Simple quantum mechanical oscillatory behavior

Well, yes, it is time to get something posted.  Here is a thought to ponder before any equations are written down:  How many different faces does the harmonic oscillator have in elementary quantum mechanics?  That is, in how many different cases does simple harmonic oscillator (SHO) behavior show up? 

In answering this question, I'm going to leave out the usual SHO Hamiltonian, which is the normal starting point for creating the quantum harmonic oscillator raising and lowering operators--that is, for quantizing the energy of the oscillator. Instead, I'm just looking for oscillatory behavior, meaning either sine waves themselves or equations whose solutions are sine waves.

The first one that pops up is in the post below, from my Weinberg class notes of 15 Sept. 1998. In parentheses next to the word "integrable!" Nope, those are exponential functions, non-oscillating, no i in the exponent.  But it is true that the wave function for the particle in a box (infinite square well potential) is a sine wave. Wave functions are not "real," however, so this is not called  a harmonic oscillator. Squaring the wave function gives the probability of finding the particle at different places in the box, but the particle hitting the walls of the box is in a way the opposite behavior of a particle in simple harmonic motion.  Or instead of opposite, let's call it complementary to the behavior of a harmonic oscillator: A constant velocity that instantly reverses itself at the walls, whereas SHO motion has the particle slowing down and then reaching zero velocity at the (unnecessary) walls, where the restoring force on the particle reaches a maximum and the particle reverses its direction.

But, in fact, since we have elastic collisions of the particle with the walls, we have the equivalent of little springs causing the interaction between the particle and the walls, and the restoring force of these reaches a maximum as the particle reverses direction.

Next, as far as simple oscillations are concerned, the time-dependent superposition of two stationary states shows sine wave behavior, even when the stationary states are not those of the harmonic oscillator.

AND, in solving the Schrodinger equation by the separation of variables method, the resulting time-dependent equation is a simple harmonic motion differential equation. Yes, it's a first order equation, but the factor of -i makes it the first-order version of the usual second-order SHO equation of motion.  This is something that is rarely pointed out--I don't recall seeing it ever pointed out--so it may be considered unimportant, or may be considered important only from a more esoteric point of view, such as  the quantum propagator point of view.  But I consider it important and will be discussing it again, soon!

So the answer to the question posed at the beginning of this post is: three. But it seems like I left one out...  Now it's December 1, and I do have to mention the coherent state, the Gaussian wave packet solution to the Schro equation that oscillates back and forth in the SHO potential energy well.  And now December 11, how could I forget to mention normal modes, since a "mode" is an abstraction of a harmonic oscillator or standing wave? I guess because normal mode analysis is not unique to quantum mechanics. But it is what Planck used when he analyzed the electromagnetic waves in a heated cavity in thermal equilibrium with the atoms in the cavity walls, thereby discovering energy quantization.

How do you get a quantum oscillator to oscillate?

9:09 a.m. Friday 11 March, still 1999+17, but who can really believe it?   

J.S. Townsend discusses the absence of oscillatory behavior in the quantum harmonic oscillator, p.261. Transitions between energy states in the quantum oscillator produce radiation—sorry, not just the quantum oscillator, but the electrically charged quantum oscillator.  That’s why it’s so strange to read these two sentences in Townsend’s book:  “A harmonic oscillator in an energy eigenstate is in a stationary state. Thus it will not exhibit the characteristic oscillatory behavior of a classical oscillator.”

The quantum harmonic oscillator is the behavior Planck discovered, the behavior on which Bohr’s H atom model depends, the very basic QUANTUM behavior itself.  So to read it stated so bluntly is strange:  the quantum harmonic oscillator does not oscillate!

Townsend is in my opinion a pedagogical genius for simply stating this idea.  Also, he writes very well and his book is organized in a way I like. After the above two statements, he says, “Time dependence for the harmonic oscillator results from the system being in a superposition of energy eigenstates with different energies.”  His emphasis, not mine. But, of course, in Q.M. the emphasis is always on the strangeness of its requirement that coherent superpositions exist. Why is this a requirement in Q.M.? (My emphasis this time.)

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This absence of time dependence relates to the elementary quantum mechanics class  at UT-Austin I took in 1998 taught by Weinberg (yes, "the" Weinberg).  I was an older-than-average student, and asked more questions than the others, who asked very few themselves. Weinberg was in a seminar lecturing mode that didn't encourage questions from the class. One of my questions was about the quantum harmonic oscillator.  After Weinberg had written the equation for the allowed energy states, and drawn the standard energy level picture on the board, I asked something about the stability of the states. He said that they were "absolutely stable."  I asked if the quantum oscillator could "emit energy," and Weinberg said yes, "if you give it a way of emitting energy."  These quotes are from my class notes of September 15, 1998.  I didn't ask for an example of giving the quantum oscillator a way of emitting energy.  Time-dependent perturbation theory is generally the way to do it, or to calculate any time-dependent quantity that can be considered to be the result of a small disturbance to the otherwise stable system.

