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Well,

Let be the Hamiltonian for the quantum mechanical system of interest. As always, we are interested in the eigenvalues and eigenfunctions which are solutions to the Schrödinger equation.

  Assume a similar system exists for which non-degenerate solutions (eigenvalues and eigenfunctions) are already known or easily obtained.

  Let the difference between and be denoted as .

Introduce a physically motivated perturbation parameter .

The Schrödinger equation for the system becomes
 
Since depends on , the eigenfunctions and eigenvalues of must also depend on . Thus and can be expanded in a power series with respect to .

The following abbreviations were employed in the expansion.

To obtain the energy or wavefunction of a system you merely sum the corrections. Obviously the convergence of these power series becomes a key issue since the summations must be truncated in practice.

Using these expansions of and in the Schrödinger equation (Eqn 10) for the system of interest we have
 
Expanding this we obtain terms with various powers of on both the l.h.s. and r.h.s. If equality is to hold for all , all terms on the l.h.s. of Eqn. 15 must equal to those on the r.h.s. for a given power of lambda. This gives rise to the so called ``kth order Schrödinger equations.'' 
    

And I'm sure my esteemed colleague Ivan will back me on that!

I have also been extremely fortunate in the tributes that have come my way. But this singular honor, the Jefferson Lecture, is for me a high point, and my gratitude could not be greater.

Among the darkest times in living memory was the early part of 1942 -- when Hitler's armies were nearly to Moscow; when German submarines were sinking our oil tankers off the coasts of Florida and New Jersey, within sight of the beaches, and there was not a thing we could do about it; when half our navy had been destroyed at Pearl Harbor. We had scarcely any air force. Army recruits were drilling with wooden rifles. And there was no guarantee whatever that the Nazi war machine could be stopped.

It was then, in 1942, that the classical scholar Edith Hamilton issued an expanded edition of her book, The Greek Way, in which, in the preface, she wrote the following:

I have felt while writing these new chapters a fresh realization of the refuge and strength the past can be to us in the troubled present.... Religion is the great stronghold for the untroubled vision of the eternal; but there are others too. We have many silent sanctuaries in which we can find breathing space to free ourselves from the personal, to rise above our harassed and perplexed minds and catch sight of values that are stable, which no selfish and timorous preoccupations can make waver, because they are the hard-won permanent possessions of humanity....

When the world is storm-driven and the bad that happens and the worse that threatens are so urgent as to shut out everything else from view, then we need to know all the strong fortresses of the spirit which men have built through the ages.

In the Rotunda of the Capitol hangs a large painting of forty-seven men in a room. The scene is as familiar, as hallowed a moment in our history as any we have.

John Trumbull's Declaration of Independence has been a main attraction on Capitol tours for a very long time, since 1826. It draws crowds continuously, as it should, every day -- about three to five million people a year. It's probably been seen by more Americans than any painting ever -- and the scene as portrayed never took place.

Trumbull said it was meant to represent July 4, 1776, and that's the popular understanding. But the Declaration of Independence was not signed on July 4. The signing began on August 2, and continued through the year as absent delegates returned to Philadelphia. No formal signing ceremony ever took place.

The scene comes closer to portraying June 28, when Thomas Jefferson submitted his first draft of the Declaration. But then, too, there was no such dramatic gathering.

The room is wrong, the doors are in the wrong place. The chairs are wrong. (They were Windsor chairs of the plainest kind.) There were no heavy draperies at the windows. The decorative display of military trophies and banners on the back wall, is purely Trumbull's way of dressing the set.

Yet none of this really matters. What does matter greatly -- particularly in our own dangerous, uncertain time -- is the symbolic power of the painting, and where Trumbull put the emphasis.

The scene proclaims that in Philadelphia in the year 1776 a momentous, high-minded statement of far-reaching consequence was committed to paper. It was not the decree of a king or a sultan or emperor or czar, or something enacted by a far-distant parliament. It was a declaration of political faith and brave intent freely arrived at by an American congress. And that was something entirely new under the sun.

