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Unformatted text preview: 1 ADDITIONAL COMMENTS ON HERMITIAN OPERATORS The remarks below are not intended to be complete. Rather, they supplement the material given in the lecture of October 5, 2010. The theory of Hermitian operators arises naturally in quantum theory. In the lectures, using functions drawn from the space of square-integrable wave functions Z ∞-∞ | ψ ( x ) | 2 dx < ∞ vanishing at infinity, we defined operators A † and A as Hermitian conjugates provided they satisfy Z ∞-∞ dx ( A † ψ ( x )) * φ ( x ) = Z ∞-∞ dxψ * ( x ) Aφ ( x ) . As examples we showed that the raising ˆ a † and lowering ˆ a operators of the harmonic oscillator are Hermitian conjugates and that ( d/dx ) † =- d/dx . On the other hand x , p =- i ~ d/dx and p 2 =- ~ 2 d 2 /dx 2 are all Hermitian in common with the Hamiltonian operator H = p 2 / (2 m ) + V ( x ). We were able to prove two simple but important theorems about Hermitian operators which follow immediately from the definition above. If we let ψ and φ be successively (i) the same eigenfunction of A with eigenvalue λ , and (ii) eigenfunctions of A corresponding to two different eigenvalues, we find that (i) the eigenvalues of a Hermitian operator are real, and (ii) the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal.are real, and (ii) the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal....
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- Fall '10
- mechanics, Hilbert space, Hermitian, Eigenfunctions, Hermitian Operators, Hermitian Operator