PS8_v2 - Physics 115A Problem Set 8 Due Tuesday November 17...

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Physics 115A, Problem Set 8 Due Tuesday, November 17 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths Section 3.1-3.4. 1 Hilbert Space 1. Show that the set of all square-integrable functions (that is, the set of functions f ( x ) for which R b a | f ( x ) | 2 dx is finite on a specified interval x [ a, b ]) is a vector space. 2. Show that h f | g i = R b a f ( x ) * g ( x ) dx satisfies the conditions for an inner product. 2 Rigged Hilbert Space In lecture, we briefly discussed the sense in which physical wavefunctions live in a Hilbert space, but in quantum mechanics we often have to consider a slightly larger vector space that contains the Hilbert space as well as non-normalizable functions such as plane waves, or distributions such as delta functions. We call this extended space a rigged Hilbert space (think sailing, i.e. “a fully-rigged ship”). 1. Consider a Hilbert space H that consists of all functions ψ ( x ) such that Z -∞ | ψ ( x ) | 2 dx is finite. Show that there are functions in H for which ˆ ( x ) ( x ) is not in H .
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