Third video sequence: Planck’s quantum of action

Here are the video and videoscript for the third sequence, which deals about the elementary quantity of action: Planck's quantum of action.



Hello, I’m Arjen, the Common Sense Quantum Physicist. My goal is to facilitate the understanding of the fundamentals of Quantum Physics. In the preceding sequences, we saw how a quantum particle could be represented by a spinning arrow-like object [image seq 2], which quantum physicists call a ket or state vector. Spinning arrow-like objects are filtered through regular gratings depending on the orientation and frequency of their spinning motion. So this helps to understand polarization and diffraction effects [image seq 1]. We could also deduce easily a generalized form of the Schrodinger equation [image seq 2] which simply states that the result of an arrow subtraction between two subsequent states of the arrow is always perpendicular to the arrow itself and proportional to the infinitesimal change in angle. We saw that this evolution equation is valid for any spinning arrow-like object, whether a microscopic quantum particle or a macroscopic rod or a needle or a wheel-spoke or a twirling baton or for this spinning mikado stick... So this evolution equation characterizes the rate of change of the orientation of the arrow.

The change of orientation of the arrow representing the quantum system is a very important concept in QM. When the orientation of the arrow varies, the arrow acts, it has ‘action’. An arrow whose orientation does not change is inactive. It does not play in the game. This does not necessarily mean that it does not exist but it simply does not act. Like this hanging mikado stick or like an immobile figurant in a movie scene awaiting for an actor to poke him.

So in QM, we could see the measure of the action of an object to be the measure of the variation of the orientation of the arrow representing the object. It is therefore analogous to an angle. When this arrow rotates over an angle alpha the action deployed by the arrow is alpha times a constant quantity. So we may express the action in units of an angle. [image]. For exemple, when the arrow has rotated one turn about a fixed axis, we may say that the deployed action during that turn was 2 pi in radians, or 360 in degrees. For elementary particles represented by arrows (like photons or electrons or quarks), the action is generally expressed in a unit called Planck’s quantum of action h. h is the measure of the action deployed by a quantum particle after a cycle in the meter-kilogram-second system [images]. So when you hear about Planck’s constant h, just think of an elementary arrow having rotated about 360°, it’s analogous.

The physicist Max Planck [picture] first showed the importance of this quantum of action, in the year 1900, because it showed up in a formula that characterized the thermal energy radiated by a body. So the concept action is at the origin of QM. When a quantum particle acts, it is often more convenient to talk about the energy of a quantum particle, which is just also a quantity of action but measured during a unitary time interval. For a unitary time interval of one second, the photons that are detected by our eyes have an energy a bit less than 10^15 times h. So this just means that the arrow representing the photon nearly accomplish a million of billion cycles during one second. The angle swept by the tip of the arrow representing the photon during one second is therefore a measure of the energy [images].

We could also measure the action of a particle when it travels over a space interval. We then speak about the momentum of the particle. Measuring the momentum is just another way to measure the action of an object. While the energy expresses the action of an arrow during a unitary time interval, the momentum expresses the action of an arrow during some space interval. For example, the momentum of a photon emitted by an object is analogous to the angle swept by the tip of the arrow while the photon travels over a unitary space interval. If distance is expressed in meters, the photons that are detected by your eyes generally have a momentum about a million times h [images].

So remember, the energy and momentum are just quantities of action. It is analogous to the measure of the angle swept by the tip of the rotating arrow, if that arrow represents the quantum particle or the quantum system.

It appears that there are various ways to express quantities of action in physics. Besides energy and momentum, angular momentum is also a quantity of action. It is a measure of the quantity of action deployed by a system of arrows if it is rotated over some angle about some axis. Temperature is also a quantity of action, it is a measure of the average action exchanged between arrows composing the environment. And you surely know the formula E=mc^2, which learns us that mass is also a quantity of action but measured over a tinier interval of time than energy. So, you may think of all those familiar physical quantities as measures of angles swept by the arrow (or set of arrows) representing the object. And they all relate to Planck’s quantum of action, which is analogous to the angle swept by an elementary quantum particle during one cycle.

So when you analyse a physical system, it helps to see it as a set of very tiny continuously spinning and interacting needles. That's the essence of Quantum Mechanics. The numerous mathematical formulas that characterize physical behaviour just work this idea out. Feynman cast this insight in a famous sentence "Things are made of littler things that jiggle".

Next time we'll look again at this mikado stick and at measurements you may perform on it.

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There is a specificity in quantum physics with respect to classical physics. You see this hanging stick, you see the whole of the object because the ambient light has been reflected from nearly every point of it and you receive a continuous flux of information via your eyes. The object looks like a continuity of matter. Now in Quantum Physics, you never see the quantum object as a whole. No, you receive the information on the position of the quantum object bit by bit. It works as if the only way to get information about this stick is to let it interact with another stick (or set of sticks), and notice how it affected the system.

So, before the interaction, the state of the mikado stick is unknown. We don't know where it is located, we don't know whether it is spinning, etc. It could be at any place depending on the conditions because you have not yet noticed an interaction with the detecting environment.

When I throw the second stick (the ‘detecting’ arrow) and that second stick collides with the hanging stick, I get information about the hanging stick. For example, I get information about the location of the hanging stick, because I know that if I throw this stick along the coordinate x1, and it doesn’t show up along the same line, there was a collision. The x-coordinate of the hanging stick therefore was x1. But wait, you’ll say. The coordinate of the hanging stick was not really x1! Both arrows collided at a point away from the geometrical center of the hanging stick. The real coordinate was x2 because the center of the stick was at x2. Well, that’s classical physics. In quantum physics, things work completely differently. Remember CM uses points, QM uses arrows or sticks. And arrows are spread out over their entire length, so there is an intrinsic indeterminacy in every measurement of location even if my quantum measurement gave the result x1. When I detect an arrow at x1, in fact its geometrical centre could be located at + or minus half the length of the arrow. So there’s always an indeterminacy “delta x” equal to the length of the arrow, even if my quantum measurement is very accurate.

Besides measurements of location, we may also try to measure the change of orientation of an arrow. Remember this quantity is just an angle, or a phase, analogous to the quantity of action. Just try to find ways to measure the change of orientation of the arrow. We could for example let the spinning arrow travel through a regular grating. If the arrow passes between the N points of the grating, we know that the angle swept is N times pi, with an uncertainty of plus or minus pi, which corresponds to an uncertainty in the action of plus or minus h/2. Whatever the experimental setup, we’ll never be able to determine precisely the action of the arrow better than with an uncertainty of h. We’ll never be able to beat this principle of quantum mechanics: "If the state of the arrow before the measurement is unknown, quantum measurements are always undetermined."

This indeterminacy principle was first formulated by the famous physicist Werner Heisenberg.