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.

Download evolution of a gaussian particle in a box

Download Mathematica version of Griffiths analytic example 2.2

Download Lecture notes 1/25/08. Review of integration by parts. Stationary states. Fourier superposition. Completeness and orthogonality of eigencuntions of Hermitian operators. Example 2.2 in book. Quantum harmonic oscillator

very nice applet illustrating the dynamics of a deep well