The US Penny as a Quantum Object

ABSTRACT: I estimate the lifetime of an upright US penny against perturbations due to thermal hopping and quantum tunneling.


 BATISTA & PETERNELJ'S "QUANTUM BAT"

            While browsing the Los Alamos physics arXiv I discovered a paper (1) by Slovenian physicists M. Batista & J. Peternelj called "Quantum Cards and Quantum Rods" which examines the assertion by John Wheeler and Max Tegmark that quantum mechanics predicts that a perfectly balanced playing card would fall in "both directions at once", i.e. as a quantum superposition of the two macroscopically distinguishable states Falling-to-the-Right and Falling-to-the-Left.

            Batista & Peternelj examine in great detail the quantum mechanics of an upright rod pivoted at its base. They conclude that for a perfectly balanced rod the Wheeler-Tegmark conjecture is correct--the rod falls in both directions at once--behaving as a kind of Schrödinger's (baseball) Bat analogous to Schrödinger's famous Cat who could exist (courtesy of quantum mechanics) in a state where he is simultaneously dead and alive. However Batista & Peternelj also show that if the table on which the rod is mounted is tilted ever so slightly, the rod will fall only towards one side. Thus the practical impossibility of achieving a perfectly horizontal table seems to nullify the Wheeler-Tegmark conjecture.


CLASSICAL PHYSICS OF AN UPRIGHT PENNY

            The pivoted rod is a manifestly unstable situation and highly artificial. I decided to examine a small realistic metastable system--the upright US penny--to explore the magnitude of quantum effects on a simple macroscopic situation.

penny.jpg            The penny balanced on its edge is stable against small perturbations because, unlike the pivoted rod which will naturally fall to the ground, the upright coin sits in a local minimum of potential energy. In order to make the coin fall, a certain minimum amount of energy must be added to the coin to lift it over an energy barrier, after which it becomes just as unstable as the balanced rod and quickly falls to the ground.

            In the absence of external forces the balanced penny is completely stable and will stay balanced forever.
However two possible internal mechanisms may cause it to fall prematurely--thermal agitation and/or quantum tunneling.

            According to the classical Equipartition Theorem, each mechanical degree of freedom possesses on the average one kT of thermal energy where k is Boltzmann's constant and T is the absolute temperature. This average thermal energy is quite small compared to the energy barrier confining the coin but thermal fluctuations may occasionally occur which concentrate thermal energy in a given mode. If this added energy exceeds the barrier energy the penny can escape its confinement by a process called "thermal hopping".

            Furthermore, the penny, considered as a quantum object, may actually tunnel thru the energy barrier in a classically impossible manner, emerging "as if by magic" on the other side where it then proceeds to fall to the ground. Quantum tunneling is a well-established physical phenomenon that forms the basis for many technological achievements including STMs (Scanning Tunneling Microscopes), tunnel diodes and computer flash drives.

            Because the penny is so massive and the associated energy barrier relatively large, the effects of thermal hopping and of quantum tunneling are expected to be small. It is the purpose of this article to estimate the magnitude of these effects in the absence of which the lifetime of the upright penny state would be infinite. How many years must we wait before a large thermal fluctuation topples the one-cent piece? How many years will pass before a lucky quantum tunneling event accomplishes the same deed?

penny potential
            The left-hand figure illustrates the magnitude and shape of the energy well that keeps the penny standing upright. Like the balanced rod the penny is a kind of inverted pendulum characterized by a pivot point (two in the case of the penny), a moment of inertia J, an energy barrier E(max) and a barrier width W.

            This energy barrier exists because one needs force to tilt the upright penny because small angles of tilt actually raise the penny's center of gravity. For small angles of tilt gravity provides a restoring force that returns the penny to its upright position. When the coin is tilted past a certain critical angle (about 3.15 degrees) gravity acts in the opposite direction to cause the coin to fall.

            In the case of the US penny the energy barrier is about 3.4 ergs high and about 6.3 angular degrees wide. To calculate hopping and tunneling probabilities these quantities must be expressed in thermal and quantum units.


THE UPRIGHT PENNY'S NORMAL OSCILLATIONS

         
  The penny is trapped in a potential well from which it is seeking to escape. The lifetime Tpenny of this metastable state is the inverse of its frequency of escape. The penny's frequency of escape FE is equal to its frequency of attempt (to escape) multiplied by its probability of success: ( FE = FA x P) where the probability P will be calculated both for thermal hopping (PT) and for quantum tunneling (PQ). Here I calculate the frequency of attempt FA.

