Particle in a box classical expectations

[sunrah]

Homework Statement

A particle with energy $E = \frac{\hbar^{2}k^{2}}{2m}$ > 0 moves in a finite potential

V(x) = 0 for x > abs(a)
V(x) = V₀ x < abs(a)

How does a particle coming from the left behave classically under the following conditions:
a) V₀ < 0, E = -V₀
b) E < V₀, E = 3/4 V₀
c) 0 < V₀ < E, E = 2V₀

Homework Equations

The Attempt at a Solution

I have to sketch the behaviour of the particle. My thoughts are that in (a) and (c) the particle is unbound and in (b) it is bound, but I don't know what the situation would be classically. Also what am I actually expected to sketch, the wave function or the particle (e.g. classical trajectory r(t))

 

[mark726]

?

Hello,

I would like to clarify a few things before addressing the question at hand. Firstly, the given potential V(x) is not finite, as it is dependent on x and thus infinite at x = ±a. Secondly, the expression for energy E is not fully accurate as it is missing a constant term, typically denoted by V∞, which represents the potential energy at infinity. This term is important as it affects the behavior of the particle in the classical limit.

Now, moving on to the question, I assume that by "classically" you mean in the limit of large energies, where the particle behaves like a classical object. In this case, the particle's behavior can be described using the classical trajectory, which is given by the equation of motion:

m(d^2x/dt^2) = -dV(x)/dx

For simplicity, I will assume that the particle is moving in one dimension (x-axis) and that V∞ = 0. With these assumptions, let's analyze the three cases:

a) In this case, V0 < 0 and E = -V0. This means that the particle has negative energy, which is not possible in classical mechanics. In this case, the particle would not be able to enter the potential region and would simply reflect off the potential barrier at x = ±a. The classical trajectory would be a straight line with a slope of zero, indicating that the particle is not moving.

b) Here, E < V0 and E = 3/4 V0. This means that the particle has positive energy but is still bound within the potential region. In this case, the particle would oscillate back and forth between the two potential barriers at x = ±a. The classical trajectory would be a sinusoidal curve with decreasing amplitude, indicating that the particle is losing energy due to the potential barriers.

c) In this case, 0 < V0 < E and E = 2V0. This means that the particle has positive energy and is unbound. In this case, the particle would pass through the potential region and continue to move in the positive x-direction. The classical trajectory would be a straight line with a positive slope, indicating that the particle is gaining energy and moving away from the potential region.

In conclusion, the classical behavior of the particle in these three cases would be different depending on the energy and potential values. I hope this helps in understanding