Shifting Constraints in the Particle in a Box System

[Domnu]

I was just wondering... if a problem involved a particle which was constrained to move from x = -a/2 to a/2 and asked you to find it's properties (not position, though), could you just "shift" the entire system from x = -a/2, a/2 to x = 0 to a?

Also, let's say that a question asked for the probability of the particle being present from x = -a/2 to a/10 (assuming the particle is constrained from -a/2 to a/2). Could we just shift the box to 0 to a and find the probability of the particle being present in the areas between 0 and a/10+a/2 = 3a/5 ?

This would be really useful, because I can still use the eigenstates of the energy function for a particle in the box scenario,

$$\phi_n = \sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}$$

By the way, this is all assuming that no potential energy is present in the system.

 

[lbrits]

Yes, the Schrodinger equation is invariant under coordinate changes $$x \to x + b$$.
You'll also find that if you do the transformation on your eigenstates, you end up with cosines, which are solutions to the SE with the new boundary condition.

 

[denian]

I would say that shifting constraints in the particle in a box system can be a useful tool in certain situations, but it is important to understand the implications and limitations of doing so.

First, let's address the idea of shifting the entire system from x = -a/2, a/2 to x = 0 to a. This essentially means we are changing the boundaries of our system, and therefore, the energy levels and eigenstates of the particle will also change. While this may be a valid approach in some cases, it is important to consider the physical significance of these changes and whether they accurately represent the original system being studied.

Furthermore, shifting the box to find the probability of the particle being present in a specific region can also be a useful tool, but it is important to note that this technique assumes a uniform probability distribution within the shifted region. If the original system had a non-uniform probability distribution, shifting the box may not accurately represent the true probabilities.

In general, shifting constraints in the particle in a box system should be used with caution and only when it is justified and accurately represents the physical system being studied. It is important to carefully consider the implications and limitations of this approach in order to ensure accurate and meaningful results. Additionally, if there is a potential energy present in the system, it may not be possible to simply shift the box, as the potential energy may also be affected by the shifting of boundaries. Overall, while shifting constraints can be a useful tool, it should be used thoughtfully and with a thorough understanding of its implications.