And there Trumbull has assembled them, men like other men, each, importantly, a specific, identifiable individual.

The accuracy is in the faces. John Adams, Thomas Jefferson, and Benjamin Franklin were painted from life. Before he was finished, Trumbull painted or sketched thirty-six of the faces from life. It took him years and he spared no expense, because he wanted it right. He wanted us to know who they were.

Adams, Jefferson, and Franklin stand front and center exactly as they did in the real drama of the Revolution.

A number who signed the Declaration are not shown. Several who did not sign are present.

Most conspicuous by his absence is George Washington who had departed Congress the year before to take command of the army.

Lord Bolingbroke, the eighteenth century political philosopher, said that "history is philosophy teaching by examples." Thucydides is reported to have said much the same thing two thousand years earlier.

Jefferson saw history as largely a chronicle of mistakes to be avoided.

Daniel Boorstin, the former Librarian of Congress, has wisely said that trying to plan for the future without a sense of the past is like trying to plant cut flowers.

One might also say that history is not about the past. If you think about it, no one ever lived in the past. Washington, Jefferson, John Adams, and their contemporaries didn't walk about saying, "Isn't this fascinating living in the past! Aren't we picturesque in our funny clothes!" They lived in the present. The difference is it was their present, not ours. They were caught up in the living moment exactly as we are, and with no more certainty of how things would turn out than we have.

Nor were they gods. Indeed, to see them as gods or god-like is to do disservice to their memories. Gods, after all, don't deserve a lot of credit because they can do whatever they wish.

Those we call the Founders were living men. None was perfect. Each had his human flaws and failings, his weaknesses. They made mistakes, let others down, let themselves down.

Washington could be foolhardy and ill-tempered. Adams could be vain, irritable, Jefferson evasive, at times duplicitous. And even in their day, many saw stunning hypocrisy in the cause of liberty being championed by slave masters.

They were imperfect mortals, human beings. Jefferson made the point in the very first line of the Declaration of Independence. "When in the course of human events..." The accent should be on the word human.

And of course their humanity is not evident only in their failings. It's there in Adams's heart-felt correspondence with his wife and children, in Jefferson's love of gardening, his fascination, as he said, in every blade of grass that grows.

Washington had a passionate love of architecture and interior design. Everything about his home at Mt. Vernon was done to his ideas and plans. Only a year before the war, he began an ambitious expansion of the house, doubling its size. How extremely important this was to him, the extent of his esthetic sense, few people ever realized. He cared about every detail -- wall paper, paint color, hardware, ceiling ornaments -- and hated to be away from the project even for a day.

The patriotism and courage of these all-important protagonists stand as perhaps the most conspicuous and enduring testaments to their humanity. When those who signed the Declaration of Independence pledged their "lives, their fortunes, their sacred honor," that was no mere verbiage. They were putting their lives on the line. They were declaring themselves traitors to the King. If caught they would be hanged.

Stephen Hopkins of Rhode Island, who suffered from palsy, is said to have remarked as he signed his name, "My hand trembles, but my heart does not."

Hopkins was a grand old figure who had seen a lot of life. You can't miss him in the Trumbull painting. He's at the back with his broad-brimmed Quaker hat on. In after-hours he loved to drink rum and expound on his favorite writers. "He read Greek, Roman, and British history, and was familiar with English poetry," John Adams wrote. "And the flow of his soul made his reading our own, and seemed to bring recollection in all we had ever read."

We must never forget either how hard they worked. Nothing came easy. Nothing. Just getting through a day in the eighteenth century meant difficulties, discomforts, and effort of a kind we seldom even think about.

But it is in their ideas about happiness, I believe, that we come close to the heart of their being, and to their large view of the possibilities in their Glorious Cause.

In general, happiness was understood to mean being at peace with the world in the biblical sense, under one's own "vine and fig tree." But what did they, the Founders, mean by the expression, "pursuit of happiness"?

It didn't mean long vacations or material possessions or ease. As much as anything it meant the life of the mind and spirit. It meant education and the love of learning, the freedom to think for oneself.

