In the previous chapter we began to discuss some of the very strange behaviour of particles/waves in the atomic and subatomic realms. We learned about Planck's desperate hypothesis (that elementary vibrators in a hot solid can only emit energy in quanta that are a whole-number multiple of an elementary quantum), and how Einstein extended this hypothesis even more boldly by declaring that electromagnetic radiation itself EXISTED in tiny bundles of energy, which became known as photons. This is extremely strange, because we're used to thinking of light as a wave, and we have centuries of evidence to back it up.

But scientists don't just make wild speculations in a vacuum, they do so to solve very specific problems. So Einstein's wild speculation did an excellent job of explaining the photoelectric effect, and together with Planck's explanation for blackbody radiation spectral curves, provided strong evidence for the hypothesis.

But no amount of evidence can ever prove a scientific hypothesis valid. (Is your marriage red or green?) The best we can do, and indeed what we must do, is amass as much evidence as we can; the weight of evidence is what wins the argument in science. And the significance of the collection of evidence is greatly increased when there is evidence from a greater variety of situations.

Thinking back to the photon hypothesis from the previous chapter, we have already discussed two quite different types of evidence in favour of photons: blackbody radiation and the photoelectric effect. The photon hypothesis explains both situations handily.

However, the main tools of science are doubt and skepticism, so inevitably, no matter how good one's explanations are, there are clever people out there who will strongly disagree with you, demanding even more and better evidence. This is both a strength and a weakness of science. The harsh treatment that most hypotheses receive is a strength because

Chapter 29 Atoms and MoleculesFriday, January 14, 2011

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treatment that most hypotheses receive is a strength because it pushes their proponents to do their very best work, and it ensures that only the best, most productive ideas are accepted. It's a weakness because sometimes the climate becomes so hostile that the scientific community loses good but sensitive people, who recoil from the cut and thrust of scientific debate. Like any community, the scientific community is full of good, kind, supportive, generous individuals, but also contains a fair share of hostile, argumentative egomaniacs.

Millikan in particular was very hostile to the photon hypothesis, and set out to design and conduct experiments to prove Einstein wrong. However, he was surprised to discover that his experiments actually supported the photon hypothesis, and so he changed his tune.

And this is another important guiding principle in science: That one honestly report the results of ALL of one's investigations, no matter what the results are, no matter whether they support your favourite theory, or your preconceptions. In this aspect, doubt and skepticism once again prove to be useful tools, for dishonest investigators are very soon found out, exposed, and their sham results are discarded and corrected.

"Extraordinary claims require extraordinary evidence," as they say, and the photon hypothesis was sufficiently surprising that it was not generally accepted for a long time. Yet another line of evidence for the photon hypothesis came from what we now call Compton scattering.

At that time J.J. Thomson explained the scattering of electromagnetic waves from free charged particles using Maxwell's theory of electromagnetism, and the theory was verified experimentally for incident light of low intensity. The classical theory predicts NO change in wavelength of the scattered electromagnetic waves.

However, in 1923 Compton hypothesized that if light were really composed of photons, then high-energy photons scattering from charged particles should experience a shift in

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their wavelength upon scattering; some of a photon's energy would be transferred to the charged particle in the collision, and because the energy of a photon is related to its wavelength, the scattered photon would have a different wavelength than before it collided with the charged particle. This is now called the Compton effect, and it ought to be noticeable for relatively high energy photons. (For low energy photons, the wavelength change is not very great, and so there might not be enough difference to distinguish between the predictions of Thomson and Compton.)

Compton observed this wavelength shift in a series of experiments beginning in 1923, and this cemented the photon hypothesis.

Joseph A. Gray, an Australian-Canadian physicist who worked at Queen's University from 1924 to 1952, also did extensive work on the scattering of photons from charged particles, and some of my older physics profs (I was a student at Queen's in the late 1970s and early 1980s) think he ought to have shared credit with Compton, going so far as to call it the "Compton-Gray"effect, instead of the Compton effect.

