E is for Eigenfunction

Shamelessly aping "Talk like a Pirate Day" (9/19), some science punks are lobbying the blogosphere to recognize "Pi Day" (3/14) as "Talk Like a Physicist Day". There's even a TLAPhizz website from which this Heisenberg Uncertainty tat was lifted.

Heisenberg's Principle has been characterized as "The only certainty is uncertainty." but this is clearly false. One of the most common terms in the physicist's vocabulary is the word "eigenfunction". In a sea of quantum uncertainty the eigenfunction (from the German "eigen" which means " personal", "private", "special") stands as a welcome island of 100% predictibility--something in quantum physics that you can always count on.

Today's physicists represent everything in the world from cats to electrons as "state vectors" dwelling in an abstract realm called Hilbert space. In Dirac bra/ket notation the state vector for a cat is written |cat> and for an electron |electron>. This abstract object contains all knowable information about the results of any conceivable observation. You choose what you want to measure (and because of Heisenberg, each choice of observable demands a renunciation of knowing some complementary observable) and by a simple operation on the abstract state vector you can calculate the outcome of a measurement of the chosen observable in real life. The enormous practical success of quantum theory is dimmed only slightly by the fact that what the state vector actually "means" is still a controversial issue. We don't really know what we're doing here--but it works like a charm! After nearly a century of ruthless testing by Nobel-hungry tinkerers, quantum mechanics has given up millions of right answers, but not one single wrong prediction. 

To calculate the results of any observation--position, momentum, live, dead--one simple forms the "bra vector" corresponding to the desired observation , to form the bra/ket expression which is a number that represents the probability that the cat is here or there. (actually you have to square this number to get probability but that's an unessential detail.) The bra/ket expression is a measure of the probability of the cat being alive. Since the world is full of uncertainty, we expect the value of these expressions to lie somewhere between 1 (high probability) and 0 (low probability).

For most observables, the probabilities do indeed lie somewhere in this range. But an eigenfunction (as its name implies) is special. For an eigenfunction of an observable, the probability of observing a particular value of that observable (called its eigenvalue) is precisely 1. No uncertainty here at all.

One of the most impressive intellectual achievements of the 20th century was the working out of the exact mathematical description of nature's simplest atomic system--the hydrogen atom. The determination of the hydrogen wavefunction by a small group of European physicists in the late 1920s was a cultural high-water mark comparable to the construction of the cathedral of Notre Dame in the Middle Ages.

The hydrogen wavefunction is NOT an eigenfunction of position--the electron is spread out in space as an extended probability cloud which means its position is uncertain. But the hydrogen wavefunction IS AN EIGENFUNCTION of the observables Energy (E), Angular Momentum (L) and M, the projection of the angular momentum upon the z-axis. If you choose to make a measurement of one of these observables, either E, or L or M (or all three together) on a hydrogen atom in one of its allowed quantum states, then you will get a precise result with zero uncertainty.

The eigenfunction (zero uncertainty for certain observables) is so important that quantum wavefunctions are customarily labeled by the observables (and their associated eigenvalues) for which the results are single-valued and 100% certain. Thus the state vector for a hydrogen atom is customarily written |N, L, M> where N = 1 corresponds to the lowest energy E, N = 2 corresponds to the next-highest energy state.

Eigenfunction is actually not a quantum concept at all but has been borrowed from the old-fashioned classical physics of vibrating systems where the EF describes the "normal vibratory modes" of violin strings, drum heads, and other oscillating media including the resonant vibrations of this rocky sphere we call home. In the classical case we know exactly what the mathematics means; in the quantum case we are not so sure.

Recently Dean Dauger (formerly of UCLA) has released an elegant shareware program (Macintosh only, and now iPhone) that produces colored, 3D animations of the hydrogen wavefunction for a large number of its E, L, M eigenvalues. Dauger's program also allows you to form simple quantum superpositions (non-eigenfunctions) of some of these states and view the results.

