Another look at the Schroedinger equation

I just read this article on PhysOrg about a paper recently published on the origins of the Schroedinger equation. One interesting thing I learned is that, in the classical wave equation for matter waves, the phase of the wave determines the amplitude. However, in Schroedinger's equation, the amplitude and phase of the wave are coupled to one another.

The authors of the PNAS article demonstrate that this coupling leads to the linearity of the Schroedinger equation, which is one of its most important properties. If it were not linear, I'm not sure that the mathematics would have turned out so relatively simple in quantum mechanics; i.e. it may not have been formulated in terms of linear algebra.

Unfortunately, I think the PhysOrg article was a bit misleading. They repeatedly referred to the classical wave equation when speaking of the Hamilton-Jacobi equation. To my knowledge, the classical wave equation and the HJ equation describe different things. More importantly, the classical wave equation is linear.

Is it better to be absolutely truthful in popular science articles or to minimize the amount of jargon and smooth over some minute but important points?

A diverse basis for science

The sum of our experiences, environment, and genetic predispositions form the basis for how we view and interpret the world. This is a principle of many philosophies and helps to explain the broad diversity in human behavior.

The practical view of science is of a rigid structure built upon basic assumptions and prior knowledge. After the proper application of "the scientific method," this foundation leads us to new discoveries. Roughly speaking, the method goes as such: we start from our current state of knowledge, make a hypothesis followed by observations, formulate a proper model that accommodates the data, then draw our conclusions while taking into account the prior information. This outline must follow the rules of logic and not contradict what we already know to be true (and if it does, we place this contradiction under extreme scrutiny until the contradiction is resolved).

This generalization of the scientific method is confounded by the inherent variability in the assumptions from which it starts. In reality, every one possesses a different set of beliefs regarding scientific inquiry. This is exactly analogous to the variety of metaphysical beliefs held by people across the globe. And, just as this variety gives rise to the diversity of people, it leads scientists to different interpretations of theirs and others' work.

Rather than enforce a common basis for the pursuit of science, scientists ought to respect the basic diversity within their own field. Science is more than rote application of a formula; it engages the scientist to the point that discovery becomes an act of self-expression. A study is flavored with the thoughts and feelings of the people involved and can not be separated from them. Once this is understood, we can see that science is a very human endeavor and not the cold, calculated formula known as the scientific method.

When is a conclusion good enough?

I've been doing a lot of reading recently on modern probability theory, Bayesian analysis, and information theory. One of the central tenets of these theories is that any proposition has associated with it a degree of plausibility, i.e. a probability of being true, that reflects the amount of information available. The proposition, "the sky is blue" is extremely plausible since it is based off of daily self-observations and confirmed by others. As another example, I judge the proposition, "thirty million Americans have blue eyes" as plausible based on the knowledge of the current population of the United States and my own observations on the frequency of encountering blue-eyed people. However, this is not as likely to be true as the first statement.

A conclusion in a scientific paper is nothing more than a proposition and thus possesses its own degree of plausibility. The information available to the reader for determining the plausibility of the conclusion is the data presented in the paper and all previously published work on the same topic. Of course, other factors weigh in, such as prejudices for particular theories and affinities or dislikes for the authors on the paper. Temporarily placing these other factors aside, I wonder, "when does a conclusion possess a large enough degree of plausibility to be considered true?"

Of course, a definite answer doesn't exist. No scientific proposition can be true with 100% certainty and it is silly to think that we can even assign a threshold probability for evaluating a scientific paper's correctness. Just imagine a paper successfully passing through the peer-review process so long as it is evaluated to be 78.63% or more true by its reviewers. But the question remains relevant. Science is a culture and every culture has criteria by which it evaluates claims.

In a closely-related post I wrote about the fallacies that workers commit while evaluating other work. But I find it much more difficult to identify the criteria for establishing the truthfulness and quality* of research. Unfortunately, I don't think I'll be able to fulfill my full potential as a scientist until I am capable of doing so.

Note: The problem of defining quality has been approached at great length by American author Robert Pirsig in his popular novel "Zen and the Art of Motorcycle Maintenance." One of his primary arguments is that Quality is actually an undefined construct present at the seminal moment when an observation is made and processed by the brain. People may know if something possesses Quality, but it is inherently undefinable.

