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problems4 - 1 = ∠∞ ∞ βˆ’ dx x(c x x x x βˆ’ = βˆ’(d x...

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Physics 4021: Introduction to Quantum Mechanics I Homework Assignment 4 Due in class: Thursday, Oct. 4, 2007 1. Problem 2.15 2. Problem 2.19 3. Problem 2.22 4. Problem 2.41 (Hint: try to represent the wavefunction as a linear superposition of stationary states of harmonic oscillator) 5. Problem 2.42 6. Dirac delta function δ ( x ) is defined in the following way +∞ = dx x x x f x f ) ( ) ( ) ( 0 0 where f ( x ) can be any continuous square integrable function. Using this definition, prove following relations: (a) 0 ) ( = x for 0 x (b)
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Unformatted text preview: 1 ) ( = ∫ +∞ ∞ βˆ’ dx x (c) ) ( ) ( x x x x βˆ’ = βˆ’ (d) ) ( ) ( ) ( x f dx dx x x d x f β€² βˆ’ = βˆ’ ∫ +∞ ∞ βˆ’ There are many examples you can write an explicit mathematical expression for Dirac delta function. Show that the following expression satisfies the above definition of ( x ). (e) 2 2 2 / 2 1 lim ) ( a x a e a x βˆ’ > βˆ’ = Ο€ (f) ∫ +∞ ∞ βˆ’ = dk e x x ik 2 1 ) ( (Hint: use Plancherel’s theorem) 1...
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