Jefferson defined happiness as "tranquility and occupation." For Jefferson, as we know, occupation meant mainly his intellectual pursuits.

Washington, though less inclined to speculate on such matters, considered education of surpassing value, in part because he had had so little. Once, when a friend came to say he hadn't money enough to send his son to college, Washington agreed to help -- providing a hundred pounds in all, a sizable sum then -- and with the hope, as he wrote, that the boy's education would "not only promote his own happiness, but the future welfare of others …." For Washington, happiness derived both from learning and employing the benefits of learning to further the welfare of others.

John Adams, in a letter to his son John Quincy when the boy was a student at the University of Leiden, stressed that he should carry a book with him wherever he went. And that while a knowledge of Greek and Latin were essential, he must never neglect the great works of literature in his own language, and particularly those of the English poets. It was his happiness that mattered, Adams told him. "You will never be alone with a poet in your pocket."

The Revolution was another of the darkest, most uncertain of times and the longest war in American history, until the Vietnam War. It lasted eight and a half years, and Adams, because of his unstinting service to his country, was separated from his family nearly all that time, much to his and their distress. In a letter from France he tried to explain to them the reason for such commitment.

I must study politics and war [he wrote] that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture in order to give their children a right to study paintings, poetry, music, architecture, statuary, tapestry, and porcelain.

That was the upward climb envisioned for the good society in the burgeoning new American republic. And Adams was himself vivid proof of the transforming miracle of education. His father was a farmer, his mother almost certainly illiterate. But with the help of a scholarship, he was able to attend Harvard, where, as he said, he discovered books and "read forever."

His Harvard studies over, Adams began teaching school at Worcester, then virtually the frontier. One crystal night, twenty years before the Declaration of independence, he stood beneath a sky full of stars, "thrown into a kind of transport." He knew such wonders of the heavens to be the gifts of God, he wrote, but greatest of all was the gift of an inquiring mind.

But all the provisions that He has [made] for the gratification of our senses ... are much inferior to the provision, the wonderful provision, that He has made for the gratification of our nobler powers of intelligence and reason. He has given us reason to find out the truth, and the real design and true end of our existence.

He had decided to study law. "It will be hard work," he told a friend. "But the point is now determined, and I shall have the liberty to think for myself."

Of all the sustaining themes in our story as a nation, as clear as any has been the importance put on education, one generation after another, beginning with the first village academies in New England and the establishment of Harvard and the College of William and Mary. The place of education in the values of the first presidents is unmistakable.

Washington contributed generously, some $20,000 in stock to the founding of what would become Washington and Lee University in Virginia. His gift was the largest donation ever made to any educational institution in the nation until then, and has since grown to a substantial part of the endowment.

Jefferson founded the University of Virginia. But then it may be fairly said that Jefferson was a university unto himself.

The oldest written constitution still in use in the world today is the Constitution of the Commonwealth of Massachusetts, drafted by John Adams in 1778, just two years after the Declaration of Independence and fully a decade before our national Constitution. In many respects it is a rough draft of our national Constitution. But it also contains a paragraph on education that was without precedent. Though Adams worried that it would be rejected as too radical, it was passed unanimously. Listen, please, to what it says:

Wisdom and knowledge, as well as virtue, diffused generally among the body of the people being necessary for the preservation of their rights and liberties. [Which is to say that there must be wisdom, knowledge, and virtue or all aspirations for the good society will come to nothing.] And as these depend on spreading the opportunities and advantages of education in various parts of the country, and among the different orders of the people [that is, everyone], it shall be the duty [not something they might consider, but the duty] of legislatures and magistrates in all future periods of this commonwealth to cherish the interests and literature and the sciences, and all seminaries of them ... public schools, and grammar schools in the towns.

And he goes on to define what he means by education. It is literature and the sciences, yes, but much more: agriculture, the arts, commerce, trades, manufacturers, "and a natural history of the country." It shall be the duty, he continues,

to countenance and inculcate the principles of humanity and general benevolence, public and private charity, industry and frugality, honesty [we will teach honesty] ... sincerity, [and, please note] good humor, and all social affections, and generous sentiments among the people.

