Exploring Time-Independent Schrödinger's Wave Equation and Particl...

Greetings to everyone! In my last blog we have studied about a very important mathematical tool i.e., gamma function. We have also discussed a rough idea about Schrödinger wave equation. Today, as promised previously, we will study thoroughly about Schrödinger wave equation. After discussing, we shall then see solution for Schrödinger wave equation for a simple system. We will restrict our discussions to particle in a one dimensional box for today.

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Erwin Schrödinger, in 1926, gave a wave equation to describe the behaviour of electron waves in atoms and molecules. In Schrödinger's wave model of an atom, the discrete energy levels or orbits proposed by Bohr are replaced by mathematical functions, Ψ, which are related to the probability of finding electrons at various places around the nucleus.

From previous discussion on my post about eigenvalues and eigenfunction, we know that we have the time-independent Schrödinger wave equation as,

ĤΨ = EΨ

I hope you remember that here E is eigen value and Ψ is eigenfunction.

From the class on operators, we know that the Hamiltonian operator Ĥ is a comination of kinetic energy and potential energy. On putting the value of kinetic energy operator, and proceeding to solve it, we finally get as,

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The above equation is the time-dependent Schrödinger wave equation.

The Schrödinger wave equation can be solved for different simple systems, such as particle in a 1d box, particle in a 2D box, 3d box, one dimensional simple harmonic oscillator or a rigid rotor. Let us see the solution of the Schrodinger wave equation for particle in a 1 dimensional box.

Particle in a 1D box

Let us consider a free particle of mass 'm' is moving in a asymmetric box which is bounded in between o to l.

There exists no force in a free particle, so F(force)=0 and also Potential(V)=0. The particle of mass 'm' is confined to move in a one dimensional box of width 'l' having infinitely high walls. This is the simplest quantum mechanical problem which represents translational motion. The particle is subjected to a potential energy function that is infinite everywhere along the x-axis except for a line segment of length 'l', where the potential is 0.

Let's mark all the three possible region that the particle could reside as A, B and C and examine them one by one.

Region B

Now, from time-independent Schrodinger equation (TISE), we have,

1D box.png

math-20240127 (1).png

Since, the walls are of infinite length, i.e., the particle is definitely present within the walls. Remember? which condition does it satisfies? Yes! correct, it satisfies the condition of normalization. That means the particle is definitely present within the region B.

Region A and C

As the potential is ∞ in regions other than B, so again from the TISE, we have,

math-20240127 (2).png

Since the avefunction Ψ(x)=0, so the probability of finding the particle outside the box is zero. Hence, the particle does not exists in region A and C.

Thus we can conclude that Ψ=0 outside the box and likewise Ψ=0 at the walls also, i.e., x=0 and x=l.

So, if the particle exists within the region B, there must be a solution to the wave equation. Let's solve and find the value of the wavefunction, Ψ(x)

Finding the solution

From the wave equation from region B, we have,

math-20240127 (3).png

The above equation is an ordinary second order differential equation and has the solution of the form, Ψ(x)= A sin kx +B cos kx, where A and B are two arbitrary constants.

We need to verify the solution, then only we can be certain that we are doing the right thing.

Proof of the solution

math-20240127 (4).png

As the values satisfies the equation, hence it is proved!

Now we have to check the boundary conditions too!

Boundary conditions

math-20240127 (5).png

A can not be zero, cuz B=0 already and if both the arbitrary constants are zero, the wavefunction value will be 0, which can not be true for region B.

If A ≠ 0, then

math-20240127 (6).png

Now, we have,

math-20240127 (8).png

To find the normalization constant A, we will do what we have studied earlier.

math-20240127 (9).png -

Finding the energy eigenvalue

From the value of k, we can find out the energy eigenvalue.

math-20240127 (10).png -

Where,m = mass of the particle
l=length of the box
h=Planck's constant
and n=energy levels.

What we learnt?

We studied the Schrodinger wave equation involving the total energy operator and the energy eigenvalue. We formulated the time independent Schrodinger wave equation. This form is called time independent as the potential, V is not a function of time, t but depends only on x.
We studied the different possible regions in a 1D box to which the particle could possibly reside. We studied each regions separately and concluded the possibility of finding the particle in region B.
We studied the solution to the Schrodinger wave equation considering the boundary conditions. We then, went onto find the important energy eigenvalue.