Why I like Dr. Ben Goldacre

I've been trying to keep my posts to once a week on Wednesdays, but sometimes I simply just want to share something interesting. In a recent Bad Science post, Dr. Ben Goldacre discussed sampling error in relation to unemployment figures in the UK. The article itself is interesting, but I found the description of systematic sampling error especially amusing:

Firstly, you’ll be familiar with the idea that a sample can be systematically unrepresentative: if you want to know about the health of the population as a whole, but you survey people in a GP waiting room, then you’re an idiot.

How not to argue in science

I'd like to expand a little on yesterday's post. I'm beginning to better understand what constitutes proper debate of a scientific work. Whether the following logic is actually practiced by most researchers is questionable, but this is nevertheless an interesting and important point.

Data from a study provides information that does one of three fairly obvious things to a conclusion: it increases, decreases, or leaves unaffected the likelihood that the conclusion is correct. I place emphasis on the word likelihood because any given conclusion can not be demonstrated as being correct with 100% certainty, and I highly doubt that conclusions can be proven false with certainty.

I think that this—the likelihood that a conclusion is correct given all information—as well as the competency with which an experiment was performed are the two objects open to debate within science. The debate becomes unscientific when researchers and journal reviewers perform the following errors:

1. Assigning too much weight to prior information, thus making the likelihood that another work's results are correct less likely then it perhaps should be.
2. As a corollary to the first point, workers would be in error if they didn't properly balance the weighting of all prior information. For example, the media, in their coverage of climate change, has been chastised by some for giving equal attention to climate change skeptics as they do to proponents. This is because the proportion of scientists against climate change is significantly fewer than the proportion who see it as a true occurrence.
3. Assuming that a finding is false given prior information or prejudices. If one accepts that a finding can not be false but rather highly unlikely, then arguing to reject a journal article because it contradicts previous findings is itself fallacious. The wider scientific community should (with its more balanced opinions) be a better interpreter of the likelihood that the claims are real.

Of course, if these errors were corrected, they could very well lead to many more published works, which would in turn dilute the field. As a result, grants may be harder to obtain (since they are in part based on published works) and the dissemination of knowledge would become greatly impaired; there would simply be too much information to analyze.

Uniqueness of probability allows for assertion

But until it had been demonstrated that [the probabilities] are uniquely determined by the data of a problem, we had no grounds for supposing that [the probabilities] were possessed of any precise meaning.--E. T. Jaynes and G. Larry Bretthorst, from "Probability Theory: The Logic of Science (emphasis mine)

I take this to mean that any experiment whose results are debated is under question because either 1) the logic of the critics is flawed or 2) there is not enough information in the data to reach the argued for conclusion to a large degree of plausibility. Uniqueness of the probabilities assures this. Furthermore, there can be no argument for the truth or falsehood of the claims.

Coming to conclusions

In the introduction of E. T. Jaynes's Probability Theory: The Logic of Science, Jaynes states

...the emphasis was therefore on the quantitative formulation of Polya’s viewpoint, so it could be used for general problems of scientific inference, almost all of which arise out of incomplete information rather than ‘randomness’.

As I learn more about the field of sensing, I find that this is the mentality, whether acknowledged by a practitioner or not, that is adopted when coming to a conclusion about the interpretation of data. The uncertainty involved in coming to a conclusion is not because the measurement process is inherently random but rather that one has not collected enough data to say whether this conclusion is true or false.

And in the light of Bayesian analysis, one will never be able to claim with 100% certainty that the conclusion is true (or false, for that matter).

Notes on Bayesian Analysis

This afternoon I encountered one of those rare moments when I had little to do since I was waiting on input from a number of collaborators, so I read through half of this site on intuitively understanding Bayes' Theorem. Here are some things that I learned or that were made clearer than my previous understanding:

1. The outcome of Bayesian analysis is a modification of the prior (the probability of an event that is known before the analysis) which produces the posterior (the probability of an event given certain conditions and dependent upon the prior).
2. Bayesian analysis requires three pieces of information: the prior and two conditional probabilities (a true positive and a false positive outcome).
3. If the two conditional probabilities are equal, the prior is unmodified and equals the posterior. This is because the result of the test is uncorrelated with the outcome.
4. The degree to which the prior is modified can be described by the concept of differential pressure, i.e. the relative difference between the two conditional probabilities. The process of changing the prior due to differential pressure is known as selective attrition.
5. People use spatial intuition to grasp numbers. Teachers can use this and the idea of natural frequencies to their advantage to teach difficult concepts.
6. Related to number 5, the way in which given information is presented in a word problem (e.g. percentages vs. ratios) will affect the percentage of correct scores.

