Lecture 1 | Modern Physics: Quantum Mechanics (Stanford)

The speaker introduces the educational program at Stanford University, highlighting its focus on continuing education for non-traditional students. The class is not a standard undergraduate or graduate course but aims to deliver theoretical physics at a full scale level. The speaker addresses the curiosity of European physicists about the nature of the class and clarifies that it is designed for adults from the community, including those over 70 and even over 90, who are eager to grasp fundamental concepts efficiently and quickly.

The lecturer outlines the series of courses, emphasizing that the current class is on quantum mechanics, following an initial course on classical mechanics. The explanation delves into the statistical nature of quantum mechanics, contrasting it with classical physics. The speaker criticizes a deterministic approach to physics, alluding to Einstein's famous quote about God not playing dice, and introduces the concept of quantum entanglement as a fundamental aspect of the course.

The discussion focuses on the unique randomness inherent in quantum mechanics, differentiating it from classical randomness. The lecturer uses the illustration of a deity throwing dice to modify Newton's laws, suggesting that while this introduces unpredictability, it does not replicate quantum mechanics' peculiar brand of unpredictability. The conservation of energy in quantum mechanics, despite its inherent unpredictability, is emphasized, setting the stage for a deeper exploration of quantum phenomena.

The lecturer introduces the two-slit experiment to demonstrate the odd behavior in quantum physics, contrasting it with classical randomness. The experiment involves particles like photons, electrons, or neutrons passing through two slits and creating an interference pattern on a screen. The speaker explains how this pattern defies classical expectations and introduces the concept of destructive interference, which is a fundamental aspect of quantum mechanics that challenges classical logic.

The concept of reversibility in physics is explored, where the lecturer discusses deterministic laws and the idea of reversing the laws of physics to test for information loss. It is explained that in classical mechanics, information is conserved, and the system returns to its original state after being reversed. However, the introduction of classical randomness leads to a loss of this conservation. The lecturer then ponders whether information is lost in quantum mechanics, setting up a comparison that will be explored further in the lecture.

The Heisenberg Uncertainty Principle is introduced as a fundamental concept in quantum mechanics that sets it apart from classical physics. The principle implies a limit to the precision with which pairs of physical properties, like position and momentum, can be known. The lecturer illustrates this with the idea of measuring an electron's position with a photon, explaining how the act of measurement affects the electron's momentum, thus introducing uncertainty.

The lecturer delves into the idea that measurement in quantum mechanics is inherently disturbing, contrasting it with classical physics where measurement can be arbitrarily gentle. The text discusses how observing a quantum system, such as an electron, during an experiment can drastically change the outcome, as opposed to classical systems that can be observed without significant disturbance. The uncertainty principle is highlighted as a key factor that prevents the simultaneous precise measurement of position and momentum.

The lecturer shifts the discussion towards the mathematical representation of quantum states, emphasizing that states in quantum mechanics are not points in a set but vectors in a vector space. This vector space is over the complex numbers, distinguishing it from classical mechanics that operates on set theory. The speaker introduces the concept of a Hilbert space, a vector space that will be central to understanding quantum mechanics, and contrasts it with the classical notion of a phase space.

The discussion continues with the properties of vector spaces over the complex numbers, including the ability to multiply vectors by complex numbers and to add vectors to form new vectors. The lecturer provides examples of vector spaces, such as complex functions and column vectors, and explains the rules for their manipulation. The abstract nature of these mathematical constructs is emphasized as a foundational aspect of quantum mechanics, which will be applied to describe quantum systems.

The lecturer concludes the segment by introducing the concept of a dual vector space, which is in one-to-one correspondence with the original vector space. The importance of complex conjugation in the context of complex numbers is also highlighted. The speaker notes that the mathematical framework presented is essential for understanding quantum mechanics and that future sessions will apply these concepts to quantum mechanical problems.