6.3 What does it all mean?
The connection between the kinetic energy associated with a wave function and its curvature. Introducing operators, eigenfunctions and eigenvalues.
The most mysterious thing about the Schrödinger equation is (as it was when we were thinking about localisation in Chapter 5) the kinetic energy term. Rearranging the equation by dividing through by ψ(x), we get:
This seems, at first sight, a bit perplexing. However there is in it an echo of the de Broglie equation. To see this, we have to pause to think about what the second derivative of a function actually means. Let’s do this in the context of a sine waveform:
Figure VI-i
While dψ/dx is the gradient of the curve, d2ψ/dx2 tells you how quickly the gradient changes as x changes. This means that it is greatest in magnitude around the peaks and troughs in the waveform where the sign of the gradient changes. d2ψ/dx2 is sometimes described as measuring the curvature of a function. The significance for us in relation to kinetic energy is that the bigger the value of d2ψ/dx2, the more quickly the gradient changes around the peak: in other words, the narrower the peak will be. We can see now the relationship to the de Broglie equation: in a sinusoidal curve, a shorter wavelength and a bigger d2ψ/dx2 are both indicative of narrower peaks and troughs. Since Prince Louis tells us that p = h/λ, we can see that it makes sense that a bigger d2ψ/dx2 in a wavefunction indicates a higher kinetic energy. This is also consistent with the uncertainty principle, which we also saw implied that greater localisation of a particle means a narrower peak in the wave function and again, therefore, implies a higher kinetic energy.
The Schrödinger equation is often written in shorthand form as:
Ĥψ = Eψ. (VI-8)
Here Ĥ is an operator. This equation reads as: the operator operates (as operators are wont to do) on the wavefunction (ψ) to regenerate ψ, but now multiplied by a simple number. Only certain functions will be compatible with this equation - hence only certain functions are possible as wave functions. The problem then is to solve the equation to find these functions.
In mathematics an equation like this is called an eigenvalue equation. A function ψ that fulfils the requirement - that it is regenerated by the action of the operator, just multiplied by some number -is called an eigenfunction of the operator, Ĥ, and E (the number it's multiplied by) is the corresponding eigenvalue. Looking at equation, VI-6, we can see that Ĥ has the form:
Ĥ is the energy operator in quantum mechanics, which is usually called the Hamiltonian operator (a ghostly Irish figure taps Cormorant on the shoulder and points out that the Ĥ is a nod to himself, William Hamilton, who set up classical equations of motion using a similar approach, long before Erwin was even a twinkle in his mother’s eye). At each point, x, the first term of Ĥ (involving the second derivative) generates the kinetic energy at that point while the second term generates the potential energy. The sum of these is the total energy, given by the eigenvalue, E.
There are many flavours of Schrödinger equation and this is the vanilla one: it is one-dimensional but for practical applications it usually needs to be generalised to three; it is based on classical mechanics but for some purposes a relativistic form is necessary and it only applies to states that are not changing with time but a more general, time-dependent version can also be constructed. Despite all these limitations, it will enable us to explore the principles of how it is used and to see how it leads to the observation we started with in chapter one – the quantisation of energy levels.
To really understand the Schrödinger equation, we need to make it work. Our starting point was curiosity about the energy levels of an electron in an atom but that is an enormous leap from where we are now. We’ll head first for the traditional nursery slopes where those who would wield the equation in earnest learn to handle it.
