WAVE-PARTICLE DUALITY
The thought experiment we carried out made us realize some very interesting things about the classical and quantum world.
The double-slit experiment shows that quantum objects can have wave-like as well as particle-like properties.
The table tennis balls hit the screen in a random manner as expected. This implies a particle-like behavior for classical objects.
The light waves showed an interference pattern on the screen after passing through the double slits. When single photons or electrons passed through the double slits, an interference pattern was observed. This would imply a wave-like behavior for quantum objects.
If there was a detector placed on one of the slits, electrons (or photons) would now hit the screen randomly without showing any interference pattern. This means that they exhibit particle-like behaviour if we try to find out which path the electron (or photon) took before it hit the screen.
SUPERPOSITION
THINK: If an electron (or any quantum object) passes through the double slits and reaches the screen, which path did it exactly take (path A or path B)?
The answer is that the electron was travelling via both paths at the same time! This is the quantum magic of superposition.
However, things change if we place a detector to find out which path was actually taken by the electron. In this case, the electron suddenly starts behaving normally like a particle. The act of measurement changes the way the electron was behaving. It seems as if the quantumness of the electron is mocking the observers!
If this was confusing for you - do not worry! This is a topic that has baffled scientists for over a century now. Perhaps, the concept of superposition is best explained by the physicist Schrödinger and his cat!!
Watch the video provided below:
Quantum Wavefunctions
CLASSICAL DETERMINISM Classical objects follow rules of classical physics (such as Newton's laws of motion). Consider a car moving in a straight line with a uniform velocity v.
We define a quantity called momentum as p=m.v where m is the mass of the car.
Classical physics tells us that at any time t, the car would have traveled a distance that is given by x=v.t.
In other words, if you were to conduct an experiment that allows you to measure the momentum of the car at any time, using some knowledge of physics, you would be able to calculate the position of the car at any given time. Alternatively, if you know where the car is at any time, you would be able to calculate its momentum. Additionally, we can measure both position and momentum of the car at the same time. There are no surprises here at all. The motion of the car is completely deterministic.
Rules of classical physics are completely deterministic.
SCHRODINGER'S EQUATION AND THE UNCERTAINTY PRINCIPLE
We looked at the wave-particle duality of electrons in the context of the double-slit experiment. If we think of electrons as particles, we can assume that they follow some kinds of equations of motion similar to the way classical objects follow Newton's laws of motion. However, we need to be able to account for all of their wave-like properties as well. The equation that captures this essence of quantum objects is Schrödinger's equation. It gives information about all quantum-mechanical properties (for example, quantized energy levels of a hydrogen atom).
Quantum objects follow rules of quantum mechanics (such as the ones described by Schrödinger's equation).
What quantity does Schrödinger's equation actually talk about? Schrödinger's equation describes the wavefunctions of quantum objects. Now, these wavefunctions are related to the probability of finding a quantum object at a particular position. This means we cannot say with certainty where the electron (or any other quantum object) will be at any given time.
Let a quantum object be described by a wavefunction ψ. The figure depicts the probability of finding the quantum object at any point along the x-direction. The shaded region represents the probability of finding the quantum object between points a and b on the x-axis. Note that this just depicts the chance of finding the quantum object in the shaded region - there are no guarantees here! This is in sharp contrast with classical physics where we can determine the past, present, and future positions of an object simply by looking at some equations.
Quantum mechanics is not deterministic like classical mechanics.
In the example of the car, we talked about how the position and momentum can both be simultaneously measured for classical objects. Quantum objects such as electrons will also have properties such as position and momentum. Now, similar to the car experiment, we can take up fancy detectors and try to measure the position as well as the momentum of the electron. However, if we pin down the position of the electron, we will be unable to measure its momentum, irrespective of how accurate our detectors are. Moreover, if our detector determines the momentum of the electron, then we will lose any information about its position. It is as if the electron is playing an eternal game of hide-and-seek with us!
Heisenberg's uncertainty principle dictates that the more precisely determines a quantum object's position is, the less precise is its momentum (and vice versa).
Superposition and Measurement
We discussed that for a wavefunction ψ (given by the Schrodinger's equation), the position is not a determinate quantity. It is given by some probability distribution such as the one shown in the top panel of the figure below. Then, the shaded region represents the probability of finding the quantum object (let's say an electron) between points a and b on x axis. However, this is all just probability - we have not actually measured the position of the quantum object yet!
When we actually use our experimental apparatus and measure the position of the electron, we may find it at any of the points along x based on the probability distribution in the top panel of the figure. Recall that higher the probability, higher are the chances of finding the electron at that point along x direction.
Suppose, after conducting one measurement, the electron is found to be near point C. Notice now that the probability distribution is now centred at point C. This means that we have now determined the location of the electron to be at point C which reduces the probability of finding the electron at any other point on x to zero.
Before reading the sub-section below, try to think about the following questions:
Where exactly was the electron just before it was measured to be in state C?
What do you think will happen if we measure the position of the electron again? Will it still be at point C or at any other point (A, B and so on)?
Where exactly was the electron just before it was measured to be in state C?
In the example above, before we actually measured the position of the electron, we were not sure where exactly the electron was. We say that this the electron is in a superposition state of all possible points that are predicted by the wavefunction. The probability of finding the electron at any specific point is determined by the wavefunction. In the example of the car, we could exactly say where the car was one second before the measurement.
We could also determine where the car will be one second after the measurement of its position. But what about the electron? So, where exactly was the electron located before we measured its position? The honest answer is that nobody knows! The probability distribution just says that the electron could be anywhere in that region (with some regions more favourable than others). The topic of where the electron was prior to measurement has been debated for over a century now and has puzzled the likes of Einstein, Bohr and many other great scientists!
What do you think will happen if we measure the position of the electron again? Will it still be at point C or at any other point (A,B and so on)?
Once we perform a measurement to determine the position of the electron, we say that the wavefunction has collapsed to that specific position (point C in the example above). Now even if we perform repeated measurements on the same electron, we will still find it only at position C. All the previous information about the probability distribution has been erased from its memory!
This section just gave a preview of the concept of superposition and measurement in quantum mechanics. We will see these two concepts again when we study qubits in the next section. There, we will not be dealing directly with the position or wavefunction of electrons (or any other material used for the qubit) anymore. Instead, we will mainly be using a very intriguing quantum mechanical property called spin.
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