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The Hamiltonian of an electron stuck within a tunnel in a dialectic cube is found to be

$$H=\frac{p^2}{2m}+\frac{1}{2}Kx^2-\frac{e\Phi_0}{a}x$$

Find the possible energies and corresponding wavefunctions of H. Write an expression for the ground state energy and normalized wavefunction.

I tried plugging this Hamiltonian into the Schrodinger Equation and completing the square, but then I get an impossible differential equation. Is this the right thing to do? If it is, how do I solve for the ground state energy and the normalized wavefunction? Any help would be welcome--I am really confused by this problem.

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1 Answer 1

up vote 7 down vote accepted

You're on the right track. Complete the square on $x$ and you'll have some newly defined Harmonic oscillator whose position operator you have found. The additional constant that comes from completing the square will add to your ground state energy.

(Once you have completed the square, you should have something of the form

$H = constant + \frac{P^2}{2M} + \frac{1}{2}M\omega^2(X-x_0)^2$

which is a Harmonic oscillator centered at $x_0$ and ground state energy

$E_0 = constant + \frac{1}{2}\hbar\omega$).

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But how do we find the possible energies and corresponding wavefunctions of H? Do we have to solve the entire differential equation? –  Bronzeclocksofbenin Sep 27 '13 at 20:29
    
You can solve the differential equation, yes. That would lead you to Hermite polynomials (see the appropriate chapter in Griffiths, for example), whose energies you can compute. Are you familiar with the creators and annihilators of a harmonic oscillator? Without solving the differential equation, you can easily define them and find the ground state energy immediately. Griffiths explains this nicely as well, as do Shankar and Sakurai. It may help you to know these operators in the future, because you can spot a HO in a Hamiltonian and immediately be able to talk about its energies. –  eqb Sep 27 '13 at 20:32
    
(or see here for a quick overview of using these operators: en.wikipedia.org/wiki/Quantum_harmonic_oscillator ) –  eqb Sep 27 '13 at 20:34
1  
I do believe you would benefit from first learning how to get the eigenvalues of the Hamiltonian in the case of a simple one-dimensional harmonic oscillator: $H=p^2/2m + \frac{1}{2}m\omega^2x^2$ –  BMS Sep 27 '13 at 22:55
    
@eqb Do I need to define some new creator/annihilator operator for this example (as opposed to the standard a_+/a_-)? –  Bronzeclocksofbenin Sep 29 '13 at 12:07

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