homework1 - ( x ) = exp ( | x | ) , use the variational...

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1. Consider a particle of mass m moving in 1D subject to the con f ning potential V ( x )= k | x | , where k is a positive constant. (a) Using a trial wave function of the form φ α ( x )= α exp ( α | x | ) , use the variational method to estimate the ground state energy of the system.[Hint: with this wave function (which has a discontinuity in its f rst derivative) it is useful to compute the kinetic energy using the positive de f nite form h P 2 i = ~ 2 R dx | ψ 0 | 2 obtained through an integration by parts of the usual expression.] (b) With a suitable modiciation to the wave function, f nd an upper bound for the energy of the f rst excited state. 2. Consider a particle of mass m moving in 1D subject to the harmonic potential V ( x )= 1 2 kx 2 , where k is a positive constant. Using a trial wave function of the form
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Unformatted text preview: ( x ) = exp ( | x | ) , use the variational method to estimate the ground state energy of the system.[Hint: with this wave function (which has a discontinuity in its f rst derivative) it is useful to compute the kinetic energy using the positive de f nite form h P 2 i = R dx | | 2 obtained through an integration by parts of the usual expression.) How close is your answer to the exact result? 3. A particle is con f ned by a spherically symmetric potential V ( r ) = r where is a positive constant. Estimate the ground state energy of this system....
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 463 taught by Professor Paule. during the Fall '10 term at Missouri S&T.

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