What a noble statement!

Years before, while still living under his father's roof, Adams had written in his diary, "I must judge for myself, but how can I judge, how can any man judge, unless his mind has been opened and enlarged by reading."

They were nearly all young men in 1776, it should be remembered, young men who believed, as Thomas Paine proclaimed, that the birth of a new world was at hand.

Jefferson was thirty-three, Adams, forty, Benjamin Rush, the Philadelphia physician, was all of thirty when he signed the Declaration of Independence. Rush, one of the most interesting of them all, was a leader in the anti-slavery movement, a leader in prescribing humane treatment for the insane, and the first to champion the elective system in higher education.

When George Washington took command of the army, he was forty-three. He had never led an army in battle before in his life, any more than the others had had prior experience as revolutionaries or nation builders.

And what of Franklin? Franklin, oldest and wisest, is for me a special case.

I met my first revisionist historian when I was six.

His name was Amos and he was a mouse, an eighteenth century church mouse to be exact, one of twenty-six children who with their mother and father lived in Old Christ Church in Philadelphia. I can never be in Old Christ Church without wondering if perhaps some of Amos's line are still there, back behind the paneling.

Amos, who took up lodging in Benjamin Franklin's fur hat, is the narrator of a little book called Ben and Me by Robert Lawson.

Most so-called historians have had Franklin all wrong, according to Amos. "Ben was undoubtedly a splendid fellow, a great man, a patriot and all that," he writes, "but he was undeniably stupid at times and had it not been for me -- well, here's the true story..."

I was six, as I say, and I was hooked. I learned all about Philadelphia, printing presses, electricity, Franklin stoves, and the Palace of Versailles. I got to know Benjamin Franklin and, like Amos, relished his company.

And that was the start. I learned to love history by way of books. There was The Matchlock Gun by Walter D. Edmonds, The Last of the Mohicans, with those haunting illustrations by the N. C. Wyeth, the Revolutionary War novel Drums by James Boyd, with still more Wyeth paintings.

That was in the day when children were put to bed when sick, and I remember lying there utterly, blissfully lost in those illustrations.

The first book I ever bought with my own money was a Modern Library edition of Richard Henry Dana's Two Years before the Mast. I was fifteen years old. In a book shop in Pittsburgh, I picked up the book from a table, opened to the start of chapter one, read the first sentence, and knew I had to have that book:

The fourteenth of August was the day fixed upon for the sailing of the brig Pilgrim, on her voyage from Boston, round Cape Horn, to the western coast of America.

Growing up in Pittsburgh, I had never seen the ocean or heard the cry of a sea gull or smelled salt air.

I read Kenneth Robert's Oliver Wiswell, Thornton Wilder's The Bridge of San Louis Rey, Frederick Lewis Allen's Only Yesterday, and A Night to Remember by Walter Lord. I thank my lucky stars for Robert Abercrombie, who taught history my senior year in high school and made Morison and Commager's The Growth of the American Republic required reading. In the college years, I must have read a half dozen novels on the Second World War, including Herman Wouk's The Caine Mutiny and Irwin Shaw's The Young Lions.

Bruce Catton's A Stillness at Appomattox, a graduation present, started me reading about the Civil War, and started me thinking that maybe some day I might try writing something of the kind.

I loved all those books, and they're all still in print, still being read, which is no mystery. They're superbly well done, wonderfully well written.

There should be no hesitation ever about giving anyone a book to enjoy, at any age. There should be no hesitation about teaching future teachers with books they will enjoy. No harm's done to history by making it something someone would want to read.

We are what we read more than we know. And it was true no less in that distant founding time. Working on the life of John Adams, I tried to read not only what he and others of his day wrote, but what they read. And to take up and read again works of literature of the kind we all remember from high school or college English classes was not only a different kind of research, but pure delight.

I read Swift, Pope, Defoe, Sterne, Fielding, and Samuel Johnson again after forty years, and Tobias Smollett and Don Quixote for the first time.

