237module7

237module7 - Quantum Mechanics I - Module 7 Getting Started...

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Quantum Mechanics I - Module 7 Getting Started 1. What are the raising and lowering operators? 2. Why are the raising and lowering operators important in quantum me- chanics? Enrichment Problem Consider the harmonic oscillator (with ~ = m = ω = 1) given by: H = p 2 2 + x 2 2 . Define: a = 1 2 ( p - ix ) a = 1 2 ( p + ix ) 1. Show that [ a,a ] = 1 . 2. Show that H = a a + 1 2 . 3. Show that [ a a,H ] = 0 . 4. Show that if | n i is an eigenstate of N = a a with eigenvalue n :
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