Problem Set 5

Problem Set 5 - mate is above the true energy. Now do the...

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PHYSICS 230 – PROBLEM SET 2 1. Consider the perturbed harmonic oscillator Hamiltonian H = p 2 / 2 m + 1 2 2 x 2 + λx 4 Calculate the ground state energy to second order in λ . Calculate the energy of an arbitrary eigenstate | n > to ﬁrst order in λ . 2. Consider the Hamiltonian deﬁned by the following matrix: 1 ± 0 ± 1 ± 0 ± 2 Find its eigenvalues to second order in ± using degenerate perturbation theory. 3. Consider the harmonic oscillator deﬁned by potential V ( x ) = 1 2 2 x 2 . Find a variational estimate of its ground state energy using the bound state wavefunction of the delta function potential as a trial wavefunc- tion. Let the width be a variational parameter. Verify that your esti-
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Unformatted text preview: mate is above the true energy. Now do the reverse: Consider the attractive delta function potential V ( x ) =- ( x ). Use the ground state wavefunction of the harmonic oscillator as a variational trial wavefunction. Let the width vary. 4. Consider the attractive square well potential: V ( x ) =-, | x | &lt; L : V ( x ) = 0 , | x | &gt; L . Use a variational argument to show this potential has a bound state for any &gt; 0 no matter how small. Argue that an extension of this argument shows that any attractive potential in one dimension has a bound state....
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