Physics 1 Problem Solutions 298

Physics 1 Problem Solutions 298 - 294 Chapt. 39 Photons and...

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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 i 2 where |ai | = |ai |2 = 1 , a∗ ai . i c) The time-independent Schr¨dinger equation for our system can be written o Hφi = Ei φi , where H is called the Hamiltonian operator or just HAMILTONIAN. For one dimensional systems −2 ∂ 2 h H=− + V (x) . 2m ∂x2 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 Emean = 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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This note was uploaded on 11/16/2011 for the course PHY 2053 taught by Professor Buchler during the Fall '06 term at University of Florida.

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