Homework 13 focuses on time independent perturbation theory and explores two examples we have explored in detail: The Infinite Square Well and the Quantum Harmonic Oscillator.
For this homework, you are welcome to turn it in by midnight on Sunday Apr. 25th
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1. Perturbing the Infinite Square Well
An inifinite square well has walls at $x=0$ and $x=L$. The potential is zero in the well and rises to infinity at the walls.
We perturb the well with a delta function potential in the middle of the well. That is, $H’ = LV_0\delta(x-L/2)$.
- Determine the first-order correction to the energy for the $n$th state of the well.
- You likely found that the energy correction is different for even and odd $n$. What physical reason would there be for why those corrections would be different?
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Find the first-order correction to the ground state wavefunction. Which state provides the largest correction?
Let’s remove the delta function perturbation and instead replace with with a rectangular barrier that is centered on the well and is of length $\epsilon L$ where $0 < \epsilon < 1$ and of height $V_0/\epsilon$.
- Sketch the setup.
- Calculate the first-order correction to the energy of the ground state.
- Compare your answers in part 5 and part 1 in the limit that $\epsilon \rightarrow 0$. Discuss the result.
2. Perturbing the QHO
Consider a particle bound in a harmonic potential with $V=\dfrac{1}{2}m\omega x^2$. We perturb the harmonic well with an anharmonic potential: $H’ = \gamma x^3$.
- Determine the first order corrections to the energy. This should require a direct calculation, use a symmetry argument instead.
- Consider the second order correction. Here we have to use the off-diagonal matrix elements of the perturbation Hamiltonian. That involves a $x^3$. Using the ket formulation is likely the easiest approach, so write down $H’$ in terms of the ladder operators.
- Calculate the second-order energy corrections to first three states.
- Find the first order corrections to the first three eigenstates.