I discussed in the previous chapter you the probability of a particle being detected in the continous range can be calculated from its wavefunction using,
Now, assuming the particle actually exists, if we essentially look for it at every point in the entire universe we’re bound the detect it somewhere right? Hence the probability of finding the particle in the range should be . In other words,
- must equal .
So, what if we get a wavefunction, say. where doesn’t equal and equals some other number instead? Then we can make the function’s probability equal bu multiplying it with a constant , such that the corrected wavefunction is:
This process is called normalizing the wavefunction. But how do we figure out what the value of is? Notice that now that ,
Now we already know that , so the above expression simplifies to:
Now to ensure , we need a value for such that
So I already mentioned that everything has a matter-wave associated with it. Developing the idea a bit further, we get the concept of a wavefunction which, by convention, is denoted by .
Each physical system is described by a state function which determines all can be known about the system. … The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.
Furthermore, a wavefunction is generally a complex function i.e it consists of both a real part and an imaginary part.
- – A wavefunction
To get the probably of a particle being at a position , you have to get a real value out of the complex wavefunction. Hence the probably of finding a particle described by a wavefunction is the product of the function at and its complex conjugate at .
- – probability of finding the particle at .
To get the probability from a range where is a continous variable, you can take the integral of from to :
One of the best explanations I’ve found of the Heisenberg uncertainty principle is in Volume III of the Feynman Lectures on Physics. Read the following sections (which I’ve linked) to understand the general concept behind it:
Essentially, there is an inverse relation between the width of a wave-packet, and the range of wavenumbers of the waves you’ll need to superpose to generate that wave-packet, . In general, the more localized a wavepacket is, the more waves you need to add to create it. Here’s an animation to show what I mean:
We can describe this relation as:
Multiplying the first relation with , we get
or more precisely,
Which is the Heisenberg Uncertainty Principle.
To create a truly localized wavepacket we need to superpose not just two but an infinite amount of sinusoidal waves whose wavelengths and amplitudes vary in a continuous spectrum. To do that, we need to learn about what’s called the Fourier integral:
Why an exponential function instead of sine and cosine? This is because we just used Euler’s formula:
and it’s derivations:
to replace the sines and cosines with the exponential function.
Essentially, this gives us a framework for expressing practically any function as a superposition of harmonic waves (just plug in the function in the expression for and then plug in into the expression for the Fourier integral). This is known as the Fourier transform.