            For small angles of oscillation the frequency of a normal pendulum does not depend on the amplitude of its oscillation, a property first observed by Galileo (allegedly in church) and useful in timekeeping. For a pendulum the potential energy well is parabolic with zero slope at the bottom and gradually increasing slope at the sides. For small displacements the restoring force of an ordinary pendulum is zero the bottom of the well and increases as the pendulum is displaced.

            In contrast to the simple pendulum, the restoring force acting on the penny is maximum at the bottom of the well and decreases as the penny is tilted. Consequently the oscillation frequency of the upright penny depends on its displacement. Expressing the penny's tilt from the vertical (a) as a fraction (x) of the critical angle S/R we have:
a = x (S/R). When x = 0, penny is upright; when x = 1, penny is about to fall.

penny parameters

            Moving bodies are subject to Newton's Law of Motion:

 F = Ma

where F is force, M is mass and a is acceleration.

            Bodies, like the coin which are constrained to rotate about a fixed point, obey Newton's analogous Law of Rotational Motion:

T = J z

where T is torque, J is moment of inertia and z is angular acceleration.

            The moment of inertia J of the penny pivoting on its edge is given by:

J = 5/4 MR2

and the torque on the penny T is given (see right-hand figure) by:

T = Mgs

            Plugging these values into the Law of Rotational motion we easily obtain a simple expression for the frequency F(x) of the penny's oscillation as a function of its tilt parameter x:

F(x) = SQRT (g/80 Rx)

            The energy E(x) corresponding to this particular frequency of oscillation is:

E(x) = 1/2 (S/R)2MgSx2

            Note that when the amplitude x of the oscillations goes to zero, the frequency F(x) becomes infinite. However the energy E(x) also goes to zero so these infinite frequencies are non-physical. Now that we know the relationship between amplitude and frequency how shall we choose an appropriate frequency of oscillation for the isolated upright penny?

            The Equipartition Theorem states that each degree of freedom contains an average of one kT of energy. If we then set E(x) = kT we can solve for the tilt amplitude xT which corresponds to an oscillation that contains exactly one kT of energy. Then F (xT) will be the room-temperature frequency of oscillation of this "hot copper oscillator". We will use this frequency as the "escape attempt" frequency. The penny is rattling around in its little gravity well with frequency F (xT) and has a certain probability to escape at each attempt--a probability we will estimate in the next section. Meanwhile the relevant parameters calculated for this "hot copper oscillator" at room temperature (T = 300 degrees Kelvin) are:

E(xT) = kT = 4.8 x 10-14 ergs

F(xT) = 11 kHz = FA

xT = 5 x 10-7

            We see that due to thermal agitation the penny is vibrating at a frequency of about 10, 000 cycles per second and the amplitude of this vibration is 5 ten millionth of the critical tilt angle S/R. We take this frequency to be the escape attempt frequency and now calculate the thermal and quantum probabilities for a successful attempt. And hence the estimated lifetime for a penny standing on its edge.


PROBABILITIES FOR A SUCCESSFUL ESCAPE

THERMAL HOPPING

           
The probability PT that the penny with energy kT will surmount an energy barrier of height NkT is just:

PT = exp (-N)

where N is the number of what might be called "thermal quanta" needed to surmount the barrier. If this number N is small,  the probability of thermal hopping is high. If N is a large number, the probability of hopping is exponentially small. In the case of the upright US penny, the energy barrier E(max) is 3.4 ergs. Expressing this energy in "thermal quanta" we obtain:

E(max) = NkT

N = 7.1 x 1013

PT = exp (-7.1 x 1013)

Multiplying this probability by the attempt frequency F (xT), and transforming from natural exponentials to powers of ten, we obtain the penny's thermal hopping frequency FT:

FT =  1.1 x 10-3.1 x10 to the 13th power sec-1

which is the inverse of the thermal escape lifetime TT --how long the penny will resist being pushed over by its internal  thermal oscillations:


TT = 0.9 x 103.1 x10 to the 13th power  secs


This estimated lifetime is an inconceivably large number, not only stretching the powers of our imagination but also pushing the limits of decent mathematical notation. The lifetime of this upright penny against the force of thermal agitation is not merely a number with 13 zeros after it but a number with 1013 zeros after it! In practical terms this number is an extremely good approximation for "forever".  Balanced on its edge, perturbed only by thermal agitation, the penny will never fall.