Dauger's program (called Atom in a Box) lets you not only talk (like a physicist) but lets you make your own home movies of quantum wavefunctions, probably the closest humans will get in my lifetime to visualizing "what atoms really look like". Sophisticated enough to satisfy a real physicist yet easy enough for a normal person to use to toy with the structure of the universe at a very basic level. It's a great way to learn about quantum mechanics. Many thanks, Dean, for this sweet little tool. 

Sexual Paradox

"[Sexual Paradox] gives rise to reproductive strategies which are polarized and divergent, although the roles are also highly complementary. There is thus a temptation for one sex to defect against the other and to try to assert reproductive dominance. Many religions and religious texts appear to be an attempt by males to gain control of culture and assert reproductive dominion over women by a variety of coercive and oppressive methods. These are contrary to nature and the evolutionary process. Although there are strong mutual and complementary forces in human sexual partnership, and male choice of female partners is a strong selective force, female reproductive choice is paramount in the sexual symmetry breaking. Astute female reproductive choice is the most powerful evolutionary dynamic in the flowering of human culture and super-intelligence and is key to human futures."

(pix of Christine Fielder & Chris King

authors of "Sexual Paradox")

Thanks to Iona Miller who briefly describes this book:

"This book is about sexual paradox, the nemesis of our pretensions, yet the genesis of our living destinies. It demonstrates that sexual paradox is at the core of all descriptions of reality, lurking in the quantum realm and in the relationship between body and mind as much as in our hormone-steeped bodies and rising pulses. It presents the idea of sexual paradox, not as an inscrutable icon for the vagaries of sexual intrigue, but as a cosmic principle spanning the widest realms, from physics, though biology, to our social futures."

Dale Pendell. the Psychobotany Guy

I met Dale Pendell many years ago in Santa Cruz, CA at his home near Neary Lagoon, a compact little wetlands sanctuary flanked by a large sewage disposal plant--odd juxtaposition of the wild and the tame. As befits a psychobotanist, Dale's house was crawling with exotic plants inside and out, his front porch crowned by a gigantic Datura plant whose sinister-looking white trumpet flowers put out a sweet, seductive scent. Mercury Press had just published Dale's first book in the celebrated "Pharmako" trilogy which he was promoting while researching book #2.

Dale later moved from Santa Cruz into the hinterlands so I now see him infrequently but each time we meet I get a lot of juice from this guy. A while back we gave a seminar together at the pagan gathering "Pantheocon" at a big hotel in San Francisco on Valentine's Day. Dale talked about plants and I presented an early version of quantum tantra. At this same conference a pretty pagan boldly opened her blouse for me to show a paw print she had had tattooed between her breasts.

Then I met Dale a second time with his partner Laura in a mock Moorish auditorium at the "Sacred Elixers" conference in San Jose where he performed "Amrita" and other poems to an appreciative audience that included Earl and Sherri Crockett as well as Sasha and Ann Shulgin.

Dale Pendell's "Pharmako" trilogy surveys the history, composition and usage of a wide variety of psychoactive plants. The books themselves, like their author, are hard to categorize, part botany, part history--poetry, dialog, chemistry, commerce and physics. Reading one of Dale's books is like exploring a labyrinthine jungle with a witty and knowledgeable native guide. This guy really knows his stuff. Pendell calls the study of plant powers "the Poison Path" for which his trilogy is an indispensable field guide. This noteworthy trio of books is strange, scholarly and full of good sense.

1. Pharmako/Poeia: Plant Powers, Poisons and Herbcraft
   
2. Pharmako/Dynamis: Stimulation Plants, Potions, Herbcraft
3. Pharmako/Gnosis: Plant Teachers and the Poison Path

The best introduction to Dale Pendell's life and work is Emily Green's LA Times interview "The Poet of Plants". Like Nick Herbert, Emily Green finds Dale Pendell impossible to categorize. Dale and his writings are in a class by themselves.

Recently Dale published "Inspired Madness", a book about Burning Man, and a collection of conversations with the controversial UCSC professor Norman O Brown, the author of "Love's Body". From a collection of Dale's plant poetry: "Living With Barbarians" comes this:

Every plant is a teacher--

but as in every crowd,

there are always

a few loudmouths.

