Introductory Chapter on Quantum Mechanics, Lecture notes of Physics

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INTROCUTION TO

QUANTAM

MECHANICS

Dated 27th July 2023

Introductory Chapter on Quantum Mechanics:

 First let’s understand the differences between classical and quantum mechanics with the help of examples of how quantum mechanics can be used to predict the behaviour of particles.  The subject also covers the uncertainty in mentioned principle and how it affects the behaviour of particles in a quantum system.

Age Distribution:

 The age distribution of the participants to understand the subject is quite diverse.  This subject is for continuing education on quantum system for people who are not currently enrolled in college.

Course Outline:

 The first introductory course is to understand on classical mechanics, which serves as the foundation for all classical physics. The next course, already available on the internet, is on quantum entanglement. It is recommended to complete the second course before diving into the full-scale quantum mechanics covered in this class.  Further to understand the differences between classical and quantum mechanics, classical mechanics is based on deterministic principles whereas quantum mechanics is based on statistical thinking and randomness.  Classical mechanics and quantum mechanics have distinct differences in their logic. Quantum mechanics is based on statistical thinking and introduces concepts such as randomness and unpredictability. While some may argue that these concepts imply a lack of determinism, it is important to note that quantum mechanics is not purely random. It operates under a specific set of laws that govern the behaviour of particles.  To illustrate the differences between classical mechanics and a theoretical modification with randomness, let's consider a scenario involving the motion of the moon. In classical mechanics, the motion of the moon around the Earth can be predicted with infinite precision using Newton's laws of motion and the law of gravity. However, imagine a modification where randomness is introduced by "God" throwing dice to perturb the moon's motion. While this introduces some level of unpredictability, it is not comparable to the randomness and unpredictability observed in quantum mechanics.  The unpredictability of quantum mechanics is very different from classical randomness. In quantum mechanics, energy is exactly conserved even though the outcomes of measurements are unpredictable. This contrasts with classical randomness, where giving a system a random kick can change its energy.  One example that highlights the oddness of quantum randomness is the two-slit experiment. In this experiment, particles such as photons or electrons pass through a small hole and create a pattern on a screen. Classically, if there was randomness, the

However, if you let it run for a longer time, there is a significant probability for a fluctuation to occur. This means that the conservation of probability is lost.

Quantum Mechanics and Information Loss

 In quantum mechanics, there is also a statistical element. For example, when an electron goes through a hole, it has a probability of getting kicked up. If we reverse the direction of time and run the system backward, the electron will move along the original trajectory every single time. However, if we interfere with the electron and detect its position, the probabilistic character of its motion gets compounded. This means that the test of reversibility fails, and information is lost.

Looking at the System

 In classical physics, looking at a system does not necessarily disturb it. We can determine the state of an object without affecting its behavior. However, in quantum mechanics, looking at the system and detecting it can disturb the system and compound its probabilistic fluctuations.  In quantum mechanics, determining the state of a system can have a significant impact on the system itself. A classic example of this is the two-slit experiment, where the interference pattern only occurs if nothing in the environment records which way the electron went through the slits. If something does record this information, such as a molecule getting disturbed or excited, the interference pattern is destroyed and the probabilities behave just like classical physics.  So, it is not possible to measure the path of the electron without disturbing the experiment and potentially changing the outcome. This is a fundamental aspect of quantum mechanics known as the uncertainty principle.  The uncertainty principle states that there is a limit to how precisely we can measure certain pairs of properties, such as position and momentum. In classical physics, it is easy to imagine a bit of uncertainty in both position and momentum due to random factors. However, in quantum mechanics, there is a deeper, fundamental obstruction to knowing both the position and momentum of a particle simultaneously.  In classical physics, with better equipment and more precise measurements, it is possible to determine both the position and momentum of a particle with increasing accuracy. However, in quantum mechanics, there is a logical obstruction to this. It is not just a matter of laziness or lack of precision; it is a fundamental limitation.  This limitation was first thought about by Heisenberg through abstract mathematics. When questioned by Bohr about the impossibility of measuring both position and momentum simultaneously, Heisenberg devised an experiment to illustrate this limitation.  Overall, quantum mechanics operates on different logic than classical mechanics, and the uncertainty principle is a key aspect of this distinction.

Understanding the Uncertainty Principle

 There is a consistent reason why it's impossible to simultaneously determine the position and the momentum of an object. Let's explore this concept further.  Measuring Properties of Particles  In 1926, Heisenberg, Bohr, and Einstein discovered that when measuring the properties of a particle, they were actually thinking about using microscopes and photons. They imagined putting the particle under a microscope and detecting its position and velocity by bombarding it with photons. The photons would scatter off the particle, and by focusing the light waves, they could determine its exact position.

Energy and Momentum of Photons

 According to Einstein, the energy of a photon is equal to Planck's constant times the frequency of the light describing the photon. Beams of light not only have energy but also momentum. When a beam of light is absorbed by an object, it can push or heat the object, demonstrating its momentum.

Relationship between Energy and Momentum

 The relationship between the energy and momentum of a beam of light is given by the equation: energy = speed of light * momentum. This equation is similar to Newton's theory of momentum and energy for ordinary particles.

The Momentum of a Photon

 The momentum of a photon is given by the equation: momentum = Planck's constant

• frequency / speed of light. The wavelength of the photon is inversely proportional to its momentum, meaning that shorter wavelengths correspond to larger momenta.

Photographing the Electron

 If one wants to take a photograph of an electron with non-fuzzy details on a certain scale, they need to use wavelengths shorter than the scale they want to capture. The wavelength and momentum of a particle are inversely related.  In quantum mechanics, there is a fundamental limit to how precisely we can measure both the position and momentum of a particle at the same time. This is known as the Heisenberg uncertainty principle. In classical physics, we can imagine doing a gentle experiment that does not disturb the system and then carry out another experiment without the first one influencing the outcome. However, in quantum mechanics, the act of measuring the position of a particle necessarily imparts a random momentum kick, making it impossible to determine both the position and momentum accurately.  To measure the velocity of a particle, we can use a gentle measurement technique by measuring the position of the particle at two different times and taking the difference in position divided by the time interval. This gives us an estimate of the velocity. However, in order to avoid disturbing the velocity of the particle, the position measurements should be done with photons of very long wavelength. By waiting a