I then began to find lines from these writers turning up in the letters of my American subjects, turning up without attribution, because the lines were part them, part of who they were and how they thought and expressed themselves.

But we do the same, more often than we realize. Every time we "refuse to budge an inch," or speak of "green-eyed jealousy," or claim to be "tongue-tied," we're quoting Shakespeare? When we say "a little learning is a dangerous thing," or "to err is human," or observe sagely that "fools rush in where angels fear to tread," we're borrowing from Alexander Pope, just as when you "slept not a wink," or "smell a rat," or "turn over a new leaf," or declare "mum's the word," you're quoting Cervantes?

When young Nathan Hale was hanged by the British as a spy in New York in 1776, he famously said as his last words, "My only regret is that I have but one life to lose for my country." That's a line from the popular play of the eighteenth century, Cato by Joseph Addison, a play they all knew. Washington, who loved the theater, is believed to have seen it half a dozen times.

Imagine how it must have been for Nathan Hale, about to be hanged. Who, in such a moment, could possibly think of something eloquent to say. I think he was throwing that line right back at those British officers. After all, Cato was their play.

I imagine him delivering the line, "I only regret that I have but one life to lose for my country."

One needs to read the great political philosophers -- Hume, Locke, Ferguson, Montesquieu -- whose writings had such profound influence on the founders. Yet there is hardly a more appealing description of the Enlightenment outlook on life and learning than a single sentence in a popular novel of the day, A Sentimental Journey by Laurence Sterne.

What a large volume of adventures may be grasped within this little span of life by him who interests his heart in everything.

The stimulation, the motivation, the hard work and pleasures of writing history are mainly in the material. It's the finding and figuring out that keep you in pursuit. And you never know... you never know where you will find something, see something that's gone unnoticed, or make some unexpected connection that brings things into focus in a new or different way.

The truth of history is the objective always. But the truth isn't just the facts. You can have all the facts imaginable and miss the truth, just as you can have facts missing or some wrong, and reach the larger truth.

As the incomparable Francis Parkman wrote:

Faithfulness to the truth of history involves far more than a research, however patient and scrupulous, into special facts. Such facts may be detailed with the most minute exactness, and yet the narrative, taken as a whole, may be unmeaning or untrue. The narrator must seek to imbue himself with the life and spirit of the time. He must study events in their bearings near and remote; in the character, habits, and manners of those who took part in them. He must himself be, as it were, a sharer or spectator of the action he describes.

"I hear all the notes, but I hear no music," is the old -piano teacher's complaint. There has to be music. History at best has to be literature or it will go to dust.

The work of history -- writing history, teaching history -- calls for mind and heart. Empathy is essential. The late J. H. Plumb, the eminent British historian, said that what is needed is more "heart-wise" historians.

What happened? And why? Who were those people? What was it like to have been alive then, in their shoes, in their skins? Of what were they afraid? What didn't they know?

Studying his face in the mirror, John Adams decided, "I am but an ordinary man. The times alone have destined me to fame." He was fishing. He was anything but ordinary and it is not possible to understand what happened in that tumultuous, protean time without knowing and understanding him and the others.

There are, of course, great sweeping tides in history -- plague, famine, financial panic, the calamities of nature and war. Yet time and again, more often than not history turns on individual personality, or character.

I am presently at work on a book about the Revolution, with the focus on Washington and the army in the year 1776, which in the last months was the nadir of the fortunes of the United States of America, when the army was down to little more than three thousand men. By December, by all signs, the war was over and we had lost. Fortunately Washington did not see it that way. Had it not been for Washington and his little ragtag army, the Declaration of independence and all it promised would have truly been "a skiff made of paper."

There are no paintings or sketches of the soldiers done at the time. Most that we have are by Trumbull, himself a veteran of the war, but his were all painted afterward.

It's in the surviving diaries, journals, letters, pension files, in descriptions posted for deserters, that those in the ranks begin to emerge as flesh-and-blood individual men caught up in something far bigger than they knew.