Heisenburger Cat

Quantumtantra blog's homage to a cute and enduring Internet theme--the LOL cat meme, also known as "I can has cheezburger?" named after the most popular LOL cat site. Today's tribute stars my cat Onyx and Werner Heisenberg who said: "In these quantum experiments we do not observe nature herself but nature exposed to our method of questioning." The aim of quantum tantra is to devise brand new ways of questioning nature with the hope of eliciting new, more intimate and more erotic human experiences of the natural world than our current ways of doing physics provide.

Forbidden Knowledge

We scientists are accustomed to investigating forces of nature ranging in strength from the whisper of a single photon to energies that rip entire galaxies apart. Our calling is to uncover the secrets of the universe no matter how difficult, terrifying or dangerous the quest might be. But are there questions too dangerous for science to ask, places no scientist should be allowed to go? Do there exist--in the words of the old B-grade monster movies--THINGS THAT NO MAN SHOULD KNOW?

Apparently so. The US Government has published a big list of them. Science keep out!

What are these realms too dangerous for science to investigate? Some Brobdingnagian force more powerful than nuclear fusion? Some hideous virus that could end all life on Earth? What monstrous horror lurks behind the forbidden door? Weak hearts should read no further. Are you ready?

The forbidden substance is---marijuana.

Yes, ma'm. Scientists can't study marijuana (without becoming criminals). And they can't investigate dozens of other perfectly normal molecules classified as Schedule 1 by our paternalistic bosses. These hysterically dangerous substances, legislated off limits to scientists, include mescaline, LSD, psilocybin. MDMA and more than a hundred other molecules you've never heard of. And marijuana too, of course. Hands off pot, Doc, unless you want to be locked up with murderers, armed robbers and rapists for a long, long time.

Are we living in the Middle Ages or what?

Fortunately some reasonable folks are trying to change this absurd situation for the better. One such group that's working to make science legal again is MAPS (Multidisciplinary Association for Psychedelic Studies) who are lobbying Congress, obtaining funding for research, and educating the public about the advantages of seeking knowledge rather than legislating ignorance about these substances "too dangerous for science to touch". 

MAPS used to be located in Florida but has recently moved to Ben Lomond, CA, only a few minutes drive from my Quantum Tantra Ashram in Boulder Creek. Last year I stopped by, introduced myself and got to know these science-friendly strangers who persuaded me to contribute an article to their newsletter about Nick's first psychedelic experience way back in the '60s. It's called "Doctor Quantum Drops Acid" and is available in pdf form at http://www.maps.org/news-letters/v17n2-html/index.html. Lots of other good information on the MAPS site about how one might successfully investigate these substances that some people find glorious, life-enhancing and even divine while others find them so fearsome they will put clamps on normal human curiosity and make outlaws of scientists for merely doing their job--which is to explore the unknown, no matter how dangerous.

It's a bad situation. Call up your Congress critter and ask why he/she hates science so much.

The Babies Are Here

My ex-Marine brother "Duke", formerly of San Diego, CA, now living in Montana on the edge of Glacier National Park sends these pictures of the new neighbors in his backyard. The moose have just had children. What do you call baby moose, anyway? Fawns? Cubs? Mooselets?

^^^My brother from Montana writes that they are elk not moose. And that the mom & dad are called cow and bull; baby elk are called calves. Thanks, Duke.^^^

What Do Quantum Tantrikas Want?

"This is the sort of connection with Nature we're looking for, Nick. 

Can quantum theory deliver the goods?"

"Well. I certainly hope so. honey.

Now would you pass me that hookah?"

[Visuals by Iona Miller]

Nick's Favorite Quantum Textbooks

I've been out of (formal) school for several decades so am not familiar with modern textbooks on quantum mechanics. There certainly must be newer books that teach the subject better than the books I learned from, but I still like and remain faithful to my first loves.

One of the best books on how to actually use quantum theory to solve practical problems is Leonard Schiff's Quantum Mechanics . In the first chapter Schiff (who was a fellow native of Columbus, OH) poses a problem which still continues to stimulate my imagination. One ping pong ball is glued to a table top; a second ping pong ball is suspended 10 ping pong ball diameters above the first. Schiff's question: when the first ball is released, what is the maximum number of bounces it executes before falling off the ball fastened to the table?