* * * * *

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This brings us to the remaining two puzzling items from last chapter, dealing with atomic spectra and atomic structure, which we'll discuss now.

Atomic Structure

What is the inside of an atom like?

There were quite a number of atomic models bandied about in the first decade of the 20th century, as physicists became more convinced in the existence of atoms. None of them explained the existing experimental data very well, though. Notable is J.J. Thomson's "plum-pudding" model, which he was developing. His discovery of the electron in 1897 gave him hope that his model would eventually be successful in explaining spectroscopic data.

Enter Rutherford, and the Geiger-Marsden experiment (1909). The results of the experiment decisively scrapped Thomson's model, and many of the others as well. But what could be put in its place?

The following diagram is an outline of the actual apparatus. Alpha particles fly through M towards the gold foil F; D and R is the microscope/detector that can be rotated through various angles about a vertical axis in the cylindrical central part of the apparatus.

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And here's a schematic diagram of the apparatus that indicates the path of scattered alpha particles:

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Rutherford's atomic model (1911) --- a good step, but it was NOT quantitative

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one of the lessons/methods of science: look for the unexpected

•

Also, Rutherford's model suffered from a fatal flaw: according to classical electromagnetic theory (Maxwell's equations), atoms should emit continuous (that is, a wide range of wavelengths) electromagnetic radiation (they don't; rather they emit "line spectra"), and they should be unstable (electrons should lose energy as they emit electromagnetic radiation, and eventually spiral into the nucleus, where they would die a dramatic death upon joining forces with the protons in the nucleus), which they aren't. So, although it was clear that Rutherford was onto something, it was also very clear that his model was wrong.

This is another lesson of science: sometimes a theory that you know is wrong is still worth pursuing, studying, and understanding deeply, because it may be just the stepping stone you need to help you discover a good theory. However, it takes courage to publish a theory that you know is wrong; of course, you would be honest about it and admit the fatal flaws in the theory.

Enter Niels Bohr.

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Bohr's atomic model; see pages 960 ff in the textbook•

What are line spectra?

emission and absorption spectra:•

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Pasted from <http://www.google.ca/imgres?q=electron+orbitals&um=1&sa=N&biw=1280&bih=576&hl=en&tbm=isch&tbnid=1zKh-g8udAwM6M:&imgrefurl=http://chemicalfacts4u.blogspot.com/2011/06/atomic-orbital.html&docid=F81z3i_N2a-TcM&imgurl=http://3.bp.blogspot.com/-7WaPCpa9mF4/TfgpuFxlFII/AAAAAAAAACA/x5_0HjUpn2Y/s1600/ch9orbitals1.jpg&w=576&h=388&ei=yyLcUej6Jq7BywHh2oH4BQ&zoom=1&iact=hc&vpx=721&vpy=165&dur=328&hovh=184&hovw=274&tx=168&ty=107&page=1&tbnh=136&tbnw=191&start=0&ndsp=18&ved=1t:429,r:4,s:0,i:98>

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Exercises

Collisional excitation: Suppose that two hydrogen atoms collide, and in the process the electron in one of the atoms is knocked into a higher orbit. Initially the electron is in the n = 1 orbit and it

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into a higher orbit. Initially the electron is in the n = 1 orbit and it ends up being knocked into the n = 3 orbit. Then the electron "cascades" down to its ground state; that is, the electron first makes a transition from the n = 3 orbit to the n = 2 orbit, and then makes a transition from the n = 2 orbit to the n = 1 orbit.

(a) Determine the energy absorbed by the electron in the initial collision.(b) Determine the wavelengths of the emitted electromagnetic radiation.

Solution:

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CP 3 The Paschen series is analogous to the Balmer series, but with m = 3. Calculate the wavelengths of the first three members in the Paschen series. Which part(s) of the electromagnetic spectrum are these in?

