PHYS4221(Fall2010)_ProblemSet_2

# PHYS4221(Fall2010)_ProblemSet_2 - PHYS4221 Quantum...

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PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.2 (Due on September 29, 2010) 1. Show that two linear operators ˆ A and ˆ B in the Hilbert space H are equal if h u | ˆ A | u i = h u | ˆ B | u i , ∀ | u i ∈ H . 2. Given the commutator h ˆ a, ˆ a i = 1, evaluate the commuator h ˆ a n , ˆ a m i . 3. The displacement operator ˆ D is deﬁned by the equation ˆ Df ( x ) = f ( x + η ) . Show that the eigenfunctions of ˆ D are of the form φ β = exp ( βx ) g ( x ) where g ( x + η ) = g ( x ) and β is any complext number. What is the eigenvalue corresponding to φ β ? 4. (a) Find an appropriate unitary transformation which diagonalizes the Hamilto- nian ˆ H : ˆ H = ˆ p 2 2 + ˆ q 2 2 + α p ˆ q + ˆ q ˆ p ) + β ˆ q +
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## This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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