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Unformatted text preview: 294 Chapt. 39 Photons and Matter Waves
b) Given that
ψ ∗ ψ dv = 1
(i.e., ψ is normalized), show that
2 where |ai | = |ai |2 = 1 , a∗ ai .
i c) The time-independent Schr¨dinger equation for our system can be written
Hφi = Ei φi ,
where H is called the Hamiltonian operator or just HAMILTONIAN. For one
−2 ∂ 2
+ V (x) .
The Hamiltonian acting on the eigenfunction φi changes it into itself times Ei . Note Ei is
just a CONSTANT NUMBER and can be taken out of any integral. Now it turns out
that the mean energy Emean of a system with H and wave function ψ has a value given by
Emean = ψ ∗ Hψ dv . Show that
i |ai |2 Ei also. For no good reason mean values in quantum mechanics are called expection values.
So Emean is the expectation value of the Hamiltonian.
039 qfull 03000 3 5 0 tough thinking: Einstein, Runyon
6. “God does not play dice”—Einstein. Discuss. ...
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