Lecture36

Lecture36 - Physics 344 Foundations of 21st Century...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TA: Dan Hartzler Office: PHYS 7 dhartzle@purdue.edu Grader: Fan Chen Office: PHYS 222 chen926@purdue.edu Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 – 4:00 in PHYS 154 Reading: Chapters 7, 10 and 11 in the text. Exam 2: Tomorrow: Thursday, December 1 at 8:00pm in ARMS B061
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Recitation Recap and Review The models of specific physical systems we’ve constructed to demonstrate how the fundamental principles of quantum mechanics work have all focused on finding the energy eigenvalue-eigenvector pairs solving a time-independent Schrödinger equation (SE) of the form 2 2 2 ( ) ( ) ( ) 2 n n n n r d E x V x x m dx ψ = - + We examined two models of an electron interacting with a massive metal film, one more idealized than the other because it neglected the tiny, but striking, new quantum mechanical effect of tunneling. We also examined the harmonic oscillator model of the oscillation of a diatomic molecule along a fixed direction. The potential energy function V(x) in the SE for electron-film system and in the SE for the harmonic oscillator system are different, so, the specific eigenvalues, E n , for these model systems are different and their corresponding eigenvectors, ψ n (x) , have different functional forms. We can, however, use the general form of the SE above to emphasize features that are common to all models of effectively one-particle systems like the ones we’ve analyzed in some detail. We want to do this becaue these features are reflections of the underlying fundamental principles of quantum mechanics and because such systems are a building block for models more complicated systems.
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2 2 2 ˆ ( ) ( ) ( ) ( ) 2 n n n n n r d E x V x x H x m dx ψ = - + The discussion of the energy (Hamiltonian) operator of a system last lecture and during recitation yesterday was framed in this general way. Each system of the kind we are considering will have its own energy operator depending upon the interactions that determine the systems potential energy function V(r) . As we’ve observed on a number of occasions, one of the remarkable features of the energy eigenfunctions of each and every system of this kind is their mutual orthogonality. For each system, no matter the specific functional forms of its energy eigenfunctions are and no matter what their corresponding eigenvalues are, the energy eigenfunctions are orthogonal * 0 ( ) ( ) a m n mn x x dx δ = The above assumes that they have been normalized in accordance with the State Vector Rule.
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The fact that the energy operator of each and every system of the kind we are considering is Hermitian, ( 29 * * ˆ ˆ ( ) ( ) ( ) ( ) H x x dx x H x dx φ ψ -∞
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Lecture36 - Physics 344 Foundations of 21st Century...

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