The Dirac Delta Wavefunction

I thought a lot about the first equation of 'Month of Equations' series this year. What should be the first equation? Then I came up with the Dirac delta wavefunction. Why? Because this equation is perfect to start learning about quantum mechanics. Today, through this equation, I will explain two main concepts: wavefunction and measurement in quantum mechanics. So let us start!

Dirac Delta Wavefunction
The Dirac Delta Wavefunction

What is a wavefunction?

The first thing you should keep in your mind is that if something does not make sense, it isn't necessarily wrong. Yes! quantum mechanics cannot be compared with what we see in our everyday life. The basic difference between classical mechanics and quantum mechanics is the nature of the systems involved. Look at the objects around you. You can precisely tell where each object is. Look at a moving car. You can tell its position at every moment of time. But when you go to the microscopic world, things start getting fuzzy. The certainty of the classical world is curtained by probability. At the quantum scale, you cannot tell where your particle precisely is. If someone comes to you and says he has precisely found the particle, don't believe that person. He/She is lying with all certainty.

Quantum physics was born as a result of the debate on the nature of light. Is light made of particles or is it a wave? Scientists found that it is both. In 1897, J.J. Thomson discovered the first particle, the electron. But then came a twist. The Davisson and Germer experiment showed that electrons, so far thought to be material particles, also have a wave nature associated with them. A wave accompanies a material particle. But there is a difference between the wave nature of a photon and that of an electron. The wave of the former is an electromagnetic wave described in terms of electric and magnetic field while that of the latter has no relation with these fields. The wave quantity of material particles is described by a mathematical function, known as the wavefunction ψ(r,t).

Also Read: How was quantum physics born?

Interpreting the wavefunction

The physical interpretation of wavefunction came from a brilliant quantum physicist: Max Born. The Born interpretation says that the square of the wavefunction over a range represents the probability of finding the particle within that range. Did you notice r and t written with ψ? This means that the wavefunction (and thus the probability of finding the particle) is different at different position and time. I will explain this point with a very simple analogy.

The interpretation of Wavefunction

Suppose you plan to give a surprise visit to your friend. The probability of your friend doing a particular activity depends on the time when you visit him. Suppose you visit him at 5 in the morning. There is a higher probability, say 80%, of finding him sleeping in the bed. There's say 15% probability of finding him taking his bath and 5% probability of finding him studying. But if you visit him at 1 in the afternoon, these numbers will change (Say, 10% sleeping, 85% studying and 5% taking his bath). So the probability is actually a spread and not a certain quantity (unless you open the door). This is exactly what a wavefunction means in quantum mechanics. The particle can be anywhere within the boundaries of the system and the probability of finding the particle at a particular position is different at different places and time.

Measurement in quantum mechanics

Now comes the twist in the story. When you open the door, what happens? You find your friend doing one particular activity. He cannot be sleeping and taking a bath at the same time. So the probability spread, which you had before opening the door, has now collapsed into just one spike. Similarly, once you make a measurement of a quantum system, that probability breaks down and you find your particle to be in one particular state. By state I mean not only the position, but it can be anything e.g. the spin of a particle. The best example of this is the double slit experiment with electrons and a detector and the Stern Gerlach experiment.

Image result for measurement in quantum mechanics
Measurement collapses the wavefunction

So we can safely deduce: Measurement changes the state of the system. Once you make your measurement, you disturb your system. After you stop your measurement, the spike again forms a probability distribution. Interesting, ins't it?

The Dirac Delta Wavefunction

I want you to go back to your high school mathematics class. Remember studying functions? What if I tell you to think of a function that is zero everywhere and infinite at just one point. Yes, that's the delta function. Ask a mathematician and he will scold you for calling it a function because it does not follow the formal definition of a function. But I'm a physicist and it does not matter to me if I call it a function or something else (mathematicians, no offence!). The plot of a delta function is given below:

Dirac Delta Wavefunction

You can clearly see the function is zero everywhere except at x = 0 where it gives a finite value. The delta function is one of the most important functions in mathematical physics. It has a lot of applications and quantum mechanics and electrodynamics.

If I convert such a mathematical function into my quantum mechanical wavefunction, what I get is the Dirac delta wavefunction. Consider a particle is at a precise position say x = a at time t = 0. This means that the wavefunction is zero everywhere except at x = a and hence such a wavefunction has a form of delta function. Such a wavefunction is represented by ψ(x) = A δ(x-a) where A is a constant (we will discuss about it in another equation) and δ(x-a) is the delta function centered at x = a. In reality, by keeping in mind the nature of quantum world, such a wavefunction is hypothetical. It does not exist in reality because there is no uncertainty in the position of the particle. But what is it important?

Importance of dirac delta wavefunction

Why place does such an unrealistic wavefunction have in theory? The answer is: Dirac delta function is a pure state of ideal position measurement. Suppose the particle is in the state given by the Dirac delta function, then if I do an ideal measurement of position, then I am sure to get the position of the particle as x = a. For each measurement, there are certain possible results and with each result, there is associated a state called pure state. If the state of the particle before the measurement is one of the pure states, then the measurement is sure to give the corresponding result.

Author's message

I wrote this article to introduce you to the basic concepts of quantum mechanics. I hope you have got some idea about the wavefunction, measurement and the Dirac delta wavefunction. If you didn't understand it completely, it is perfectly fine but having some idea about these 3 concepts is a good start to quantum theory. There will be some more equations on quantum mechanics in 'Month of Equations' series. My team and I will try our best to explain everything in the easiest way. I hope you enjoy this series!

Also Read: All the 30 articles of Basics of Astrophysics series.

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