Particle in a box

Welcome to another edition of SOLVE THAT ACTIVPHYSICS PROBLEM. Our contestant today must answer 10 questions related to the particle in a box simulation at:
http://wps.aw.com/aw_young_physics_11/13/3510/898597.cw/index.html

Lets get this contest started.
Questions in order:
Question 1: Standing Waves
From your study of mechanical waves, what is the longest wavelength standing wave on a string of length L?

The longest wavelength for a standing wave is twice the length of the string, or twice the distance between the two hard boundaries.

Question 2: The de Broglie Relation
What is the momentum of the longest wavelength standing wave in a box of length L?

 de Broglie hypothesized that the momentum of a particle must be equal to plank's constant divided by the wavelength, which in this case is 2L.

Question 3: Ground State Energy
Assuming the particle is not traveling at relativistic speeds, determine an expression for the ground state energy. 

Ground state energy here refers to the kinetic energy stored in the particle, if we are operating at non-relativistic speeds then:
p=mv
K=.5mv^2
p^2 = (mv)^2
p^2/(2m) = K

then, applying p = h/2L
K = (h/2L)^2/(2m)
K = h^2/(8mL)
Question 4: Increasing L
If the size of the box is increased, will the ground state energy increase or decrease?
It will decrease, as the relationship places L as the divisor of the energy equation.


Question 5: The Correspondence Principle: Large Size
In the limit of a very large box, what will happen to the ground state energy and the spacing between allowed energy levels? Can this result explain why quantum effects are not noticable in everyday, macroscopic situations?

As L approaches the scale of human objects, the ground state energy takes on a magnitude of 10^-37 Joules, which is negligible on macroscopic scale.

Question 6: The Correspondence Principle: Large Mass
In the limit of a very massive particle, what will happen to the ground state energy and the spacing between allowed energy levels?

Massive particles also act as divisors in the energy equation, thus large particles have low levels of ground stage energy.


Question 7: Ground State Probability
If a measurement is made of the particle's position while in the ground state, at what position is it most likely to be detected?

The particle is most likely to be found in the middle of the trap.

Question 8: Probability: Dependence on Mass and Size
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation depend on either the mass of the particle or the size of the box?

No, the wave function of the particle is independent of mass, and always has a maximum at L/2.


Question 9: Probability: Dependence on Energy Level
The most likely position to detect the particle, when it is in the ground state, is in the center of the box. Does this observation hold true at higher energy levels?

No, for instance the first excited state is more likely to be found at L/4 and 3L/4.

Question 10: The Correspondence Principle: Large n
In the limit of large n, what will happen to the spacing between regions of high and low probability of detection? Does this agree with what is observed in everyday, macroscopic situations?

As n approaches large values, the probability density function becomes almost continuously flat, or equally likely in every location. This is consistent with a free particle, which has no definite location.