The Quantum "Particle in a Box"

Here an introduction to the one dimensional Schrodinger Equation with a very interesting example of a particle in a box. Let us first see what the Scrodinger Equation looks like generally:

And now for the one dimensional time independent case we find that:

and the 'E' is nothing but the total energy of the system, which here is an eigenvalue of the Hamiltonian operator. Rewriting the first equation in these terms gives us:

I will not go into the general properties of the one dimensional Scrodinger equation, but let's directly take a look at a physical case, the potential well. Now why it is called so is pretty simple, it is just a particle trapped in a bounded (in this case a ) one dimensional region. In classical mechanics all the particle could do is bounce off these walls of the well, and move to and fro along a line segment, i.e (Take a look at 'A') :

Now B, C, D, E and F exibit the quantum behaviour of the particle. Here, the blue line represents the real part of the wavefunction, and red, the imaginary part. Now here's what the Potential well looks like(note that 'V' instead of 'U' has been used here to denote potential energy):

now, we find that there are two Schrodinger Equations for the parrticle within the well, (where its a free particle) and when its outside the well, whcih ofcourse, it cant be, but analysis of that equation leads to the much required boundry condition. So let me denote the potential energy term outside the well as U₀, so then our Schrodinger equation looks like:

the prime denotes differentiation with respect to x. Now the solution will be of the form:

where 'C' is an arbitrary constant. Now the minus/plus term there refers to regions to the left and to the right of the well respectively. The Kappa is:

now we find that:

the Kappa bloats up to infinity outside the well, which means that, at the boundry of the well, the wavefunction has to vanish (that way, the numerator over infinity is 0 in the above's R.H.S). This is the boundry condition. Taking this into account, one could see for him/herself that the solution to the free particle Schrodinger equation within the well must be given by(hint, see the general solution of the second order differential equation y''+y=0):

now the condition ψ=0 for x=0 gives us δ=0, and when x=L (length of the 1D box) goves sin(kL)=0 thus kL=nπ, where n is a positive number, plugging this back into the wave number, we find that the energy states of the Box are given by:

Now plugging all we have back into the solution, we can find that c=√(2/L)

so the wave function(eigenfunctions of energy) now becomes:

we can now extend this to 3 dimensions and can actually call this a "box":

And there you have it folks,  a particle in a box!

By Vasudev Shyam (Admin at PDEP)