Consider a particle in a one dimensional box (infinite potential well). we can work out its wave function and energy levels without solving the schrödinger equation, by regarding its motion as standing waves subtended in the box of length l. take x 0 to be at the center of the box, so that the box extends from x =-l/2 to x = l/2. a. (6 points) write down an expression for the wavelength ? n of the nth harmonics standing waves, with n1 being the fundamental mode and n - 2 the first harmonic, etc. b. (6 points) write down the form of the wave function ? (x) for the nth state, setting the amplitudes to be a. also sketch y(x) for n = 1 and for n = 2. c. (8 points) from the normalization condition calculate the amplitude a d. ( 10 points) calculate the probability of finding the particle in its ground state (fundamental mode) in the x position range between -l/4 and l/4. sin2 x =-[1-cos(2x)] cos2 x = [1 + cos(2x)] possibly useful identities: sin(2x) 2 sinx cos x cos(2x) = cos 2 x-sin2 x 2