Physics 1 Problem Solutions 298

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

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294 Chapt. 39 Photons and Matter Waves b) Given that integraldisplay ψ ψdv = 1 (i.e., ψ is normalized), show that summationdisplay i | a i | 2 = 1 , where | a i | 2 = a i a i . c) The time-independent Schr¨odinger equation for our system can be written i = E i φ i , where H is called the Hamiltonian operator or just HAMILTONIAN . For one dimensional systems H = h 2 2 m 2 ∂x 2 + V ( x ) . The Hamiltonian acting on the eigenfunction φ i changes it into itself times E i . Note E i is just a CONSTANT NUMBER and can be taken out of any integral. Now it turns out that the mean energy
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