Particle in a one-dimensional box... Most of the motion of the particle is confined within the box. Hey, do you have a model for this? Let me elaborate on the wave equation.
Say, we're considering some model. Hey, at time T=0, with the length of the box denoted by L and X as the direction. We have an infinite potential at the boundaries, with B=0.
Okay, now let's consider a particle of mass m moving in this box along the X-axis. Outside the box, the potential is infinite, while inside it's 0, preventing the particle from moving outside.
Do you follow so far?
Okay, now let's derive the wave equation for this scenario. Outside the box, where B equals infinity, we get the equation:
Del^2 PSI Del x^2 + 8 Pi^2 H^2 PSI = 0.
Since Infinity is very large, it's approximately equal to negative infinity.
Therefore, Del^2 PSI Del x^2 + 8 Pi^2 m H^2 PSI = 0.
Hence, PSI is 0 everywhere outside the box, implying no probability of finding the particle there.
Now, inside the box where B=0,
we get the equation: Del^2 PSI Del x^2 + 8 Pi^2 m e H^2 PSI = 0.
By making the homogeneous substitution, let's denote 8 Pi^2 m e H^2 as Alpha^2. Hence, the equation becomes Del^2 PSI Del x^2 + Alpha^2 PSI = 0. This is a second-order differential equation with a general solution of the form a sin(Alpha x) + b cos(Alpha x).
To satisfy the boundary conditions, at x=0, PSI=0, which implies b=0. At x=L, PSI=0, which implies a sin(Alpha L) = 0. Since a cannot be 0, sin(Alpha L) must be 0, leading to the condition Alpha L = n Pi, where n is a positive integer. Therefore, Alpha = n Pi / L, where n = 1, 2, 3, .... Using this, our wave function becomes PSI = a sin(n Pi x / L).
The energy is given by Alpha^2 = 8 Pi^2 m e / H^2. Substituting Alpha, we get En = n^2 Pi^2 H^2 / (8 m L^2).
Now, for normalization, since there's no existence of the particle outside the box, the probability inside the box must be 100%. This leads to the condition a^2 / 2 * L = 1, giving us a = sqrt(2 / L). Therefore, the normalized wave function is PSI_n = sqrt(2 / L) * sin(n Pi x / L), with energy En = (n^2 Pi^2 H^2) / (8 m L^2).
So, this describes the characteristic behavior of a particle in a one-dimensional box.
Member since February, 2024
{{ comment.student.firstname }} {{ comment.student.lastname }}