20. Quantum Mechanics II

The professor begins an informal discussion on quantum mechanics, emphasizing the importance of asking questions to understand the strange phenomena that don't follow the rules of daily life. He stresses that quantum mechanics, unlike classical physics, is not about making the world reasonable but about stating the rules and calculating outcomes. The professor summarizes the last lecture, explaining that everything is made of particles such as electrons, photons, protons, and neutrons. He delves into the double slit experiment, which highlights the quantum mystery and contradicts Newtonian physics, to illustrate the particle nature of electrons and the problems that arise when attempting to observe their behavior.

The paragraph discusses the double slit experiment with electrons, which leads to a contradiction with Newtonian physics. When one slit is blocked, electrons behave as particles. However, when both slits are open, an interference pattern emerges, indicating a wave-like behavior. This experiment demonstrates that electrons do not follow a single trajectory, contradicting the predictions of Newtonian mechanics. The professor illustrates this with the example of a machine gun firing bullets through two holes in a wall, where opening the second hole would intuitively double the number of bullets reaching the target, but with electrons, opening both slits can sometimes result in no electrons reaching certain areas, a phenomenon that would never occur with classical particles.

The professor explains that the observed patterns in the double slit experiment are reminiscent of wave interference, such as with water or sound waves. This leads to the concept that an electron is associated with a wave function, where the square of the wave function's amplitude gives the probability of finding the electron at a certain location. The wavelength associated with the electron is derived from its momentum, and the experiment shows that the wave function is essential for calculating the outcomes of quantum events. The professor also discusses the implications of a single electron being described by a wave, and how this wave function gives the odds of where the electron will land upon observation.

The professor introduces Heisenberg's Uncertainty Principle, which arises from the wave nature of particles. When attempting to localize a particle's position by passing it through a very small slit, the wave function spreads out, leading to greater uncertainty in the particle's momentum. This is demonstrated through the single slit experiment, where the uncertainty in position (Dx) and momentum (Dp) are related by the equation Dx * Dp ≥ ħ/2. The professor calculates the uncertainty in the y-component of momentum (Dpy) for an electron passing through a slit and shows that it is related to the slit width (d) and the electron's wavelength (λ), emphasizing the inherent uncertainty in both position and momentum at the quantum level.

The professor discusses the probabilistic nature of quantum mechanics, contrasting it with the deterministic outcomes of classical mechanics. In quantum mechanics, the wave function provides the probability of finding a particle in a particular location, but it does not determine a single outcome. This is likened to statistical predictions in classical mechanics, such as the odds of a coin landing heads or tails, which in practice are determined by numerous trials. However, unlike classical mechanics, where the exact outcome is theoretically calculable but impractical to determine, quantum mechanics inherently involves probabilities due to the wave-particle duality and the act of measurement affecting the outcome.

The professor emphasizes that while macroscopic analogies can sometimes help illustrate quantum mechanical concepts, they can also be misleading. Quantum mechanics deals with probabilities and the superposition of states, which have no direct analog in the macroscopic world. The act of measurement in quantum mechanics is crucial as it collapses the wave function and determines the particle's position. This is unlike macroscopic objects, which always have definite positions that are simply being observed. The professor warns against accepting any macroscopic analogy that suggests otherwise, as the quantum world is fundamentally different.

The professor discusses the concept of a quantum bit or qbit, which unlike classical bits that are either 0 or 1, can exist in a superposition of states, representing both 0 and 1 simultaneously. This property allows quantum computers to perform many calculations at once, offering exponential speedup for certain problems like factorization. However, building a quantum computer is challenging because qbits must be isolated from the environment to maintain their quantum state, yet they must also be interacted with to perform calculations and read the results. The professor humorously suggests that the ability to build a quantum computer could lead to fame, tenure, or the ability to go on a significant shopping spree.

The professor explains the concept of normalization in quantum mechanics, which is the process of scaling the wave function so that the integral of the probability density function over all space is equal to 1. This ensures that the total probability of finding the particle somewhere in space is 100%. The professor provides examples of how to normalize different types of wave functions, such as a rectangular wave function and a Gaussian wave function, and emphasizes the importance of this procedure for making accurate probabilistic predictions in quantum mechanics.

The professor prompts students to consider how information about momentum, a key quantity in classical mechanics, is contained within the wave function in quantum mechanics. While it might be tempting to think that a separate function is needed to describe momentum probabilities, the professor hints that the wave function Y(x) itself contains all the necessary information. This sets the stage for a deeper exploration of how various physical quantities, including momentum, energy, and others, can be derived from the wave function, which will be discussed in subsequent lectures.