Particle in a Box: Solving Qs on Lowest Energy & Highest Probability

[Physicsiscool]

Can someone please explain to me how I should go about solving the following 2 questions?

#1 The lowest energy level in a particle confined to a one-dimensional region of space with fixed dimension "L" is Eo. If an identical particle is confined to a similar region with the fixed distance (1/4)L, what is the energy of the lowest energy level that the particles have in common? Express in terms of Eo.

#2 Consider a particle in a box of width "L" and let the particle be in a state n = 11. What is the first value of x, larger then 0, where the probability of finding the particle is the highest?

 

[Galileo]

If these are questions concerning the infinite square well, use the fact that the energy levels are:
$$E_n=n^2<br /> \frac{\pi^2\hbar^2}{2mL^2}$$
for n=1,2,3,...

 

[jtbell]

#2 Consider a particle in a box of width "L" and let the particle be in a state n = 11. What is the first value of x, larger then 0, where the probability of finding the particle is the highest?

Write down the probability distribution for the general case with arbitrary n. (You may need to start with the wave function and find the probability distribution from that.) Substitute n = 11. For the particle in a box, it should be pretty obvious where the maxima are, from the general form of the probability distribution. Sketching a graph of it might help.

 

[Dr Transport]

FOr the first part of the problem

$$E_{0} = \frac{\pi^2\hbar^2}{2mL^2}$$

for the smaller width box

$$E_{n} = 4n^{2}\frac{\pi^2\hbar^2}{2mL^2} = 4n^2 E_{0}$$

from here figure it out.