Where the Weird Things Are

Abstract

Up to this point, I have been careful to emphasize mostly the experimental facts of quantum physics—together with a theoretical structure that is useful for predicting those facts. This might be regarded as the “what” of quantum physics—and in this regard, the story that has emerged over the last 100 years is remarkably clear, comprehensive, and unambiguous.

Appendices

Appendix I: Looking Under the Hood: The Mathematical “Method” Behind the “Madness”

In this Appendix, we present some of the mathematical equations that form the bedrock of what we have many times referred to as the “standard quantum theory.” Warning! Not for the faint of heart.

The most important equation in all of quantum mechanics is also the simplest:

$$\displaystyle{ \hslash = 1.0545718 \times 10^{-34}\mbox{ J s} }$$

(6.1)

However, like all equations, it requires an explanation for a proper understanding. The quantity ℏ is called (the reduced) Planck’s constant. It is a fundamental constant of nature, representing the “size” of the quantum. In Eq. (6.1), ℏ is given in standard macroscopic units called “SI” (système international) units—in terms of which, it is seen to be very small indeed. The dimensions of Eq. (6.1) are also significant; we see that ℏ has units of energy × time, also known as action.

More generally, pairs of quantities whose product has dimensions of action are called conjugate variables. Other examples include position × momentum and angle × angular momentum. Conjugate variables are important because they are incommensurate and can therefore be used to formulate uncertainty principles. Thus, the usual HUP as described in Sect. 2.3 is actually the position–momentum uncertainty principle:

$$\displaystyle{ \varDelta x\varDelta p \geq \hslash /2, }$$

(6.2)

where Δx is the uncertainty (technically standard deviation) in position, and Δp is the uncertainty in momentum (where momentum is just mass × velocity). The energy–time and angle–angular momentum uncertainty relations have a similar form.

From a wave mathematics point of view, the uncertainty principle is nothing new. Given any “wave” function ψ(x), the Fourier transform function, $\tilde{\psi }(k)$, is defined:

$$\displaystyle{ \tilde{\psi }(k) ={ 1 \over \sqrt{2\pi }}\int _{-\infty }^{\infty }\exp (-ikx)\psi (x)\,dx }$$

(6.3)

This represents the decomposition of the original wave into its Fourier (i.e., sinusoidal) components. It is well known that a narrow ψ(x) implies a broad $\tilde{\psi }(k)$ and vice versa. In particular,

$$\displaystyle{ \varDelta x\varDelta k \geq 1/2. }$$

(6.4)

Equation (6.4) suggests the identification of p with ℏk, which has the right dimensions for momentum (since k must have dimensions of 1∕x, and x has dimensions of length). Note that λ = 2π∕k is the wavelength for the Fourier component k. Thus,

$$\displaystyle{ p = \hslash k = 2\pi \hslash /\lambda; }$$

(6.5)

momentum in quantum mechanics is inversely proportional to wavelength.

Using wave properties of ψ as discussed above, we can estimate the distance f between adjacent “fringes” in the double-slit interference pattern that appears on the far wall in Fig. 3.5 First note that the phase of the ψ branch that passes through the upper slit is given by ϕ ₊ = kx ₊, where x ₊ is the distance from the upper slit. Likewise, ϕ ₋ = kx ₋ for the lower slit. Since relative phase is what causes interference (Sect. 4.2), f must correspond to a relative phase change of one cycle, i.e.,

$$\displaystyle{ \varDelta (\phi _{+} -\phi _{-}) = k\varDelta (x_{+} - x_{-}) = 2\pi. }$$

(6.6)

If s is the distance between the two slits, and D the distance between the two walls, with s∕D ≪ 1, then simple geometry shows Δ(x ₊ − x ₋) ≈ f(s∕D), or

$$\displaystyle{ f \approx 2\pi D/sk =\lambda (D/s). }$$

(6.7)