Is Bayesian analysis related to Kant's a priori and a posteriori knowledge?

The coffee-spilling uncertainty principle

If, when carrying your cup of coffee back from the coffee machine to your office, you hold the full cup close to you, you're less likely to spill any over the sides of the cup, but if you do spill, it is more likely to spill on your clothes. However, if you hold it further out from you, the coffee is much more likely to spill over the side of the cup, but much less likely to spill on your clothes.

The uncertainty in spilling your coffee multiplied by the uncertainty in spilling your coffee on you is always greater than Planck's constant times the amount of time spent at work.

Absolutes vs. relatives

Though my advisor has stressed this for the entirety of my grad school career, I today finally appreciated the significance of relative measurements over absolute ones.

An absolute measurement is one in which a value is extracted from a data set that is physically important in a particular model. A relative measurement, on the other hand, is one that extracts the effect of varying a parameter amongst two or more data sets.

Model specific parameters are obtained from absolute measurements. Curve fitting is usually performed to find the values of parameters. Alternatively, relative measurements establish relationships between two variables. For example, the reading on a scale will increase proportionally with the mass added on top of it. From this observation, one can infer that weight is linear with mass. A constant, namely the acceleration of a particle due to gravity at the earth's surface, is needed to obtain the absolute value of the weight from a single measurement.

The practical problem with absolute measurements is that they require certain standards to have any significance. At the start of graduate school, I would often puzzle over why a parameter from a curve fitting routine would so often differ from theory. I would often vary different parameters in my calculation and struggle in vain to determine which independent quantity I had measured "wrongly." However, I failed to realize that each measurement was against some standard. A time is measured relative to an internal clock in a circuit; a length is measured relative to a ruler; mass is measured relative to a scale which was calibrated relative to some mass standard.

From the above it seems that the nature of measurement itself is a relative process, and as such a measurement can not be "wrong." If standards differ between two measurements, the measured variable will differ as well. And no one can say which measurement produced the "correct" value. Both conclusions are correct so long as they are logically consistent with how they are derived from the measurement.

I am aware of the definitions of the second and other fundamental quantities, but the definitions are simply agreed to based upon the precision of the measurement that produced the standard. They are arbitrary.

If I have to assert anything from this, it is that I value relative measurements above absolute measurements in scientific papers. Relative measurements reveal physical truths where as absolute ones tell us how well data fit into some theory.

I hope to write more on this in the future once my thoughts have more fully materialized.

Lunchtime reading: The Observational Foundations of Physics

I have started reading "The Observational Foundations of Physics" by Sir Allan Cook during my lunch breaks. The book's purpose, as Cook states in the first sentence of Section 1.1, "is to attempt to unravel some ways in which the practice of physics determines the form and content of physics and physical theory." In other words, Cook wishes to understand how the practices found in physics affect physical theories and the practices themselves. It is as if there existed a feedback loop such that performing experiments changed not simply the theory used to describe a phenomenon but the nature of theory itself.

Further in Section 1.1, he poses these questions that are central to his analysis:

1. "Why should physics be so effective, and what does that tell us about the world of physics and our ways of gaining knowledge of it?"
2. "Is there a real world that exists independently of whether I or anyone else is looking at it, or are all the ideas I have about a world external to me just the construction of my mind?"

He defers a thorough answer to the second question until the end of the book, but does offer that he believes that most physicists, while working at the bench or on a computer, act as if an external world existed independent of their attention.

Section 1.2 deals with observations and sets many of the premises of his arguments. Observation and experiment are decided to be equivalent. Observations also consist of two aspects: objective and subjective. Of the subjective aspect, only the communal nature of observation is of consequence to his arguments. Science is a social construct and scientists hold great influence over each other such that the act of observation is never truly independent of people other than the experimenter.

