Potential well solutions question

[physicsjock]

Hey,

I've been trying to work out how, for a finite well of high Vo and width L, the interior solution has the form L Sin(kx + d),

I see that if d=0 then the solution resembles an infinite well, so that implies d depends inversely on the wells potential. But I can't work out what d comes from, and why the constant at the front is the length of the well (why is the amplitude of the wave the length of the well)

d is just the phase of the wave, so does it represent an adjustment to assure all the continuity requirements are satisfied?

Ive also been trying to work out what the k is, when I go through the process of finding the wave function inside I end up with (s/k)Asinkx + Acoskx after applying the boundary conditions.

where k²=2mE/$\hbar^{2}$

and s²=2m(Vo-E)/$\hbar^{2}$

i've been trying to work this out because I've also read that the below equation should also satisfy the internal solutions

ka = n$\pi - 2Sin^{-1}(\frac{k\hbar}{\sqrt{2mV_{o}}})$

this makes it seem like d resembles the inverse sin term, which makes sense because if Vo ->infinity you would get the solution to an infinite potential well

Anyone have any ideas?

 

[eljose79]

Hello,

Thank you for your interest in this topic. It seems like you have a good understanding of the basics of the problem. Let me try to explain the significance of the parameters d and k in the interior solution of a finite well.

First, let's consider the general form of the interior solution for a finite well:

Ψ(x) = A sin(kx + d)

Here, A represents the amplitude of the wave, which in this case, is related to the length of the well. This is because the amplitude of the wave is directly proportional to the probability of finding the particle in a particular region. In other words, the larger the amplitude, the higher the chance of finding the particle in that region. Since the length of the well represents the region where the particle is confined, it makes sense that the amplitude is related to it.

Now, let's look at the parameters k and d. As you mentioned, k is related to the energy of the particle through the equation k2=2mE/\hbar^{2}. This means that the value of k is determined by the energy of the particle, which in turn, is influenced by the potential energy of the well. So, different values of potential energy will result in different values of k, which will affect the shape of the wave function.

The parameter d, on the other hand, is related to the phase of the wave. It represents the starting point of the wave function, which is important in determining the continuity requirements of the wave. As you mentioned, if d=0, the solution resembles an infinite well. This is because in an infinite well, the wave function starts at 0 and repeats itself indefinitely. In a finite well, however, the wave function needs to be adjusted to satisfy the continuity requirements at the boundaries of the well. This is where the value of d becomes important.

Now, let's consider the equation ka = n\pi - 2Sin^{-1}(\frac{k\hbar}{\sqrt{2mV_{o}}}). This equation is derived from the boundary conditions of the well and it helps us determine the values of k and d for a specific energy level. This equation is valid for both the internal and external solutions, and it helps us determine the values of k and d that will satisfy the continuity requirements at the boundaries.

I hope this explanation helps clarify the significance of d and k in the interior solution of a finite well. Let me know