The above “kinematic” description is useful for interpreting ψ(x, t) once we have it but does not tell us how ψ(x, t) actually changes over space and time. A complete theory requires such a dynamical rule—which in this case should take the form of a partial differential equation (PDE). Fourier analysis can be used to guess the correct PDE. In particular, k is associated with the partial derivative of its conjugate variable:

$$\displaystyle{ k \rightarrow -i(\partial /\partial x),\quad \mbox{ or}\quad p = \hslash k \rightarrow -i\hslash (\partial /\partial x). }$$

(6.8)

Likewise, energy, E, can be associated with its conjugate variable, i.e., E → iℏ(∂∕∂t). Substituting these relations into the classical “kinetic-plus-potential” energy expression

$$\displaystyle{ E ={ p^{2} \over 2m} + V (x), }$$

(6.9)

and applying the result to ψ(x, t), we obtain the desired quantum PDE:

$$\displaystyle{ i\hslash \left ({ \partial \psi \over \partial t}\right ) = -{ \hslash ^{2} \over 2m}\left ({ \partial ^{2}\psi \over \partial x^{2}}\right ) + V (x)\psi. }$$

(6.10)

Equation (6.10) is the celebrated (time-dependent) Schrödinger equation, in 1D (one spatial dimension). Note the presence of $i = \sqrt{-1}$ on the left hand side, which ensures that ψ(x, t) is necessarily complex valued. In 3D space, Eq. (6.10) generalizes to

$$\displaystyle{ i\hslash \left ({ \partial \psi \over \partial t}\right ) = -{ \hslash ^{2} \over 2m}\left ({ \partial ^{2}\psi \over \partial x^{2}} +{ \partial ^{2}\psi \over \partial y^{2}} +{ \partial ^{2}\psi \over \partial z^{2}}\right ) + V \psi = -{ \hslash ^{2} \over 2m}\left (\nabla ^{2}\psi \right ) + V \psi. }$$

(6.11)

Here, the potential V = V (x, y, z), and ψ = ψ(x, y, z, t). Of course, this describes only a single-particle wavefunction. The real, i.e., many-particle Schrödinger equation is

$$\displaystyle{ i\hslash { \partial \psi \over \partial t} = -{ \hslash ^{2} \over 2m_{A}}\nabla _{A}^{2}\psi -{ \hslash ^{2} \over 2m_{B}}\nabla _{B}^{2}\psi +\ldots +V \psi, }$$

(6.12)

with ψ = ψ(x _A , y _A , z _A , x _B , y _B , z _B , …, t) and V = V (x _A , y _A , z _A , x _B , y _B , z _B , …, t).

In addition to spatial coordinates (x, y, z), quantum particles also possess intrinsic spin attributes. For any direction in space—e.g., such as the vertical z axis, (0, 0, 1)—a spin measurement yields (for most particles) one of two possible outcomes: “up” or “down.” Spin can be measured in any direction; however, different spin measurements are incommensurate. Note that this situation corresponds closely to the quantum coin toss experiment of Sect. 5.4. One minor technical difference is that the two entangled quantum particles in an EPRB experiment are prepared with opposite, rather than identical, spin values. The three detector settings correspond to three different spin directions, 120^∘ apart in a single plane. It can be shown that if the detector settings are different, and the first detector registers “spin up” for particle A, then ψ collapses partially (as in Sect. 5.1) to the following superposition over B states only:

$$\displaystyle{ \psi _{B}^{(A=\mbox{ up})} = \left ({1 \over 2}\right )\psi _{\mbox{ up}} + \left (i{\sqrt{3} \over 2} \right )\psi _{\mbox{ down}} }$$

(6.13)

Since probabilities are obtained as square amplitudes, the probability that subsequent measurement of B will find it to be in a spin-up state is p _B (up) = | 1∕2 |² = 1∕4. Likewise, $p_{B}(\mbox{ down}) = \vert i\sqrt{3}/2\vert ^{2} = 3/4$. Note that these are the probabilities for the same and opposite bulbs flashing on the two detectors, respectively.