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CP 10 The allowed energies of a simple atom are 0.0 eV, 4.0 eV, and 6.0 eV. (a) Draw the atom's energy-level diagram. Label each level with the energy and the principal quantum number. (b) Which wavelengths appear in the atom's emission spectrum? (c.) Which wavelengths appear in the atom's absorption spectrum?

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CP 12 A researcher observes hydrogen emitting photons of energy 1.89 eV. What are the quantum numbers of the two states involved in the transition that emits these photons?

Consider instead the related problem where we use

CP 13 A hydrogen atom is in the n = 3 state. In the Bohr model, how many electron wavelengths fit around this orbit?

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the idea of electron energy levels and "shells" was later adapted to help describe atomic nuclei as well

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Exercises

CP 26 Predict the ground-state electron configurations of Mg, Sr, and Ba.

CP 31 Explain what is wrong with each electron configuration.(a) 1s22s22p83s23p4

(b) 1s22s32p4

CP 30 Identify the element for each electron configuration. Then determine whether this configuration is the ground state or an excited state.(a) 1s22s22p63s23p64s23d9

(b) 1s22s22p63s23p64s23d104p65s24d105p66s24f145d7

CP 34 Hydrogen gas absorbs light of wavelength 103 nm.

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Afterward, what wavelengths are seen in the emission spectrum?

Some concluding remarks about quantum conundrums

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In classical mechanics, the state of a system of particles is described by specifying the position and momentum of each particle as functions of time. Typically one solves Newton's second law of motion for the system of particles (a second order differential equation for each particle, which results in a system of differential equations), which then gives us the position function of each particle. Then you differentiate the position function to obtain the velocity function, from which you can obtain the momentum function by multiplying by the mass. Thus, if you specify the position and momentum of each particle in your system, you know all about the system. For example, you can calculate any other quantity of interest about the system (kinetic energy, angular momentum, etc.) from the position and momentum functions.

The situation in quantum mechanics is totally different. For fundamental reasons (think about the Heisenberg uncertainty principle), one can't know simultaneously and precisely the position and momentum of a particle. Thus, it's impossible to specify the position function and momentum function of a particle in quantum mechanics. (In fact, one could argue that they don't even exist, but more about that later.) How then is one supposed to even specify the state of a system in quantum mechanics?

After a presentation on de Broglie's ideas about matter waves by Born in 1924, Debye asked Born: if we are to take this wave idea seriously, then what is the differential equation that is satisfied by them?

That was a good question, and undoubtedly several people sought answers. Schrodinger was the first to bear fruit, creating a suitable wave equation late in 1925. "Create" is the right word; such equations are not derived, they are "guessed," although guessing hardly does justice to this highly imaginative activity.

Here is a plausible story for how Schrodinger might have guessed the differential equation that now bears his name. Recall from earlier in the course that a transverse wave (a "plane" wave) can be expressed as (see page 485 of the textbook)

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or equivalently, as

As you'll learn in upper-year courses involving waves, it's possible to subsume both expressions into a single complex-valued exponential expression as follows:

The sine and cosine descriptions can be recovered as the real and imaginary parts of the complex function describing the wave (look up Euler's formula if you don't know it yet).

Note that it's customary in quantum mechanics to use the Greek letter Psi to represent this rather curious mathematical animal, which is called a wave function. According to the standard interpretation of quantum mechanics, the wave function "encodes" all that it is possible to know about a physical system; therefore, the state of a physical system in a quantum-mechanical description is its wave function. Contrast this with the description of the state of a system in classical mechanics in terms of the position and momentum functions of all of particles in the system.

Schrodinger guessed a differential equation satisfied by the wave function. He ensured that his guess is consistent both with the de Broglie wavelength of a particle's "companion wave" and Einstein's relation for the energy of the wave associated with a particle, as follows:

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Now observe what happens when you differentiate the wave function with respect to both x and t:

In other words, differentiating the wave function with respect

to x and then multiplying by the constant iћ is the same as just multiplying the wave function by p. A more sophisticated way to say this is that in quantum mechanics, momentum is an operator, and specifically its action on a wave function is related to the action of partial differentiation with respect to x as follows:

Sometimes a "hat" is placed above a letter to emphasize that it's being considered as an operator, as we've done above, but this usage is not universal.