In classical physics it is theoretically possible for you to locate the second ball EXACTLY above the first so the number of bounces would be infinite. But in quantum physics (Heisenberg's Uncertainty Principle), if the position of the ball is exactly specified, the momentum uncertainty is infinite so you cannot be sure of getting even one bounce. What you must do is find the right mix of position and momentum uncertainties that maximizes the number of bounces. I solved this problem a long time ago and forgot the answer.

The British physicist P.A.M. Dirac's classic textbook The Principles of Quantum Mechanics introduced the whimsical and useful Dirac "bra" and "ket" notation for quantum states. The quantum state of, say, an electron can be represented as a ket |electron> which represents a vector in Hilbert space which encodes all that can physically be known about that electron. Now you get to choose what you want to know. If you want to know about the electron's position x, you multiply the electron ket by the position bra <x| to obtain the bra-ket expression <x|electron> which is the electron's QUANTUM WAVEFUNCTION in position space. Likewise if you want to know the electron's momentum p, you multiply the electron ket by the momentum bra Ep| to obtain the bra-ket expression <p|electron> which is the electron's quantum wavefunction in momentum space. Like so many other underappreciated intellectual inventions (Arabic numerals, for example), the Dirac bra/ket notation allows lesser minds than Paul Dirac to do very smart things without having to be very smart.

<x|electron> is Dirac notation for electron wavefunction in position space.

<p|electron> is Dirac notation for electron wavefunction in momentum space.

This simple notation conceals piles and piles of complicated mathematics that you don't really want to think about--and thanks to Dirac, you don't have to.

In person, Dirac had the reputation of acting "extremely rational"-- like Spock in Star Trek. One of many Dirac stories has him writing equations on the board and then asking the class if there were any questions: "Yes" said a student, "I am confused about your derivation of XYZ". "That is not a question, " Dirac replied, "That is a fact."

In graduate school none of us understood what quantum theory actually meant as we were learning how to use it to solve problems increasingly more difficult than Schiff's ping pong balls.

One text book that attempted to deal with what quantum theory meant was David Bohm's Quantum Theory which for a physics book has an exceptionally high ratio of explanatory text to mathematical equations. One of the themes of Bohm's book was to expand on John von Neumann's proof that the statistical predictions of quantum theory could not be reproduced by some underlying deterministic hidden variable theory. Bohm shows in his book why hidden variables are impossible. Ironically, later in his career, David Bohm constructed a successful hidden variable model of quantum theory that still has its defenders and which inspired Irish physicist John Stewart Bell to devise his justly famous non-locality theorem

A few years ago, Australian philosopher of mind David Chalmers put out the call for the best book on quantum theory suitable for beginning philosophy students (my own Quantum Reality was rightly judged as too advanced). My vote (and the book that won) was for the book Ghost in the Atom which is a series of BBC interviews with various physicists about the meaning of quantum theory. The lucid introduction by physicist Paul Davies is alone worth the price of this book. A wonderful introduction for ordinary people to the dilemmas faced by modern physicists as they attempt to understand "how Nature does it."

Quantum Dating

Many Worlds Dating: Ever feel like you are living in several parallel worlds at once? Also known as the Player's Universe, this model is the source of the saying, "So many dates; so little spacetime." Can you say "Sex Addict"? This is what is meant by "turn up the volume." Victimizers may be former victims of Einsteinean dating where "it's all relative." High maintenence inhabitants of this dating universe include rap and rock stars. movie stars, and celebutantes. Or, those who are legends in their own minds.

Many Worlds is a realm of infinite possibilities, likely probabilities (scoring), lots of uncertainty (rejection, being kicked to the curb, dissed) with many futures (criminal charges, undesirable pregnancy, STDs) depending from each thread. This universe requires lots of protection. You can get it at the drugstore or a handy public bathroom. Also be sure to always carry a toothbrush.

