By now, we know the infinite square well pretty well. So let’s perturb it.

Consider an infinite square well of length $L$ defined between $0<x<L$.

  1. Write down the Hamiltonian ($H_0$) for this system.
  2. Determine the energy eigenstates and eigenvalues for this system.

Now, let’s perturb the system by adding a linear ramp: $V(x) = \beta x$ from $0<x<L$.

  1. Write down the Hamiltonian ($H$) for this system.
  2. Can you write the Hamiltonian as $H=H_0+H’$? If so, what is $H’$ (the perturbing Hamiltonian).
  3. Find the first order correction to the eigenenergies. Comment on the role of $\beta$ in your results.
  4. Setup the calculation to find the correction to the ground state wavefunction ($\ket{1} \doteq \varphi_1(x)$).
  5. Is there a term that you suspect will contribute the most to your ground state correction?