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Now we've done the ground work to show you Schrodinger's equation. Start with the Newtonian expression for mechanical energy, and substitute the quantum operators for momentum and energy:

but this usage is not universal.

The energy operator can be derived similarly, by observing what happens when we differentiate the wave function with respect to t:

In other words, differentiating the wave function with respect to t and then multiplying by the constant iћ is the same as just multiplying the wave function by the energy operator. In quantum mechanics, the energy operator is symbolized by H (it's called the Hamiltonian operator). Thus,

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We argued in terms of the wave function for a plane wave just as a way of making the guess for Schrodinger's equation plausible; this does not imply that the wave function for every single system is a plane wave. No, you solve Schrodinger's equation for each particular system, and the solution is the wave function for the system. And you specify the system that you're dealing with by identifying the potential energy function V(x).

Once you've solved the Schrodinger equation to obtain the wave

For a system that's free to move in three dimensions, the Schrodinger equation is:

All that wonderful mathematics you'll be learning in second year (multi-variable calculus, vector calculus, differential equations, linear algebra, complex numbers, etc.) will be put to good use when you study quantum mechanics in more depth!

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function, then the only things that you can know about the system are things you can derive from psi by applying the right mathematical operations to it, such as energy levels, transition probabilities, angular momenta, etc; the results are real numbers that you can then compare to experiments.

Everyone agrees about how to solve the Schrodinger equation, and everyone agrees with how to extract the values of physical quantities from the solution, and everyone agrees with how to compare the values to experimental results. Everyone agrees that the whole process works excellently, and agrees excellently with experiment. Furthermore, the thought process that goes along with quantum mechanics has been used to develop some pretty snazzy technology (lasers, solid state devices, microminiaturized solid state devices, semi-conductors, some nanotechnology, etc.).

The problem is that many people don't like the philosophy behind this program; they feel uncomfortable with its departure from "local realism" that we are so used to in classical physics; but maybe this is just the way it is, and we should just get used to it, and embrace it and learn to look at the world in this way.

But then how does the classical world of our macroscopic reality emerge from this strange microscopic reality?

Furthermore, certain physicists just don't like the quantum philosophy, and don't accept it; they figure there is a deeper theory out there, that would remove some of the quantum puzzles, and return us to a more palatable interpretation.

Many other physicists ascribe to the "Shut up and calculate!" philosophy; i.e., don't worry about interpretational issues, just get down to business and focus on practical calculations; who cares about "deeper meaning."

Now let's discuss what local realism is, and we'll also discuss some of the most troublesome interpretational issues in quantum mechanics.

The "collapse" of the wave function

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The "collapse" of the wave function

So far, we've discussed an analytical (i.e., calculus-focused) approach to the Schrodinger equation. There is another very fruitful perspective, which we might call the algebraic approach (and which highlights the need to get a solid grounding in linear algebra). In the algebraic approach, the wave function is considered to be a vector in an abstract "representation" space, which might be finite dimensional or infinite dimensional, depending on the system being studied.

In this approach, the state of a quantum system can be considered to be one of the vectors in the abstract representation space, and so vectors in this space are also often called "state vectors." It turns out that one can choose a basis for the abstract representation space consisting of vectors that each have a definite energy associated with them; such basis vectors are called "stationary states" or, equivalently, "energy eigenstates."

Because the energy eigenstates form a basis for the representation space, an arbitrary state of the system can be expressed as a linear combination (i.e., superposition; quantum mechanics is a linear theory!) of energy eigenstates. Furthermore, the Schrodinger equation can be expressed in the language of linear algebra as follows

where H is the Hamiltonian operator and E is a number. All you linear algebra fans will recognize this as an eigenvalue equation, and the solution vectors are the eigenvectors, or energy eigenstates as we have been calling them. The eigenvalues are the numbers E for which the eigenvalue equation has a solution.

