Central Forces: Spring-2026
HW 03: Due W2 D3

1. Recurrence Relations For each of the problems below, suppose you have been solving a differential equation using power series methods around the indicated point and you have derived the indicated recurrence relation. Write out the first five nonzero terms in the power series expansion. If the recurrence relation allows two solutions, write out the first four nonzero terms in each such solution.

   1. In an expansion around the point \(z=1\), the recurrence relation is: \[a_{n+1}=\frac{1}{n+1}\, a_n\]

   2. In an expansion around the point \(z=0\), the recurrence relation is: \[a_{n+2}=-\frac{(5-n)(6+n)}{(n+2)(n+1)}\, a_n\]
2. ODE Power Series Solutions One

   Consider the differential equation \(y^{\prime\prime} - 2y^{\prime} + y = 0\).

   1. Use the power series method to find the first six terms in each of two independent solutions to this differential equation.
   2. Solve this differential equation using a different method and show that your answers are the same as part a.

3. Quantum Particle in a 2D Box with Time Dependence

   The eigenstates for a quantum mechanical particle inside a 2-dimensional infinite potential well located between \(x=0\) and \(L_x\) and \(y=0\) and \(L_y\) are: \[\left|{n_x,n_y}\right\rangle \doteq \sqrt{\frac{2}{L_x}}\sin{\frac{n_x\pi x}{L_x}} \sqrt{\frac{2}{L_y}}\sin{\frac{n_y\pi y}{L_y}}\]

   At \(t=0\), the system is in an initial state:

   \[\psi(x,y,0)= \sqrt{\frac{30}{L_x^5 L_y^5}}(L_x x-x^2)(L_y y-y^2)\]

   Find an exact expression for the wave function as a function of time.

   Fun Extension: Plot an approximation for the probability density at \(t=0\) and at an interesting later time.