There was Jabez Fitch, for example, a Connecticut farmer with eight children, who liked soldiering and kept a diary describing the war as he saw it day by day. There was young John Greenwood, a fifer boy, who at age sixteen walked all alone 150 miles from Maine to Boston to join up with Washington's army. And Mathias Smith, a deserter, who was described as: "a small smart fellow, a saddler by trade, grey headed, has a younger look in his face, is apt to say, 'I swear, I swear!' And between his words will spit smart; had on a green coat, and an old red great coat; he is a right gamester, although he wears something of a sober look."

"Greece," wrote Edith Hamilton, "never lost sight of the individual." And neither should we, ever.

They were hungry, starving some of them, and without warm clothes as winter set in. Not all were patriots. Not all were heroes. Not all came home. But they were once as alive as you and I.

"Posterity who are to reap the blessings," wrote Abigail Adams, "will scarcely be able to conceive the hardships and sufferings of their ancestors."

History is -- or should be -- a lesson in appreciation. History helps us keep a sense of proportion.

History teaches that there is no such thing as a self-made man or woman, that we are all shaped by the influences of others, including so many we've never seen because they are back there in history.

History teaches that nothing happens in isolation, or without cause and effect, and that nothing ever had to happen as it did.

History teaches tolerance, and the value of common sense, and as Voltaire (and who knows how many others) observed, common sense is anything but common.

History is about high achievement, glorious works of art, music, architecture, literature, philosophy, science and medicine -- not just politics and the military -- as the best of politicians and generals have readily attested. History is about leadership, and the power of ideas. History is about change, because the world has never not been changing, indeed because life itself is change.

History is the course of human events. And it must therefore be, if truthful, about failure, injustice, struggle, suffering, disappointment, and the humdrum. History demonstrates often in brutal fashion the evils of enforced ignorance and demagoguery. History is a source of strength, a constant reminder of the courage of others in times more trying and painful than our own. As Churchill reminded us, "We have not journeyed all this way... because we are made of sugar candy."

History is filled with voices that reach out and lift the spirits, sometimes from the distance of centuries.

Is it possible to imagine not learning from the wisest, most thoughtful people who shaped the world, or to fail to take heart from manifest courage?

Is life not infinitely more interesting and enjoyable when one can stand in a great historic place or walk historic ground, and know something of what happened there and in whose footsteps you walk?

For a free, self-governing people something more than a vague familiarity with history is essential, if we are to hold on to and sustain our freedom.

But I don't think history should ever be made to seem like some musty, unpleasant pill that has to be swallowed solely for our civic good. History, let us agree, can be an immense source of pleasure. For almost anyone with the normal human allotment of curiosity and an interest in people, it is a field day.

Why would anyone wish to be provincial in time, any more than being tied down to one place through life, when the whole reach of the human drama is there to experience in some of the greatest books ever written.

I guess if I had to boil it down to a few words, I would say history is a larger way of looking at life.

One of our innumerable advantages as a nation and a society is that we have such a specific moment of origin as the year 1776. And that we know who the Founders were -- indeed we know an immense amount about an immense number of those at all levels who in that revolutionary time brought the United States of America and the reality of freedom into being.

But while it is essential to remember them as individual mortal beings no more perfect in every way than are we, and that they themselves knew this better than anyone, it is also essential to understand that they knew their own great achievements to be imperfect and incomplete.

The American experiment was from its start an unfulfilled promise. There was much work to be done. There were glaring flaws to correct, unfinished business to attend to, improvements and necessary adjustments to devise in order to keep pace with the onrush of growth and change and expanding opportunities.

Those brave, high-minded people of earlier times gave us stars to steer by -- a government of laws not of men, equal justice before the law, the importance of the individual, the ideal of equality, freedom of religion, freedom of thought and expression, and the love of learning.

From them, in our own dangerous and promising present, we can take heart. As Edith Hamilton said of the Greeks, we can "catch sight of values that are stable because they are the hard-won possessions of humanity."

Blessed we are. And duty bound, to continue the great cause of freedom, in their spirit and in their memory and for those who are to carry on next in their turn.

There is still much work to be done, still much to learn.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness ....

Who's with me on this?