It may be difficult at first to recognize that this eigenvalue equation is equivalent to the Schrodinger differential equation; remember that the Hamiltonian operator is a differential operator, and stands for the whole left side of the analytical Schrodinger equation. In linear algebra class we're used to

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Schrodinger equation. In linear algebra class we're used to thinking of linear transformations (also called linear operators) as matrices; we have to expand our perspective and realize that all sorts of (initially) strange looking mathematical animals can be considered to be linear operators acting in the right vector space.

If you've been able to follow the previous development, we're now ready to discuss the issues around the "collapse of the wave function" concept, also known as the quantum measurement problem. As a system evolves in time, its state is described by its wave function (state vector, if you prefer), whose evolution in time satisfies the Schrodinger equation. However, as soon as you make a measurement of, let's say, the energy of the system, the wave function makes a sudden change (not described by the Schrodinger equation) to one of the energy eigenstates. That is, what was a superposition of eigenstates has suddenly been projected into one single eigenstate. Furthermore, the value of the energy measured in the experiment is necessarily the energy associated with the eigenstate.

The same sort of thing happens for all other observable quantities, not just energy. In each case, the wave function goes merrily along, evolving according to the Schrodinger equation, until one decides to make a measurement, in which case the only possible results of the measurement are the eigenvalues of the corresponding operator, and the measurement suddenly collapses the wave function into the matching eigenstate of the operator. Furthermore, once the measurement is made, the collapsed wave function now continues to evolve according to the Schrodinger equation, and subsequent measurements involve further collapses of the new wave function. Thus, the measurement has definitely disturbed the system being measured, and subsequent measurements reflect this disturbance.

This was extremely puzzling, and indeed annoying, to many physicists, including some of the founding fathers, such as Einstein, Schrodinger, and de Broglie. The standard interpretation described here is now called the Copenhagen interpretation, because it was elaborated under the guidance of

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interpretation, because it was elaborated under the guidance of Bohr (who was based in Copenhagen), and its adherents included Heisenberg, Born, and many others.

Einstein and Schrodinger, in particular and among others, tried their hardest to come up with logical arguments for why this situation is unacceptable. Bohr and Heisenberg argued that the value of a measured quantity does not even exist before the measurement is made, and they used this principle to good effect in winning all of their arguments with Einstein and Schrodinger. As Einstein famously said, "Do you really believe that the Moon is not really there until you look at it?" Indeed, even if you believe that the microworld is beset with these bizzare (to our macroscopically informed sensibilities) behaviours, does the macroworld also behave in these bizarre ways? And if not, then why not? Why does the microworld behave bizarrely and the macroworld behaves "normally?" These are troubling questions that have been plaguing physicists (and philosophers of science) for nearly a century, and although progress has been made in untangling the mess, the waters are still rather murky.

Schrodinger devised a picturesque thought experiment with the intention of showing how absurd the Copenhagen interpretation of quantum mechanics is, which nowadays goes by the name of "Schrodinger's cat." Maybe he liked cats, maybe he just wanted to tar the reputations of the Copenhagen contingent by subtly suggesting that they would be animal abusers; I don't know why he chose a cat instead of some other animal.

Schrodinger suggested that an elaborate Rube-Goldberg-like device be placed in a sealed container along with a cat. In the device, a radioactive atom is placed with a detector, and the detector is connected to a mechanism that smashes open a vial of poison. Once the vial is smashed the cat will die instantly.

The radioactive atom is in a superposition of two states, one in which the atom has not yet decayed, and one in which it has decayed. Schrodinger argued that if you believe the Copenhagen interpretation, you must conclude that the cat is also in a weird superposition state of being both alive and dead. Only by

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superposition state of being both alive and dead. Only by opening the container and looking (which amounts to making a measurement) will you then collapse the wave function of the cat and project it into one of the eigenstates, either dead or alive. Schrodinger's argument is equivalent to Einstein's rhetorical question, "Do you really believe the Moon is not there until you look at it?" In other words, do you really believe the cat is in this weird superposition alive/dead state until you look at it? Do you really believe that the act of observing the cat kills it (or renders it alive, from its previous ambiguous state)? Schrodinger and Einstein certainly did not believe this.

