Quantum Mechanics: Particle in a Box

Question 1: Standing Waves
From your study of mechanical waves, what is the longest wavelength standing wave on a string of length L?


wavelength = 2L


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


p = h/2L
L = ph2


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


E= P^2/m         E= h^2/8mL .


Question 4: Increasing L
If the size of the box is increased, will the ground state energy increase or decrease?


if the size of the box was increased the ground state energy  would decrease 


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?


if the size of the box was small say on a scale where it can fit electrons in, then we can make the ground state energy almost close to zero.


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?


yes if the particle is moving fast enough we have to taken in account of the rest energy as well as the kinetic 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?


to be in the center of the box

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 at the ground state the particle is always most likely to be at the center of the box


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 when the energy leave ground state the probability of the particle is no longer more likely to be at the center it is then random, and can be at places where it matches the energy level present