Schroedinger's equation

The famous theoretician Paul Dirac wrote in his paper published in 1929 as follows:

"The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation." (6 April 1929)

(to see a completely different perspective, read more is different by the theoretician P W Anderson "More is Different" article published in Science Magazine.


Schrodinger's equation is the equation of motion for the matter waves. Now, what is matter waves? Waves associated with the matter particles (eg. electron, proton, ect) are called matter waves. So, the equation of matter waves will describe the entire property of the matter waves. The solution to this wave equation will give you a function that is called "wave function". Wave function itself is a complex entity. It has been very difficult to define the meaning of the wave function. However, the square of the wave function (the product of the complex conjugate and the wave function which is termed as the probability density) gives the probability of finding the particle at a given location at a given time. Note that, the wave function does not have any interpretation, whereas its square has the physical interpretation. 

Why do we need Schrodinger's equation?

Can we describe the entire universe with Schrodinger's equation?
In principle, yes. The former director of the Perimeter Institue Niel Turok gave a fantastic public lecture on the topic "The Astonishing Simplicity of Everything". Watch the full talk here.

If we solve the Schrodinger's equation, we can get the wave function (solution to the Schrodinger's equation). In principle, this wave function contain all information about the system of interest. Note that to for the equation, we need the potential acting on the system/electron/particle.

The realistic problem that we can use Schrodinger's equation to solve is the hydrogen (or hydrogen-like atoms He+, Li2+, Be3+, B4+, C5+ ) atom where there is only one electron. We can solve the Schrodinger's equation for atoms (or molecules or solids) having more than one electron (not even He) accurately. This is technically named as analytical solution is not possible (or too difficult to solve). But, we need solution (we need wave function to calculate the properties of the atom/molecules/solids). Here comes the approximation techniques. We can use approximation techniques to solve Schrodinger's equation and get the properties that we are interested in.

The textbook problems we learn using Schrodinger's equation are:

1. Free particle (potential=0)
2. Potential step
3. Finite potential well
4. Infinite potential well
5. Tunneling through the barrier
6. Particle in a 1D box (linear molecules)
7. Particle in a 3D box
8. Simple harmonic motion
9. Hydrogen atom model

A crucial development in this field is the formulation of Hohenberg-Kohn theorems. There are two theorems. These theorems the basis for the formulation of Kohn-Sham Density functional theory (see here in other post on this topic and in Wikipedia).

The Kohn-Sham density functional theory for which Walter Kohn (and John A Pople) has been awarded is based on those theorems. We saw that on solving the Schrodinger's equation, we get wave function. From this wave function, we can get the properties of the system (atom/molecule/solid). However, HK theorems proves that the density (which is a physical quantity and can be observed) is a fundamental variable from which we can get the energy (and hence all other properties).

Suppose we have N number of electrons in the system. The KS-DFT formalism reduces a complex N-electron problem in to N number of 1-electron Schrodinger-like equation. Note that we writ Schrodinger-like equation. Yes. Why? Because, we replace the actual potential in Schrodinger equation with an effective potential in KS-DFT. Wonderful. Right? But wait.

This simplification comes with a trouble (???). There is an important ingredient (exchange and correlation functional) which has been proven to exist but nobody (to this day) knows its mathematical form. The exchange and correlation functional is approximated and calculations are performed to predict the properties of materials (from gas to liquid to solids). Generally it works and give good results. However, many problems remain and these problems are attributed to the approximation of the important ingredient. To know more about the theory, see this post on "Density Functional Theory".