QUANTUM TUNNELING

            So thermal agitation will never succeed in freeing the penny from its gravity well, but perhaps quantum mechanics can come to the rescue. The quantum probability for tunneling thru a barrier is related to the dimensions of the barrier as measured in units of "action" where action (momentum x distance) is expressed in units of Planck's constant. If the barrier consists of a small number of action units, the probability for tunneling will be large. If the barrier consists of a large number of action units the tunneling probability is exponentially small. Because the US penny is a macroscopic object we might reasonably expect that the number of action units that describe the gravity barrier is large. The only question then is: how large? How close to "forever" is the penny's quantum lifetime TQ?

            The probability for quantum tunneling depends both on the height H and width W of the barrier and on the barrier's shape. For a rectangular barrier, the shape factor G = 2. For a triangular barrier, G = 4/3. For a parabolic barrier, G = pi/2. The penny's gravity well is an example of a parabolic barrier so G = pi/2.

PQ = exp-G(HW)/h

where the product HW has the dimensions of action and can be expressed as a multiple M of Planck's constant of action h:

HW = Mh

Analogous to the thermal hopping probability which depends on the number of "thermal quanta" N needed to surmount the barrier, quantum tunneling depends on the number of "action quanta" M of which the barrier is composed. Expressed in terms of the number M of action quanta composing the barrier, the quantum tunneling probability is:

PQ = exp-GM

For a quantum particle in a box ("hampered displacement") the quantity of action that composes the barrier consists of the product of a momentum term times the barrier width in centimeters. For quantum "hampered rotation" (the case of the penny) the quantity of action is the product of an angular momentum term times the barrier width in radians. For the penny the barrier action HW is given by:

HW = SQRT[2J(E(max) - E(min)] x 2(S/R)

Setting E(min) = kT and expressing this barrier's strength in units of Planck's constant h, we arrive at the number of "action quanta" M constituting the penny's confining gravity well:

M = HW/h = 1.1 x 1026

We can now calculate the tunneling frequency FQ and its inverse TQ the tunneling lifetime:

FQ = FAPQ = FAexp-GM = 1.1 x 10-4.7 x 10 to the 25th power seconds-1


TQ = 1/FQ = 0.9 x 104.7 x 10 to the 25th power seconds


We calculated the thermal hopping lifetime to be immensely long, essentially "forever" but the quantum tunneling lifetime TQ is an even longer kind of "forever"--not a number with 25 zeros after it but a number with 1025 zeros after it!

            To compare this penny decay "forever" with ordinary eras, aeons and millenia, a year is composed of pi x 107 secs, the universe is supposed to be 14.5 billion years old (5 x 1017 secs) and the measured lifetime of the proton is greater than 1034 sec (a 1 with 34 zeros after it)--each an infinitesimal instant compared to an upright penny's quantum lifetime!

            When you stand a penny on edge there is a small chance that it will quantum-tunnel its way out of its gravity well. But this calculation gives new meaning to the term "small chance" and illustrates just how irrelevant quantum effects are to ordinary-sized objects. Quantum mechanics completely dominates the behavior of electrons, photons, atoms and molecules. As our technology improves we have been able to prepare and measure larger and larger systems whose behavior is non-classical, systems such as beakers of superfluid helium, room-sized superconducting niobium magnets, billions of rubidium atoms forming a Bose-Einstein Condensate, and bright non-classical beams of light.

            Some claim that quantum mechanics is fundamentally involved in every conscious experience (2) and that this intrinsic quantumness of their minds may allow conscious beings to perform what would be regarded in a classical world as miracles. One possibility put forth is that consciousness might be able to hasten or retard radioactive decay which is governed by quantum laws similar to those we have invoked here to study the "decay of a penny". Conscious control of nuclear decay, for example, would allow us to clean up nuclear waste by controlled burning and could turn nuclear weapons stockpiles into liabilities if the lifetimes of their radioactive plutonium cores could be shortened by focused conscious intent.  

            For developing our "psychic muscles", it is probably best to start small. What is the tiniest, what is "the most quantum", what is that system in the world most sensitive to psychic influence? What is the ultimate PK toy? Is it the uninhibited neural synapse, one particular radioactive nucleus, some special enzymatic chemical reaction we might influence by "prayer"? The above calculation shows that even an object as small as a US penny standing on edge is "not quantum enough" to serve as an easy training weight for a tentative psychic muscle gym.

            On the other hand, if someone could learn how to knock over pennies with their mind and it had something to do with quantum mechanics, this calculation would serve to quantify the "miraculousness" of such an feat.


NICK HERBERT
Boulder Creek, CA
quanta@cruzio.com
Dec 15, 2006


(1) Milan Batista & Joze Paternelj "Quantum Cards & Quantum Rods" http://xxx.lanl.gov/abs/quant-ph/0611036

(2) See, for example, Nick Herbert "Elemental Mind" Dutton (1993)