Appendix II: Further Reading

Less Advanced Books (alphabetical order by author)

I. Asimov, The New Intelligent Man’s Guide to Science (Basic Books, New York, 1965)

J.E. Baggott, Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth (Pegasus Books, New York, 2014)

A. Einstein, The World As I See It (Citadel Press, 2000)

R.P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, Princeton, 1988)

R.P. Feynman, Six Easy Pieces (Perseus Books, Cambridge MA, 1995)

R.P. Feynman, Six Not-So-Easy Pieces (Perseus Books, Cambridge MA, 1997)

R.P. Feynman, The Character of Physical Law (British Broadcasting Corp., 1965)

B. Greene, The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos (Alfred A. Knopf, New York, 2011)

G. Greenstein, A.G. Zajonc, The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics (Jones and Bartlett, Sudbury, 1997)

M. Kafatos, R. Nadeau, The Conscious Universe (Springer, 1990)

D. Mermin, Boojums All the Way Through (Cambridge University Press, Cambridge, 1990)

A. Pais, Inward Bound: Of Matter and Forces in the Physical World (Clarendon Press, Oxford, 1988)

R.A. Bertlmann, A. Zeilinger (eds.), Quantum [Un]speakables: From Bell to Quantum Information (Springer, 2002)

L. Smolin, The Trouble with Physics (Mariner Books, 2007)

D. Sobel, A More Perfect Heaven: How Copernicus Revolutionized the Cosmos (Walker & Company, New York, 2011)

S. Weinberg, The Discovery of Subatomic Particles (Cambridge University Press, Cambridge, 2003)

F.A. Wolf, Parallel Universes (Simon and Schuster, New York, 1988)

More Advanced Books (alphabetical order by author)

J.E. Baggott, The Meaning of Quantum Theory: A Guide for Students of Chemistry and Physics (Oxford University Press, 1992)

J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1988)

D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, 1951)

E. Commins, Quantum Mechanics: An Experimentalist’s Approach (Cambridge University Press, New York, 2014)

P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 1958)

R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Vols. I–III (Addison-Wesley, 1965)

M.S. Longair, Theoretical Concepts in Physics (Cambridge University Press, Cambridge, 1984)

L. Pauling, E.B. Wilson, Introduction to Quantum Mechanics With Applications to Chemistry (Dover, New York, 1963)

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1983)

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950)

Internet Sites (in order referenced, plus additional)

http://www.amazon.com

http://www.mvjs.org

http://www.google.com

http://www.emqm15.org/presentations/speaker-presentations/

http://www.nature.com/nature/journal/v474/n7350/full/nature10120.html

http://redd.it/1xxmfl

http://redd.it/1r6kfc

http://forteana-blog.blogspot.com/2013/05/spooky-action-at-distance.html

http://www.acoustics.salford.ac.uk/feschools/waves/diffract3.php

http://www.huffingtonpost.com/bill-poirier/quantum-weirdness-and-many-interacting- worlds_b_6143042.html

http://physics.stackexchange.com

Appendix III: Glossary

• Bell’s Theorem: Theorem derived by John S. Bell in 1964 establishing the experimental decidability between quantum theory and local hidden variable theories; quantitatively it is expressed as an inequality known as “Bell’s inequality.”

• Classical Mechanics: Theory of physics epitomized by the work of Sir Isaac Newton in the seventeenth century that describes a “clockwork” motion of objects; characterized by precision and “commonsense” determinacy; later superceded (at very small scales) by quantum mechanics.

• Double-Slit Experiment: Experiment demonstrating the wavelike nature of quantum particles, by firing a large number of identically prepared particles, one at a time, at a screen with two separated parallel slits, and observing a pattern of interference fringes on a second screen placed beyond the first.

