Discussion9 - H = =<T> +...

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Physics 486 Fall 2007 Discussion Problems 9 1). Consider an electron in a gravitational potential, V = mgz z ≥ 0 = z < 0 as you did on the mid-term. While it is difficult to obtain the exact form of the wavefunctions, this is a place where variational methods can be used. (a). Sketch the potential and the eigenstates of the electron for the three lowest energy eigenvalues. (b). Calculate the expectation value of the kinetic energy with respect to the trial wavefunction ψ tr (z) = zexp(-az) {Hint: / 1 0 ! n x a n x e dx n a - + = } (c). Calculate the expectation value of the potential energy with respect to ψ tr . (d). Obtain a "best" upper bound on the ground state energy using ψ tr . ( ) E a
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Unformatted text preview: H = =&amp;lt;T&amp;gt; + &amp;lt;V&amp;gt; (from above): Physics 486 Fall 2007 2). Consider a particle in an isotropic 3D harmonic oscillator potential, V(x,y,z) = m 2 x 2 + m 2 y 2 + m 2 z 2 . (a). Write the energy eigenvalues for a particle confined to this potential in terms of the angular frequency and the quantum numbers n x , n y , and n z . (b). Write, in the table below: (i) the total energy, (ii) the states (n x ,n y ,n z ) associated with each total energy, and (iii) the degeneracy g n (where n=n x +n y +n z ) associated with each total energy, for the lowest 4 energies of the 3D harmonic oscillator. n=n x +n y +n z 0 1 2 3 E n (n x , n y , n z ) g n...
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This note was uploaded on 09/20/2010 for the course PHYS 486 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Discussion9 - H = =&amp;amp;amp;lt;T&amp;amp;amp;gt; +...

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