GITAM, Department of Engineering Physics

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Schrodinger Equation

We discussed wave particle duality and it is therefore natural to search for a wave equation to describe the behaviour of particles. In this section we introduce Schrödinger's equation, which does just this job, then look at some properties of the equation and its solutions.

 The origin of the equation

We begin with a traveling monochromatic wave in one-dimension

                       

Then

               

Thus, given that the wave velocity n = w/k, we have

                   

All classical waves satisfy this equation and previous courses have explored the equation and some of its solutions. Examples include waves on a string of length L fixed at both ends:

                   

An important feature of this equation is that wn can only take on certain values . The many different solutions to the wave equation arise because of the boundary conditions which are applied when solving the equation to get the constants of integration. Note that we have not derived the wave equation; we started with a special solution and found an equation that it satisfied. We will now carry out a similar exercise to end up with Schrödinger's equation. The starting point is a non-relativistic particle with total energy E given by the sum of its kinetic and potential energies, i.e.

                       

This simple equation is one of the most important in the course; note it well. If we want to associate a wave with this particle the wave must satisfy the equation. But for a wave E = hn = ħw and p = h/l = ħk. Thus substituting we obtain

                       

Then taking V = 0 for now (a free particle) we can write equation 2.3 as

                       

suggesting that we need to look for a wave equation containing terms in  w and k².

If we try our classical wave solution (equation 2.1) then

                       

If we try equating these two terms to satisfy equation 2.4 we find ourselves with a sin term on one side and a cos term on the other. These cannot possibly be equal.

However, recall that in optics we can frequently use complex exponentials to make the maths simpler and so let's try

                   

Then

                   

i.e. both terms are proportional to y.

It is now simple (as you should show) to derive the equation

                   

If instead of assuming V = 0 we keep V in equation 2.3 we end up with

                   

which is known as the time-dependent Schrödinger equation (TDSE). We will not discuss this equation any further in this course but you will come across it again in later quantum mechanics courses.

If we take equation 2.2 in the form

                   

               

or, in its more commonly written form

                   

This is known as the time-independent Schrödinger equation (TISE). It is applicable when the total energy of a system is constant, the only case we will be considering in this course.

 

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