Cook goes to some length to explain that physics is empirical, "with observation primary and theory secondary," but he concedes that rarely can observation be performed without some theory underlying the act of observing. He gives the example of reading a voltage from a digital multimeter. The direct observation is of figures on a LCD readout, a consequence of numerous electronic circuits that respond to potential differences between two probes and relates to the potential energy difference of electrons between two points in a circuit. Of course, electrons are theoretical constructs. The theories underlying an observation can in some ways assure an experimenter that the results are telling us something of the real world and not subject to some extraneous errors or misinterpretations. For simplicity, an observation is defined as the operations that lead to a measurement and result in  "raw data." The data is considered "raw" regardless of the complexity of the measurement.

Finally, theories are models of observations, not a model of the real world itself. "I take a theory to be a mathematical realisation of an abstract system that has properties corresponding to those of a set of observations... It is in that sense that I take a theory to be a model of the world of observations, with the implication that there is a more fundamental correspondence than just giving the right answers..." Theory is an abstraction of the real world, not vice versa.

If you swim after eating, your stomach will cramp

As a student of the physical sciences, the importance of experimentation for determining the true principles behind many natural phenomena is impressed upon me on a near daily basis. However, I am becoming increasingly convinced that carefully designed experiments are even more important for the social sciences.

Within the the social sciences, there are (to my untrained eye at least) few theories to predict the behavior of individuals or groups. Furthermore, their behavior is often influenced greatly by the interests of other groups. For example, McDougall's Born to Run contains a chapter about the drastic increase in foot and knee injuries that occurred following the development of the athletic shoe in the 1970's. Despite an enormous amount of evidence that running shoes are the cause of many running-related injuries, companies such as Nike create a "false truth" for the public: the more cushioned a running shoe is (and the more expensive), the better it is for your feet and knees. Though this is a misconception perpetrated by a company in the field of sports medicine, the idea can be carried over quite easily to the social sciences (see Levitt's Freakanomics). Thus, common wisdom in the social sciences can be attributed to a lack of predictive power and conflicting interests.

The importance of these fields to society is enormous when compared to the physical sciences. After all, if the common wisdom is wrong in the physical sciences, the general public is likely to be affected by not having a new iPod or smart phone until the misconception is discovered and the science is applied to new technologies. However, if misconceptions exist in the social sciences, large groups of people could go without health care, school curricula could be poorly engineered by state governments (New Math, anyone?), and governments could be buried by incredible deficits.

Thus, carefully designed and controlled experiments in the social sciences, and really any science, are important for everyone. Without them, the truth might remain buried in speculation and deception.

A hierarchy of concepts

Richard Feynman, in his Lectures on Physics, had a habit of discussing both the philosophical and practical issues of the science that he taught. One such issue was on the idea that waves could possess particle-like properties, such as momentum and position. As he notes in his Lectures, Vol. 3,

"Only measurable quantities are important to physics. This is false. We need to extend current concepts to unknown areas and then test these concepts. It was not wrong for classical physicists to extend their ideas of momentum and position to quantum particles. They were doing real science, so long as they then checked their assumptions."

What I believe is of value in this statement is the idea that understanding new phenomena is achieved by applying concepts from already well-understood processes and things. So what if a wave didn't traditionally possess momentum or a position? These two concepts (waves and particles) could at least be used to further our understanding of quantum entities, which possess both wave and particle-like properties but do not act entirely like one or the other of these classical constructs.

The same idea I think can be applied in teaching. First find a concept that students are familiar with, then show how this concept can be extended to describe a new phenomenon. However, to be self-consistent and complete, a discussion of how the concept fails to completely describe the phenomenon is required as well. Momentum and position obviously can't describe quantum interference of particles. In this manner, a knowledge of the world is built up of a patchwork of prior understanding.

Addendum
I found this statement particularly enlightening:

"When a data set is mutilated (or, to use the common euphemism, ‘filtered’) by processing according to false assumptions, important information in it may be destroyed irreversibly. As some have recognized, this is happening constantly from orthodox methods of detrending or seasonal adjustment in econometrics. However, old data sets, if preserved unmutilated by old assumptions, may have a new lease on life when our prior information advances."