The problem with the arguments of Schrodinger and Einstein is that experiments with atoms and subatomic particles always behave in the way predicted by the Copenhagen interpretation! So they are certainly wrong at the micro level; if the Moon is really there even if I don't look at it, then they are right at the macro level. But this is puzzling; how on Earth does the macroworld conspire to be different from the microworld?

Many attempts have been made to sort out this mess; to learn more, search on the terms "quantum decoherence" and "consistent histories;" these approaches were motivated by the earlier de Broglie-Bohm interpretation of quantum mechanics.

Wigner extended the Schrodinger cat thought experiment in an attempt to show that consciousness is essential to collapsing wave functions, and consciousness is essential to resolving the measurement problem. However, this raises yet more difficult issues; if consciousness is necessary, then how conscious does the observer have to be? Would a dog do? How about a worm? Would a dolphin or a monkey be sufficient? Many physicists are not comfortable with the idea that a conscious observer is necessary to spring reality into existence. To learn more about this, search on "Wigner's friend."

Local Realism

The discussion so far calls into question what reality is. All of this could be dismissed as a philosophical discussion that is a pointless waste of time, except that every experiment with micro

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pointless waste of time, except that every experiment with micro objects (atoms and subatomic particles) confirms the Copenhagen interpretation!

In 1935 Einstein gave it his last shot at punching a hole in the Copenhagen interpretation. Together with Podolsky and Rosen they created another thought experiment that they figured showed, finally and emphatically, how absurd the Copenhagen interpretation is. The world could not possibly be this way, they argued. Their thought experiment is now known as the EPR problem (or the EPR paradox, although it's not really a paradox).

Here's a sketch of the EPR argument: Suppose you prepare a system containing two interacting particles which then separate and move far apart from each other. The fact that the particles interacted initially means that they have to be described by a single wave function (i.e., they are described with a single quantum state). Now consider the position and momentum of each particle. From Heisenberg's uncertainty principle, we know that it's not possible to know precisely the value of both the position and momentum of a particle. If you measure one quantity very precisely, then you lose all possibility of measuring the value of the other quantity for the same particle.

When Heisenberg proposed his uncertainty principle, he suggested that it was a consequence of inevitable disturbance in the measurement process; for example, to measure the position of a subatomic particle, you have to shine light on it and the photon "kicks" the particle in an indeterminate way, thereby disturbing the momentum of the particle. However, nowadays Heisenberg's uncertainty principle is understood more subtly, as a consequence of the wave aspect of particles. The point of the EPR argument is an attempt to show that quantum mechanics is incomplete, and they attacked Heisenberg's interpretation of his uncertainty principle.

* * * * to be completed * * * *

photon experiments, and the analogue of Heisenberg's uncertainty principle for other pairs of non-commuting variables

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entanglement, non-locality, the EPR "paradox", Bell's inequality, the Aspect experiments, and the demise of "local realism"

Conclusion: Quantum mechanics and its successor theories (quantum field theories, such as QED (quantum electrodynamics) and QCD (quantum chromodynamics)) are the most accurately verified physical theories we have. There is clearly something "right" about them, but they contain unresolved interpretational issues that trouble many people. There appears to be lots of room there for fresh ideas for both further development and also for resolving some of the troubling issues.

Maybe some of you will decide to devote some time to thinking about and working on some of these problems. To do this, you'll have to learn a lot more mathematics and physics, which is fun in itself. Others may prefer to continue with other fields of study; in your case, I hope what you've learned in physics will be of use to you in your favourite field of study.

To all of you, I wish you the best in your future studies and in all of your future endeavours.

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