Homework3 - Physics 401 Homework#3 1 A square...

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Physics 401 - Homework #3 1) A square wavefuction (three points each). Suppose we have a free particle whose wavefuction at t = 0 is described by a square function: otherwise L x L L t x , 0 2 2 , 1 ) 0 , ( a) Sketch this wavefunction. b) Show that this wavefunction is properly normalized. c) The position operator in quantum mechanics is x x ˆ . (In other words, the position operator is an instruction to multiply by (x).) Calculate the expectation value for position for this wavefunction. d) The momentum operator is dx d i p ˆ . Calculate the expectation value for momentum for this wavefunction. e) Suppose we change the wavefunction to this: otherwise L x L x ik L t x , 0 2 2 ), exp( 1 ) 0 , ( 0 where k 0 is a constant. What is the expectation value of position now? f) What is the expectation value of momentum now? 2) A Gaussian wavefunction (three points each). The Hamiltonian operator for a particle of mass (m) on a spring (a simple harmonic oscillator) is 2 2 2 2 2 1 2 ˆ Kx dx d
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