What could better describe the world of quantum dating than these parallel possibilities? Swinging? Fetish? Austin Powers, Bayyyby? In the bountiful multiverse, everything that can happen, even the improbable (cybernetic devices, virtual sex), does happen. Actually, the forbidden is mandatory. Kinky. Once two systems interact, their possibility waves become entangled, so changes in one instantly change the other (voodoo, mood swing, phone sex).

^^^ A brief excerpt from Iona Miller's Quantum Dating. Pix is from Iona's Magick album.^^^

The Wordsworth Conjecture

Despite the quibblings of philosophers, it's a good bet that other humans possess minds like your own. And it's also highly probable that animals (like these 5 hummingbirds feeding from the hand of Abagail Alfano of Pine, LA) also enjoy some sort of inner experience--as in William Blake's memorable lines:

Who knows but every bird 

that cuts the airy way is

an immense world of delight 

closed to our senses five?

But how far down the scale of life do we have to go before there is no inner experience and we are dealing with a mere machine? Are plants conscious? What about bacteria, algae or viruses? What about the parts of me? Is my hand or my heart or my stomach conscious? Are the mitachondia or ribosomes inside my cells enjoying an inner experience "closed to my senses five"? One of the greatest unknowns of our present science is our ignorance concerning the existence and the extent of other minds. 

One of the major mistakes of medieval thinkers was their gross underestimation of the size and variety of the physical world. This cozy Earth, the seven celestial spheres above, plus Dante's circles of Hell beneath our feet: that was the full extent of the universe in the medieval imagination. In 1600 the Church burnt Bruno at the stake for preaching a plurality of worlds. Now we take for granted the real existence of billions upon billions of worlds scattered through a universe of almost inconceivably large dimensions.

Might there exist inconceivably large numbers of mental worlds all around us hidden from human view by our lack of a science of mind to mind communication? In his famous lyric "Intimations of Immortality", Romantic poet William Wordsworth expressed this very notion of the existence of many invisible minds:

And never for each other shall we feel

As we may feel, till we have sympathy

With nature in her forms inanimate,

with objects such as have no power to hold

Articulate language.

In all forms of things there is a mind.

One aim of quantum tantra is to take seriously this "Wordsworth conjecture" and to look for ways to verify it by devising a "Wordsworth machine" based on quantum theory which would permit direct connection between your own mind and the mind of another. My intent is to make Star Trek's "Vulcan Mind Meld" real and to use this mind-merge machine to explore the invisible universe of other minds just as our advanced telescopes and accelerators are exploring the hidden wonders of the purely physical realm. 

Physics for Beginners

PHYSICS FOR BEGINNERS

I remove

Her outer coverings

She shows me

the very center of Her Being.

When words falter

I reach for my mathematics

Mostly She eludes description.

I remove Her inner coverings

She shows me

a deep Nothingness

simpler and more powerful

than all of my Somethingnesses

put together.

I catch Her eye

She smiles

She opens Her Paradox

and takes out Her Mystery.

First Exposure

Today was interviewed at Bruce Damer's "God Farm" in Boulder Creek by Andy and Jennifer from Bend, OR (friends of Iona Miller's) who are making a video on the entheogenic origins of Christmas. Background music was a rare (for Santa Cruz) thunderstorm which blackened the sky. (Pix = Bruce and Jennifer). Bruce, Allan Lundell and I played the 3 Wise Men bringing gifts to the Infant God--and hoping to get gifts back, gifts that reflect our deepest wishes. Allan's wish: how would an enlightened community actually behave? Bruce (who lived for a spell in Prague) told of the traditional Czech visit of Saint Nicholas accompanied by two devils to whom you were supposed to give propitiatory gifts. I wanted from the Little God some new physics tool that would open new doorways into Nature--brand new applications of quantum physics that would directly connect our human minds to the several other minds around us. I wished to be given some concrete ways to use quantum physics to expand human consciousness. I wanted some new ideas for this blog which is intended to be a record of my efforts (as one of BC's token Wise Men) to create a totally new physics in which human consciousness is an essential ingredient. Bon voyage. Bon appetite. Bon chance.