Lecture 1 234 Quantum 1 PDF

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Physics 234 Spring 2018! Budakian

Lecture 1

May 2, 2018

• Newton thought light was made of particles. Later, it was discovered that light behaved like waves. In the early part of the twentieth century, Einstein showed that light behaves like particles.! • Electrons were thought to be particles. Later, people discovered that it also behaved like waves. Now we know that it can behave like both depending on how we measure it.! • So in reality, electrons and photons (particles of light) are neither particles nor waves, although they can behave like either one. In fact, this duality applies to all matter on the atomic scale, and even larger structures like molecules. In fact, no one really knows if the laws of QM stop on some length or mass scale, or whether they persist up to the macroscopic world.! • Different observations regarding the nature of light and matter created much confusion up to the first quarter of the twentieth century, until Schrödinger, Heisenberg and Born came along and provided a solid theoretical framework that unified the field, and allowed us to accurately describe quantum phenomena. Along the way, it completely upended our understanding of reality.! • It is the goal of this course to introduce you the basic theoretical framework to explain quantum phenomena, which we still use today.$

Physics 234 Spring 2018! Lecture 1 May 2, 2018 Budakian “‘Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.” — The Feynman Lectures on Physics Vol. III Let’s consider the following Gedanken (or thought) experiment — You have a collection of 4 types of sticks:! ! !

Short red stick Long red stick

Short blue stick

Long blue stick

You have 50 of each type of stick. You sort your sticks into two bins, with all the long sticks in Bin#1 and all the short sticks in Bin#2.!

Our intuition tells us that if we reach into Bin#1 and pick a stick at random, then we have a 100% chance of picking a long stick, with 50% chance of finding either blue or red. Likewise, if we reach into Bin#2, we have a 100% chance of picking a short stick, with 50% change of finding blue or red.! In the quantum world, things work a very differently however. Imagine we have “quantum sticks” that obey the rules of quantum mechanics. In the

Physics 234 Spring 2018! Lecture 1 May 2, 2018 Budakian quantum world, if we look into Bin#1 and pick a particular color, either red or blue, then we have a 50% chance of picking either a short or a long stick. Likewise, if we look into Bin#2 and pick a particular color, then we again have a 50% chance of picking either a short or long stick. It would seem that the mere act of observing the color of a stick makes the stick “forget” whether it was long or short.! Suppose, we choose to sort based on color and put all red sticks of either length into Bin#1 and all blue sticks of either length into Bin#2.!

If we close our eyes and reach into Bin#1 and feel for a short or a long stick, we find that we have a 50% chance of finding either a red or a blue stick. Likewise, if we reach into Bin#2 and feel for a short or a long stick, we again have a 50% chance of finding either a red or a blue stick. Once again, our knowledge of the length of the stick erases all information about the color of the stick.! In fact, our quantum sticks are even stranger than you might think. Going back to our previous example where we sort by color, if we reach into the bin with our eyes closed and “feel” for a short or long stick, we only have information about its size. It is not until we look at it that we “see” its color. Until the moment that we observe its color, the stick has equal probability of being either color. ! Of course “quantum sticks” are not real. However, particles on the atomic scale, like photons (particles of light), all subatomic particles e.g.,

Physics 234 Spring 2018! Lecture 1 May 2, 2018 Budakian electrons, protons, neutrons, and even atoms themselves obey the weird rules of the quantum world.! There can be different physical attributes that characterize a particle, like the size and color of our quantum sticks. In the quantum world, we are not allowed to possess full knowledge about these physical attributes. By measuring one particular physical characteristic, we loose information about the other.! This is completely contrary to the way Physics works in the classical world. For example, in our first Physics course, we learn how to calculate the trajectory of a projectile.!

Classically, if we are given the initial position and velocity of an object along with the magnitude and direction an external gravitational field, we can calculate, with arbitrarily high precision, the position and velocity of the particle for all later times. This analysis completely fails if we are describing a quantum particle, because our knowledge of the particle’s position introduces uncertainty in the particle’s momentum. Likewise, if we measure its velocity (momentum) with very high accuracy, then we lose all information about its position. It turns out that position and momentum are incompatible observables in the quantum world, like the color and size of our quantum sticks example—much more on this later in the course.! Don’t worry if you don’t understand how this can be. The truth is no one really knows “how” or “why” quantum mechanics works. Although we don’t necessarily intuitively understand quantum mechanics, we have a theoretical framework that lets us accurately calculate quantum phenomena. The foundations of quantum mechanics are encapsulated in 6 postulates that we will explore in this course. We will begin by studying the Stern-Gerlach (S-G) experiment, which will introduce the 6

Physics 234 Spring 2018! Lecture 1 May 2, 2018 Budakian foundational postulates of quantum mechanics. The S-G experiment is discussed in Chapter 1 of McIntyre.! It is worth stating that these postulates are not proofs. They are merely conjectures that encapsulate our understanding of how matter and light behave on the atomic scale. To date, no experiment has found a violation to these postulates. Even today, there are many research groups around the world who are exploring new paradigms to test the foundations of Quantum Mechanics, some right here in Waterloo.!

The Stern-Gerlach Experiment! The Stern-Gerlach experiment was designed to measure the magnetic moment produced by the motion of electrons around the nucleus of the atom. We can consider the motion of an electron around the nucleus as a tiny loop of current. According to the laws of electricity and magnetism, we know that current loops produce a magnetic moment, just like the north and south poles of a bar magnet. If a magnetic moment is placed in the gradient of a magnetic field, it will feel a force. Thus, one might gain insight into the motion of electrons in an atom, by studying the deflection of atoms sent through a magnetic field gradient.! It was precisely this idea that occurred to Otto Stern and Walther Gerlach in 1922, when they set out to measure the orbital magnetic moment of silver atoms. What they found was totally unexpected and changed Physics forever.! Before explaining their experiment, however, I have to step back and give you some background.!...