(Solved): 1. (15 points) Schrodinger's equation for a particle of mass \( m \) in a 2-D box. Consider a pote ...

1. (15 points) Schrodinger's equation for a particle of mass \( m \) in a 2-D box. Consider a potential \( V(x) \) such that: \[ V(x, y)=\begin{array}{cc} \infty, & x, y \geq L, x, y \leq 0 \\ 0, & 0E \) and justify them. Make a sketch of the general form of the wavefunctions for all of the regions. d) Apply the boundary conditions corresponding to each region interface and establish the associated transcendental equations that determine the constants (do not evaluate the constants). Comment on the physical meaning of the boundary conditions. 3. (10 points) Consider the wavefunctions shown in the information section below as solutions for Schrödinger's equation with a 1-D harmonic oscillator potential. The subscripts indicate the principal quantum numbers. Find the expectation value of \( x \), \( \langle x\rangle \) for a harmonic oscillator in its ground state 4. (15 points) Consider the hypothetical atom " quarkonium" constituted of one down quark \( \left(q_{D Q}=-1 / 3 \mathrm{e} ; \boldsymbol{m}_{D Q}=5.0 \mathrm{MeV} / \mathrm{c}^{2}\right) \) and one top quark \( \left(q_{T Q}=+2 / 3 \mathrm{e} ; \boldsymbol{m}_{T Q}=1.7 \mathrm{x}\right. \) \( 10{ }^{5} \mathrm{MeV} / \mathrm{c}^{2} \) ). Use an approach similar initially used by Bohr's to describe the hydrogen atom to find the quarkonium's ground state energy and the radius associated with that state. Express your result in terms of the ground state energy of the hydrogen atom and Bohr's radius respectively. Comment about your result.