deg Celcius has a capital "C" for the adjective Celsius … see http://www.bipm.org/en/si/si_brochure/chapter5/5-2.html

 malawi_glenn @ 02:34 AM Feb9-09 cool

 tiny-tim @ 03:52 PM Feb8-09 Placed definitions of Hermitian transpose Hermitian matrix and Hermitian operator in the Definition section.