From my notes made after class on September 3rd, I found this comment of mine:  "How was The Weinberg? Accessible, articulate in a simple-language kind of way, and thoughtful. Not pedantic. So far (3 days worth), so good."

I plan to devote a lot of writing in this blog to how you can give an electrically charged harmonic oscillator a way of emitting electromagnetic energy.  I'll start by finishing up the Planck normal modes electromagnetic cavity radiation discussion sometime soon. This is not a perturbation problem, it's an equilibrium problem, and as discussed in the above  journal entry of mine, involves a superposition of states.

It's interesting to think about the difference in the time dependence of the oscillator resulting from a superposition of different energy eigenstates and the time dependence of a system like a Schrödinger's cat state, which results from the system's inherent time dependence--the unstable nature of the radioactive atom in the box--rather than the imposed superposition time dependence of the oscillator.

Here's an update on 22 July 1998+18:  I've been reading in Griffiths Intro to QM book again recently, and the very first and very simplest example of time dependence is discussed in his Example 2.1, page 29, 2nd ed.  Using an initial, t=0, wave function ɸ(x,0) = a superposition of two states, and multiplying this by the usual time-dependent complex exponential function exp[-itE_n/hbar]  gives a probability density with cos[(E₂ – E₁)t/hbar] time dependence. If we allow (E₂ – E₁) = hbarω, this is a quantum harmonic oscillator.

Griffiths says, "notice it took a linear combination of states (with different energies) to produce motion."  Well, yeh, and look what kind of motion!  But is it emitting energy?  No.  It's just a stable, single-frequency oscillator. What is oscillating?  The probability density.  Can you call that motion? Of what?  "The system."  Later!

N*3245 announced 50 years ago. Ratner's particle?

Nuclear Scientists Find New, Very Heavy Particle

CHICAGO (AP) — The most massive nuclear particle yet known—nearly four times as massive as the proton—has been discovered by the Argonne National Laboratory scientists.

The team of physicists who made the discovery call the new particle N-asterisk-3245.  They say N-asterisk-3245 is a mass of frozen energy—and the number “3245” stands for the amount of its energy, 3,245 million electron volts.

Its discoverers—Alan D. Krusch, John R. O’Fallon, Keith Ruddick and Steven Kormanyos, all of the University of Michigan, and Lazarus G. Ratner of Argonne—published their discovery in Physical Review Letters, a scientific journal.

They said N-asterisk-3245 is a proton in an energized state.  It belongs to a family of particles called nucleon resonances, they said.

Its life is only one-ten thousandth of a millionth of a millionth of a millionth of a second.

So far as is known, nucleon resonances do not exist in nature—only in atom smashers.

--Associated Press article printed on page 2 of the Pine Bluff Commercial on 18 April 1966. The date is the 11th anniversary of Albert Einstein's death.  The 3,245 MeV energy or "mass" of N*3245 is better expressed today as 3.245 GeV (3.245 billion electron-volts).

A Higgs boson, which is considered to be experimental proof for the existence of the Higgs field, is also a "resonance," lasting only about 10-to-the-minus-24th of a second and identifiable only by its predicted decay products such as muons, photons and other detectable particles. More specifically, the Higgs boson is a resonance in the proton-proton scattering cross-section, occurring at an energy of 125 GeV.

More on Kirchhoff's law & black-body radiation

There are a few important distinctions to be made when considering the meaning of Kirchhoff's radiation law. Firstly, it describes only thermal or black-body radiation, but these two terms themselves are not always used as synonyms as I'm using them here. So this is one point of confusion.  

People who work in the field of quantum optics tend to use "thermal spectrum" as a shorter substitute for "black-body spectrum." A true black-body spectrum is only present when the body in question is in radiative equilibrium with its surroundings. People other than quantum optics theorists consider thermal radiation to be electromagnetic radiation emitted due to an object's temperature, with no requirement of equilibrium.

So lets say "thermal spectrum" refers to a true black-body spectrum, and "thermal radiation" refers to any radiation produced by an object due to its temperature. Thermal radiation in this sense has a spectrum that is nearly identical to the black-body spectrum.

