3. Particle in a 2D Box. A quantum mechanical particle is confined in side a square 2D box, with side length L. Inside the box V=0 and outside the box V=infinity. Let the wave function to be (x,y). (a) write down the Schrodinger equation of (x,y). (b) Use the separation of variable method solve (x,y) (let the quantum numbers to be nx and ny.) (c) What is the energy for the state (nx, ny)? (d) What is the probability density p(x,y) for the state nx=3 and ny=3? Sketch this p(x,y) in a square.

3. Consider a particle in an infinite square well potential trapped between 0 < x < a. What is the energy spectrum of the particle, and specify the allowed quantum numbers. What are the normalized energy eigenstates for this particle (i.e. the solutions to the TISE)? HINT: You should have memorized this. You may derive it, but it is time consuming! The two wave functions are not solutions to the Schrodinger Equation. For each case, explain why. (i) yn (x, t) = e-ist cos (ii) ý, (x, t) = cos(wt) sin

3. An electron is trapped in an infinitely deep potential well 0.300 nm in width. a. If the electron is in its ground state, what is the probability of finding it within 0.100 nm of the left-hand wall? b. Repeat a. for an electron in the 99th excited state. c. Are your answers consistent with the correspondence principle?

6. One electron is trapped in a one-dimensional square well potential with infinitely high sides. Assume that the well extends from x = 0 to x = L. The width of the well is L= 306.6pm. Find the energies (E,,E,,E,,E,) of the four lowest levels in electron volts (eV). а. b. Sketch the probability density functions for the two lowest energy states: P(x), P,(x). Draw an energy level diagram for the for the first four energy levels. C. d. What are the wavelengths, å,, 4, , of the two longest wavelength photons that could be absorbed by an electron in the first excited state (the n=2 state) in this trap. Assume that one-dimensional electron traps can absorb and emit photons. Indicate the transitions involved in part "d." on the energy level diagram. е.