Week of 1/21/08

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Latest revision as of 17:53, 25 January 2008

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 $z = x + I y$ , then $z * z = (x - I y)(x + I y) = x 2 + y 2 = z z *$

@@ Line 42: / Line 42: @@
 {{Mathematica|filename=Griffiths_example2.2.nb|title=Mathematica version of Griffiths analytic example 2.2}}
-{{PDF|filename=1-25-08.pdf|title=Lecture notes 1/25/08.  Review of stationary states.  Fourier superposition.  Completeness and orthogonality of eigencuntions of Hermitian operators.}}
+{{PDF|filename=1-25-08.pdf|title=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}}
+[http://eve.physics.ox.ac.uk/Personal/artur/Keble/Quanta/Applets/quantum/deepwellmain.html very nice applet illustrating the dynamics of a deep well]