So: Let's say "thermal radiation" is light or other electromagnetic radiation being emitted by an object solely due to the object's temperature, and that this radiation has an approximately "thermal spectrum," and not worry about it not being exactly a black-body spectrum.

There are three common sources of visible thermal radiation that I can think of:  the sun, incandescent light bulbs, and heating elements on electric stoves and ovens.  The objects we see illuminated by the sun and incandescent bulbs preferentially absorb or reflect different parts of the spectrum, so when we look at the objects around us we don't see light that has a thermal spectrum. We see whatever regions of the spectrum the objects reflect. And you should see the spectral composition of sunlight at earth's surface section of the Wikipedia entry on sunlight for more info on the spectrum of visible light we normally observe from the sun.

Every "body" and everybody does produce some type of thermal radiation, however.  Usually, it's not in the visible spectrum.  Near-black-body radiation is produced by objects that have the same temperature as their surroundings.  When we're in a room held at constant temperature--which is usually the case--the temperature of the walls of the room and the air in the room is about the same.  This walls are producing very low frequency thermal radiation.

We warm-blooded creatures produce our own heat, and thus are not in equilibrium with our surroundings. Like other "hot" objects that are sources of their own heat, including the three mentioned above, we produce thermal radiation.

So the take-home message about thermal radiation is that a lot of care and effort go into producing an ideal absorber-radiator whose thermal radiation has the black-body spectrum.  One way of producing such a spectrum is the radiation-in-a-box method, which I've discussed previously here and will be discussing again.

See this site for a thermal spectrum graph that changes as you enter different temperatures.  If you enter room temperature of about 300K the graph shows no spectrum because the emissive power is so low that it doesn't register on the scale of the graph.  (Okay, 300K is a lower temp than the slider scale allows, but the interactive nature of this particular site is worth trying out.)

Thermal and black-body radiation are most often described in terms of emissive power: the rate of emission of electromagnetic energy per unit area of the emitting object, expressed as a function of the temperature of the body and the wavelength (or frequency) of the radiation.  But in discussing Kirchhoff's law, we can also talk about coefficients of emission and absorption, and it's often not clear whether a coefficient or emissive power or some combination of the two is being discussed.  For instance:

"Kirchhoff's law of radiation  A law stating that the emissivity of a body is equal to its absorptance at the same temperature"

--A Dictionary of Physics, Oxford University Press, 2nd edition, 1996.

This unhelpful definition says what is often said in regard to Kirchhoff's radiation law, that a good absorber of radiation must also be a good emitter.  It may also make you wonder "What's the point?"  We have equal emission and absorption of radiation as a required equilibrium condition, so in this sense it's a circular definition. Also, the most interesting thing about Kirchhoff's law is its universal applicability.  It applies to all materials.  A definition needs to take that into account.  (There is a more recent definition given at the Oxford dictionary of physics website, but the newer part of the definition is not readily viewable.) 

The Wikipedia definition of Kirchhoff's radiation law was updated recently and gives the best definition I've seen.  Planck himself calls it the law of proportionality of the emissive to the absorptive power. The coefficient of absorption mentioned in the Wikipedia definition (and shown in the Russian encyclopedia definition in my previous post) provides this proportionality.

One more thing.  You may be wondering about the universal nature of Kirchhoff's law--that it applies to all materials--and how it can apply materials that just catch on fire or melt when they get very hot.  Wood and plastic should immediately come to mind.  Kirchhoff's law still applies to these within the range of temperatures where the material isn't beginning to change its molecular structure due to being heated.  That's why viewing thermal radiation in the higher temperature region, where the material becomes red hot, requires some kind of metal that can survive being heated to high temperatures. 

The significance of Kirchhoff's radiation law

"Next to Kirchhoff's theorem of the proportionality of emissive and absorptive power, the so-called displacement law, discovered by and named after W. Wien, which includes as a special case the Stefan-Boltzmann law of dependence of total radiation on temperature, provides the most valuable contribution to the firmly established foundation of the theory of heat radiation."  --Max Planck, 1901.  (Apparently he was kind of like William Faulkner in his use of commas to extend a sentence almost to its breaking point.)

I haven't discussed Kirchhoff's law until now, because different discussions of it have seemed to be saying different things, and I couldn't get a clear idea of its significance. Now I'm ready to discuss it, starting with quotations from various authors.

From the statement above, copied from Planck's paper

On the Law of Distribution of Energy in the Normal Spectrum

Max Planck

Annalen der Physik
vol. 4, p. 553,

and as you can see from the quotes below, Kirchhoff's law was directly responsible for various attempts to find the mathematical form of the thermal radiation spectrum, culminating in Planck's formula derived from his idea of abstract, normal-mode, quantized electromagnetic oscillators.

