20.2 Quantum Mechanics: Particle in a Box

 In this activity we will learn about the probability of finding a particle in a box and how these 'particles' have a wave-like nature which can be defined by the de Broglie relation. We will also learn about the ground state energy of these particles.

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

Answer: The longest wavelength a standing wave can have on a string that is L long is 2L.

Question #2: What is the momentum of the longest wavelength standing wave in a box of length L?

Answer: De Broglie stated that the momentum of a particle is plank's constant divided 2L.

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

Answer: If the particle is not traveling at relativistic speeds then we know that for the momentum, p=mv and for the kinetic energy,  K = 1/2mv^2. If we use De Broglie's equation for momentum (p = h/2L) we see that K = h^2/(8mL).

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

Answer: As L increase we can see from the equation above that the ground state energy will decrease.

Question #5: 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?

Answer: As L increases to macroscopic levels the ground state energy decrease to such a insignificant amount that it is almost impossible to detect by conventional methods for measuring energy.

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

Answer: Since mass is also in the divisor in the above equation very massive objects also a negligible ground state energy.

Question #7: 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?

Answer: The particle is most likely to be found  in the center of the box.

Question #8: 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?

Answer: No because the wave function is independent of the mass and the particle is always most likely to be found at the center of the box.

Question #9: 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?

Answer: No, for for example the most likely place to find the particle when it is in the first excited state is at L/4 and 3L/4.

Question #10: 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?

Answer: In the limit of a large n the probability density function gets "squished" together and appears to be constant. This means that the particle has become a "free particle" and has an equal probability to be found in all locations.