Week of 1/21/08

Definition of complex (Hermitian) inner product

It's a matter of convention that the anti-linear term is the second one in the inner product:

$\langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle$

$\langle u,v+w \rangle \equiv \langle u,v \rangle + \langle u,w \rangle$

$\langle \alpha u,v \rangle \equiv \alpha \langle u,v \rangle$

$\langle u,\alpha v \rangle \equiv \alpha^* \langle u,v \rangle$

$\langle u,v \rangle \equiv \langle v,u \rangle ^*$

$\langle u,u \rangle = 0,$ with equality only if $u \equiv 0$

The basic example is the form

$h(z,w) \equiv \sum z_i w^* _i$

However, Griffiths uses a different convention with the complex conjugate on the first term: cf page 94. So, I will change mine to conform to his.

NB if z = x + Iy, then z ^* z = (x − Iy)(x + Iy) = x² + y² = zz ^*

Download Animation of the superposition of two eigenstates for a 1D string

Download Lecture notes 1/21/08. Sep. of Variables of TDSE. Eigenstates. Stationary states.

Download Lecture notes 1/23/08. Review of Hermitian matrices/operators. Particle in an infinite square well.

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