Ch3_SJP_version - Lecture notes (these are from ny earlier...

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3-1 Lecture notes (these are from ny earlier version of the course - we may follow these at a slightly different order, but they should still be relevant!) Physics 3220, Steve Pollock. Basic Principles of Quantum Mechanics The first part of Griffith's Ch 3 is in many ways a review of what we've been talking about, just stated a little formally, introducing some new notation and a few new twists. We will keep coming come back to it - First, a quick review of ordinary vectors . Think carefully about each of these - nothing here is unfamiliar (though the notation may feel a little abstract), but if you get comfortable with each idea for regular vectors, you'll find it much easier to generalize them to more abstract vectors! Vectors live in N-dimensional space. (You're used to 3-D!) We have (or can choose) basis vectors : ˆ e i (N of them in N-dim space.) (Example in an "older Phys 1110 notation" of these would be the old familiar unit vectors: ˆ i , ˆ j , ˆ k They are orthonormal : ˆ e i ˆ e j = δ ij (This is the scalar , or inner , or dot product.) They are complete : This means any vector v = v i ˆ e i i is a unique linear combo of basis vectors. The basis set spans ordinary space. This is like completeness, but backwards - every linear combination of basis vectors is again an N-Dim vector, and all such linear combos generate all possible vectors in the space. We can choose a specific representation of v , namely {v 1 ,v 2 3 , v n } , but it is not unique, it depends on the choice of basis. (e.g. polar vs. rectangular, and even which particular rotation of the rectangular axes.) Each number v i is the projection of v in the ˆ e i direction, and can be obtained by the formula v i = v ˆ e i . (This involves the same scalar product, again, as we used above in the statement of orthonormality. ) You can prove the last formula by using orthogonality and completeness. Addition, or multiplication by a scalar (number), keeps you in the same N-dim "vector space". (adding or scaling vectors gives another vector.)
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3-2 We can make very powerful analogies to all of the above in the world of square integrable functions: Note the one to one correspondence between each of the following statements about functions, with the preceding ones about vectors. Square integrable functions live in Hilbert space. (Never mind what this means for now!) We have (or can choose) basis functions : u n ( x ) ( Infinitely many of them.) (This infinity might be countable (discrete), or it might be uncountable, in which case you can't use integers as labels, but need real numbers.) We have already met some examples of both such types of u n 's, as eigenfunctions of operators. They are orthonormal : dx u n ( x ) u m ( x ) = δ nm . This is apparently our new way of writing the inner product. (If the labeling is continuous, the right side will be a Dirac delta function!) They are complete : Any function ψ ( x ) = c n u n ( x ) n is a unique linear combo of the basis vectors. (If the labeling is continuous, then ( x ) = dE c ( E ) u E ( x ) ) The basis spans Hilbert space. This is similar to
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.

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Ch3_SJP_version - Lecture notes (these are from ny earlier...

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