The Famous Schrödinger Equation

Yes, yes, I know, it's been a while since I wrote something here. This year I'm devoting myself a bit more to writing in this blog, I want to spread some knowledge and that can't be done if I don't write something on a weekly basis. 


Today I'm going to talk about the Schrödinger equation, this well known equation is the heart of quantum physics and quantum chemistry. As a quantum/theoretical chemist the aim is to solve this powerful equation. First I'll tell you a story...

Schrödinger wrote a paper in 1926, it was called Quantisierung als Eigenwertproblem, it came out on Annalen der Physik (the same magazine that published Albert Einstein's work). In this paper he stated that the old rules of the quantization of the energy could be substituted (in the simplest case: the non-relativistic, unperturbed hydrogen atom) by another postulate that didn't have any whole numbers but it introduces integers that arise in the same natural ways as in the old quantum theory. In a series of paper after this one Schrödinger applied his new wave equation to various cases, such as: 

• The harmonic oscillator

• The rigid rotator

• The diatomic molecule 

• The H atom in an electric field

For the hydrogen atom in an electric field Schrödinger developed a new way, different to that of Newton, Lagrange, and Hamilton, to describe a dynamic system. The Perturbation Theory differs in that one does not seek equations to describe the system at a given point but one finds a function of the coordinates of the system and the time. With these type of functions probable values of the coordinates and other dynamic properties can be predicted. The word probable gives us a hint of a fundamental rule of quantum mechanics, we can't describe in exact detail the behavior of a system. With this arises the famous, Heisenberg's uncertainty principle. 

With the Schrödinger equation we can determine a function ψ of the coordinates of the system and the time. These functions are called the Schrödinger wave functions or probability amplitude function. If you find the square of the absolute value of the Schrödinger wave function you find a probability distribution function for the coordinates of the system represented in the wave function. By similar means, the wave function can also be used to determine the energy of the stationary state. 

What if we include time? 

Well...it gets complicated.

I hope you remember your classical mechanics...

As a start we will consider a Newtonian system with one degree of freedom, so it only consists of a particle of a mass m moving along a fixed straight line, we will take the x-axis in this case. Another assumption is that the system can be described with a potential energy function V(x) that goes from minus infinity to infinity, so

Now, in order to simplify this equation we'll have to do some tricks. 

This equation is closely related to a fundamental equation in classical Newtonian Mechanics, 

this equation states that the total energy W of the system is equal to the sum of the kinetic energy T and the potential energy V, thus this is equal to the Hamiltonian function. Now if we introduce the momentum and the coordinate the equation becomes

Now, we replace the momentum and the energy by their differential operators arbitrarily and introducing the wave function we obtain

And this equation is identical to the classical Newtonian equation and is thus written in the simple form

This definition has only formal significance, remember that!

I think that's enough for this post, to be continued...

References: 

Quantisierung als Eigenwertproblem, Schrödinger, E. 1926 Annalen der Physik (Erste, Zweite, Dritte und Vierte Mittteilung) 

Introduction to Quantum Mechanics, Griffiths.

Classical Dynamics of Particles and Systems, Marion.