(a) Sketch out the first five eigenstates of the particle in a box. Do you notice any symmetry in the center of the box? Comment. (Hint: In general, there are two different kinds of functions, based on symmetry about the center. Do you see this? Comment. (b) Asnincreases the difference between neighboring energy levels www increases. Prove this statement. Hint: Calculate(E_n+1-E_n). What happens to this expression increases? How about when does to infinity? (C) Write down the probability distribution pn(x) =Tyn(x)|². How does this change increase? Comment on the oscillatory nature of this function with increasing. (Does it get more or less oscillatory?) Sketch out pn(x)for n= 1,2,3,4. What do you think happens for largen? What does this remind you of?

(a) Sketch out the first five eigenstates of the particle in a box. Do you notice any symmetry in the center of the box? Comment. (Hint: In general, there are two different kinds of functions, based on symmetry about the center. Do you see this? Comment. (b) Asnincreases the difference between neighboring energy levels www increases. Prove this statement. Hint: Calculate(E_n+1-E_n). What happens to this expression increases? How about when does to infinity? (C) Write down the probability distribution pn(x) =Tyn(x)|². How does this change increase? Comment on the oscillatory nature of this function with increasing. (Does it get more or less oscillatory?) Sketch out pn(x)for n= 1,2,3,4. What do you think happens for largen? What does this remind you of?