Lecture 6 | Modern Physics: Quantum Mechanics (Stanford)

TL;DR

Photon polarization is the simplest system for grasping how quantum thinking differs from classical thinking. A photon carries a polarization vector, a little flag pointing in any direction in the plane perpendicular to its motion. Yet when measured by a polarizer, it either passes through or does not, revealing a conflict between continuous symmetry and discrete outcomes.

Transcript

This program is brought to you by Stanford on iTunes at stanford University. Please visit us at iTunes.stanford.edu. They behave oddly. I've showed you some uh interference of light waves when they go through slits and that sort of thing. uncertainty principles for particles and a photon is a particle. So it uh shares the uncertainty principle with... Read More

Key Insights

• Photon polarization is the simplest quantum system for studying the fundamental difference between classical and quantum thinking, because once direction and momentum are fixed, the only remaining property to discuss is the direction the photon is polarized.
• A photon carries a polarization vector, described as a little flag that can point in any direction in the xy plane, the plane perpendicular to the photon's direction of motion along the z-axis.
• A polarizer acts as both a preparer and a measuring apparatus: it prepares a photon with a particular polarization and detects whether a photon's polarization matches its axis by whether the photon passes through.
• The official definition of polarization is the direction of the electric field of the electromagnetic wave, which a grid of wires establishes by only allowing current to oscillate in one direction.
• A wire grid reflects waves whose electric field aligns with the wires like a mirror, but is effectively absent for the perpendicular direction, so the transmitted wave emerges polarized.
• Basic polarizer behavior is fully deterministic and not inherently non-classical: a vertically polarized photon passes a vertical polarizer, a horizontal one is blocked, and classical flagged bullets could mimic this.
• Quantum weirdness begins with rotational symmetry: because a polarizer can be held at any angle, a photon polarized at an angle may or may not pass through, doing so only with a probability.
• The core mystery is a conflict between continuity and discreteness: a photon can be polarized in a continuum of directions, yet measurement yields only two possible outcomes, pass or not pass, without splitting into pieces.

Questions & Answers

Q: Why is photon polarization the simplest system for studying quantum mechanics?

Once you fix a photon's direction of motion down the z-axis and its momentum, the only thing left to discuss is the direction in which it is polarized. This single polarization degree of freedom forms a kind of mini quantum system. Susskind argues more insight can be gained from it than from almost any other simple quantum mechanical system, because it captures the fundamental difference between classical and quantum thinking without the full complexity of all of quantum mechanics.

Q: What is a photon's polarization vector?

The polarization vector is described as a little flag that the photon carries with it, pointing in a direction within the plane perpendicular to the photon's motion. If the photon moves along the z-axis, the flag lies in the xy plane and can point in any direction there. The polarization vector is always perpendicular to the direction of motion of the photon, and its direction is officially defined as the direction of the electric field.

Q: How does a wire grid polarize light?

In a grid of wires, the electric field can only drive current in one direction, along the wires. As far as that direction goes, the grid acts like a mirror, reflecting waves whose electric field is aligned with it, because the driven currents produce a reflected wave. In the perpendicular direction the current cannot flow, so it is as if the grid is not there at all and the wave passes straight through, leaving the transmitted wave polarized.

Q: How can a polarizer act as both a preparer and a detector?

When a photon passes through a polarizer, it comes out polarized along that polarizer's axis, so the device prepares a photon with a particular polarization. Sending that photon through a second polarizer checks its polarization: if the axes match, the photon passes through again. By monitoring whether the photon goes through a given polarizer, you measure whether its polarization is aligned, so the same apparatus works as a measuring device or detector.

Q: What happens when a vertically polarized photon hits a horizontal polarizer?

If a photon is created vertically polarized and the apparatus is horizontally polarized, the photon will not go through the apparatus. It may get reflected or it may just get absorbed, one or the other. Whichever it does, by monitoring whether the photon passes through, you have measured whether its polarization is vertical or horizontal. This behavior is entirely deterministic and, on its own, contains nothing especially non-classical.

Q: Why is basic polarizer behavior not yet considered quantum weirdness?

For photons aligned or perpendicular to a polarizer, the outcome is completely deterministic: a vertically polarized photon passes a vertical polarizer, and a horizontal one is blocked, end of story. Susskind notes you could easily imagine a classical setup, like funny shaped bullets with little flags that pass through matching slits and get blocked otherwise. Because classical particles could reproduce this, there is nothing out of the ordinary until rotational symmetry is added to the picture.