(Just an aside to be investigated later: an oscillator is also a clock. This is something to consider when thinking about the difference in classical and quantum oscillators.)

The first description below is from Russia, last one is from France, and in between are two from our well-known, oft-quoted writers in Los Alamos and Austin, repsectively.

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Kirchhoff's radiation law says that the ratio of emissive power ε(λ,T) of bodies to their absorptivity α(λ,T) is independent of the nature of the radiating body.

This ratio is equal to the emissive power of the black body ε₀(λ,T) (because its absorptivity is equal to 1) and depends on the radiation wavelength λ and on the absolute temperature T:

The function ε₀(λ,T) is given by Planck’s radiation formula.

Kirchhoff’s radiation law is one of the fundamental laws of thermal radiation and does not apply to other types of radiation. The law was established by G. R. Kirchhoff in 1859 on the basis of the second law of thermodynamics and was subsequently confirmed experimentally. According to Kirchhoff’s radiation law a body that, at a given temperature, exhibits a stronger absorptivity also exhibits a more intensive emission. One example:  when a platinum plate partially covered with platinum black is heated to incandescence, the blackened end will glow much brighter than the light end.

--from The Great Soviet Encyclopedia, 3rd Edition. S.v. "Kirchhoff's Radiation Law." Retrieved February 18, 2016 from http://encyclopedia2.thefreedictionary.com/Kirchhoff%27s+Radiation+Law

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In 1860 Kirchhoff derived a general relation between the radiative and absorptive strengths of a body held at a fixed temperature T.  According to Kirchhoff’s law the ratio of the radiative strength to the absorption coefficient for radiation at wavelength λ is the same for all bodies at temperature T, and defines a universal function F(λ, T).  This led to the abstraction of an ideal blackbody for which the absorption coefficient is unity at every wavelength, corresponding to total absorption.  Thus F(λ, T) characterizes the radiative strength at wavelength λ of a blackbody at temperature T.  The problem was to determine the universal function F(λ, T).  

--Peter W. Milonni, The Quantum Vacuum, pages 1 & 2. © 1994 Academic Press Inc.

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Physicists in the last decades of the nineteenth century were greatly concerned to understand the nature of black-body radiation—radiation that had come into thermal equilibrium with matter at a given temperature T.  The energy ρ(υ,T)dυ per volume at frequencies between υ and υ+dυ had been measured, chiefly at the University of Berlin, and it was known on thermodynamic grounds that ρ(υ,T) is a universal function of frequency and temperature, but how could one calculate this function?

--Steven Weinberg, Lectures on Quantum Mechanics, p. 1., 1st ed., Cambridge University Press, © S. Weinberg 2013.

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The great conceptions of Nature are those which are free from specifical properties of objects, at the price of an abstract formulation which offers dreadful problems of interpretation. This is the case for the fundamental law of dynamics, for the law of gravitation, for the law of conservation of energy, for the second principle of thermodynamics.

…

The Kirchhoff theorem on the independence of the law of black-body radiation upon the material nature of the cavity is some kind of “miracle” which has to be included in every theory, even if some explanation for it is not given. This is indeed the gateway through which Planck proceeded when he introduced abstract oscillators whose physical nature was left undefined.

These oscillators are in some way present in Quantum Mechanics which appears in many circumstances as a general model of harmonic oscillator.  A striking example being given by the universality of the response theory of quantum systems: there is a unique general form for the fluctuation-dissipation theorem. This simplicity of the quantum case is in sharp contrast to the with the high specificities of the classical case.

--Simon Diner, from “The wave-particle duality as an interplay between order and chaos,” a presentation given at a symposium held at the University of Perugia during April 1982 in honor of the 90th birthday of Louis de Broglie. The proceedings of the symposium were published as The Wave-Particle Dualism: A Tribute to Louis de Broglie on his 90th Birthday, © 1984 by D. Reidel Publishing Company.

Post Script:  Kirchhoff is also known for his current and voltage laws, which might better be called rules, used in analyzing electric and electronic circuits:  "sum of currents into a junction is zero" (ingoing current equals outgoing current, a consequence of electric charge conservation), and "sum of voltages around a closed loop is zero" (voltage drops around loop must be equal to and opposite in sign from voltage sources such as batteries in the loop; a consequence of energy conservation).  I started my physics training in basic electronics, so I learned about and used Kirchhoff's current and voltage rules before I learned about Kirchhoff's radiation law. Planck (top of page) referred to the latter as Kirchhoff's theorem.