• Einstein Podolsky Rosen [Bell] (EPR[B]) Experiment: Thought experiment originally conceived by Albert Einstein and coworkers in 1935 (EPR), to demonstrate flaws in quantum theory, and prove the existence of hidden variables; later, in the wake of Bell’s Theorem, EPRB became a bona fide laboratory experiment—which ruled out local hidden variables and vindicated quantum nonlocality.

• Entanglement: In quantum mechanics, refers to two or more particles that are statistically correlated, even across vast distances, such that they cannot be described by separate single-particle wavefunctions.

• Heisenberg Uncertainty Principle (HUP): Principle formulated by German physicist Werner Heisenberg in 1927 that describes the indeterminacy of quantum mechanics; qualitatively, it states that there is a fundamental limit to the precision with which both the position and velocity of a quantum particle can be simultaneously determined; quantitatively it can be expressed in various mathematical equations, generally as an inequality.

• Hidden Variables: Dynamical attributes purported by some to be missing from quantum theory, which would render it complete and deterministic; local hidden variables of the sort advocated by Einstein are now ruled out by EPRB experiments, although global (nonlocal) hidden variables may still exist.

• Misconception: A mistaken view; an erroneous notion; impossible for human beings to avoid completely.

• Nonlocality: Principle of physical theories that postulates “action at a distance,” i.e., the remote influence of one particle on another, either instantaneously, or at speeds faster than light; in quantum mechanics, this concept can be a bit more subtle (see Entanglement).

• Particle: Refers to “localized” (point-like) objects in the physical world; in classical physics, particles trace out a definite “trajectory” (one-dimensional curve through space) over time; in quantum physics, a “particle” can manifest delocalized wavelike behavior, when its position in space is not being observed.

• Probability Wave: Mathematical function describing the likelihood that a particle is located at a given point in space (may also refer to sets of particles); used in both classical and quantum mechanics, though the behavior is quite different in each case.

• Quantum Mechanics: Theory of physics originating in the early twentieth century that describes the mechanics of atoms, molecules, etc.; replaces the earlier “classical” theory of mechanics; characterized by indeterminacy; impossible for human beings to understand completely.

• Schrödinger’s Cat: Thought experiment devised by Austrian physicist Erwin Schrödinger in 1935, to demonstrate the absurdities that arise from quantum theory when the notion of a superposition state is extended up to the macroscopic realm.

• Wave: Any entity that is delocalized over a broad region of space.

• Wavefunction: Complex-valued mathematical function that describes the state of a quantum system; the square amplitude of the wavefunction is the probability wave.

• Wavefunction Collapse: Process by which measurement of a quantum system by an outside observer causes the probability wave to “collapse” to one specific state from among a superposition of states—seemingly at random.

• Wave-Particle Duality: Refers to the notion that quantum objects may exhibit different behaviors, under different circumstances, reminiscent of either classical waves or classical particles.

Credits

Photos:

A. Einstein: Arthur Sasse / AFP-Getty Images / Wikimedia

R. Feynman: The Nobel Foundation / Wikimedia / Public Domain

Y. Berra: Baseball Digest / Wikimedia Commons / Public Domain

W. Heisenberg: Friedrich Hund / Wikimedia Commons / CC-BY-3.0

W. White, aka “Heisenberg”: “Breaking Bad” / Vince Gilligan / AMC /

Wikimedia

E. Schrödinger: The Nobel Foundation / Wikimedia / Public Domain

Prudy (cat): Photos by the author.

Other Images:

skiier: Quantum Skiier / Courtesy of Anne Longo / derived from a cartoon by

Charles Addams / The New York Times (December 3, 2006)

Yogi Bear: Hanna-Barbera Productions / Wikimedia

cat: Diagram of Schrödinger’s Cat / D. Hatfield / Wikimedia / CC-by-SA 3.0

Quotes: All quotations are cited where used.

Other:

“Separated at Birth?”: Former Reg’d Trademark / Spy Publishing Partners

cartoon: Bill Watterson / Calvin and Hobbes © (October 29, 1989) / Reprinted with permission of UNIVERSAL UCLICK / All rights reserved