Q: How does rotational symmetry introduce quantum behavior?

Because you can hold a polarizer vertically or horizontally, the rotational symmetry of physics lets you construct a polarizer pointing at any angle, producing photons polarized at an angle. The kicker is that a photon polarized in some non-vertical, non-horizontal direction may or may not pass through another polarizer, going through only with a probability. Setting up a theory of these probabilities and how they depend on polarizer orientation is quantum mechanics itself.

Q: What is the conflict between discreteness and continuity in photon polarization?

A photon can be polarized in a continuum of directions, so there seem to be infinitely many possible states. Yet when you detect it with a polarizer, there are only two possibilities: it either goes through or it does not. The photon is never broken into pieces with part passing and part not. This tension between continuous polarization directions and discrete two-outcome measurements, symmetry versus discreteness, is described as the central mystery of quantum mechanics.

Summary

In this video, the instructor discusses the concept of polarization in quantum mechanics. He explains how the polarization of a photon can be described by a vector, and how it can be measured using polarizers. The video also explores the properties of polarization vectors and how they relate to classical wave behavior. Additionally, the instructor introduces the concept of observables and their eigenvalues and eigenvectors in the context of photon polarization.

Questions & Answers

Q: What is the simplest system for studying the difference between classical and quantum thinking?

The polarization of a photon is considered the simplest system for studying the difference between classical and quantum thinking.

Q: How can the polarization of a photon be visualized?

The polarization of a photon can be visualized as a flag perpendicular to the motion of the photon in a plane perpendicular to the motion.

Q: What happens when a photon passes through a polarizer?

When a photon passes through a polarizer, it becomes polarized in the direction of the polarizer's axis.

Q: How can the polarization of a photon be measured?

The polarization of a photon can be measured by sending it through a polarizer and observing whether it passes through or not.

Q: What is the significance of symmetry in analyzing photon polarization?

The symmetry of rotation in the XY plane allows for the construction of polarizers pointing at any angle, resulting in different polarization outcomes for the photon.

Q: What are the mathematical representations of polarization states?

The X polarization state can be represented by the vector [1, 0] or as the state labeled X. The Y polarization state can be represented by the vector [0, 1] or as the state labeled Y.

Q: What is an observable and how is it related to polarization?

An observable is a quantity that can be measured or detected in an experiment. In the context of polarization, the observable is associated with the polarization direction and its measurements correspond to definite outcomes.

Q: How can the observable linked to polarization be represented mathematically?

The observable associated with polarization in the XY plane can be represented by the matrix [1 -1] when it acts on the X polarization state. Similarly, it can be represented by the matrix [0 1] when it acts on the Y polarization state.

Q: What is the physical significance of orthogonal polarization states?

Orthogonal polarization states represent mutually exclusive outcomes of an experiment and are used to determine probabilities of specific polarization directions.

Q: How can the probability of a photon passing through a polarizer be calculated for an arbitrary angle?

The probability of a photon passing through a polarizer at an arbitrary angle can be calculated by taking the inner product (or dot product) between the initial polarization state and the polarization state associated with the polarizer, and squaring the result.

Takeaways

In this video, we learned about polarization in quantum mechanics and how it can be analyzed using polarizers. We saw that the polarization of a photon can be represented by vectors, and that different polarizers can produce different polarization outcomes. We also saw the importance of symmetry and observed how observables are related to polarization measurements. Overall, polarization provides a simple system for studying quantum mechanics and highlights the differences between classical and quantum thinking.

Summary & Key Takeaways

• Susskind introduces photon polarization as the simplest quantum system for contrasting classical and quantum thinking. If a photon travels along the z-axis with fixed momentum, the only remaining feature is its polarization, a little flag pointing somewhere in the perpendicular xy plane.

• Polarizers work like a wire grid: the electric field drives current only along the wires, so that direction reflects like a mirror while the perpendicular direction passes freely, producing a polarized wave defined by its electric field direction. A polarizer both prepares and measures polarization.

• Deterministic pass-or-block behavior is not non-classical, but rotational symmetry allows polarizers at any angle. Angled photons pass only probabilistically, never splitting, yielding just two outcomes. This conflict between continuous polarization directions and discrete measurement results is the heart of quantum mechanics.