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*Previously, on PHY 343… As the newest recruit to Dr. G’s research group in the class of 2030,*

*you were enlisted to help plan and be part of a two-person crew on a one-way voyage to the nearest*

*exoplanet, Proxima b, in orbit around the star Proxima Centauri, 4.24 light-years from the Sun.*

*Ten Earth-years after take-o↵, you were awakened in the spacepod prior to your arrival in the*

*Proxima Centauri system. The onboard computer reported an anomalously suddenly loss in acceleration from the laser, as well as the passage of an explosive device traveling toward Earth. Using*

*the on-board 121.567 nm laser, you measure a Doppler shift of the reflected beam which revealed*

*that the object was moving at 0.5c in the spacepod’s frame, and -0.17c in Earth’s frame. A quick*

*calculation showed that the citizens of Earth had 9.3 years to mount a defense. You discovered that*

*your traveling companion was none other than Dr. G. You filled him in on the readings from the*

*spacepod’s instruments, and your calculations regarding the explosive device headed toward Earth.*

*Dr. G looked over your work and agreed. “Things are worse than we thought… It looks like there*

*was a coup of the government from the faction calling themselves the Orange Party. The new*

*regime has expounded an anti-science policy, shut o↵ the laser, and incarcerated all scientists. All*

*our colleagues could do was send out this before they were sent to interment camps. They will have*

*no way of knowing about the bomb. It seems we will either be the only humans left in the Galaxy,*

*or we have to somehow figure out how stop that device.”*

*And now…*

*“There’s something else odd, Dr. G.” you reply informing him of the hot star that was detected*

*by the instruments. “I thought Proxima Centauri was the closest star to the Sun. How could we*

*find another star on our journey?” Dr. G seems impressed at your recollection of local Galactic*

*astrography. “Indeed, that is curious. Let’s take a look at the data you were looking at.” You show*

*him your analysis of the spectrum and your inference of the presence of hydrogen-like molybdenum*

*(Mo*41+*). “My young apprentice – excellent work on two accounts.” Dr. G remarks looking over*

*your analysis. “This may be exactly what we need to save Earth. But we both have work to do. I’ll*

*work on figuring out how the star got here. You work on collecting some DougAds from this star.”*

*Before you can give him a blank stare, Dr. G apologizes. “Sorry, you need a bit more explanation.*

*DougAds are the name given to the 42 **µ**eV/c*2 *particles that were created after the discovery of*

*Higgs. Particle physicists couldn’t resist the joke. Anyway, we discovered subsequently that these*

*particles actually occur naturally in some hot stars, and that they have the same electric charge as a*

*proton. The amount of *42*Mo in this star indicates this is one of those stars. We can use DougAds*

*to help save Earth. Go get us some.”*

1. In order to bone up on this new particle, you decide to do some calculations regarding

DougAds and 42Mo. You recall that the strong force is important within a distance of 10*!*15 m

of the nucleus, while the Coulomb force is more important outside that radius.

(a) For a star with a central temperature of *T *= 2*⇥*107 K, how close can an average DougAd

get to a 42Mo nucleus? Given the extreme di↵erence in the masses, you figure that the

molybdenum nucleus can be taken to be at rest in the frame of the star’s core.

(b) The rate at which DougAds would interact with 42Mo, giving something measurable,

depends on the probability of a DougAd tunneling into a 42Mo nucleus. Determine this

probability for an average DougAd. You recall the work of Banner & Stark (2012),

surmising that this temperature is unlikely to yield a “high and wide barrier” for which

you had previously worked out an approximate tunneling probability. You’ll have to use

the full-blown version that Dr. G gave you in the class notes.

*You check your calculations with Dr. G before proceeding. After all, the fate of the Earth is at*

*stake. “Yes, that looks good.” Dr. G replies. “Now go construct a particle-in-a-box device to trap*

*DougAds with that energy.”*

2. Determine the size of a one-dimensional infinite-square well box whose ground-state energy

is the requisite energy of a DougAd from 1b.

(a) Do this in the non-relativistic limit…

(b) … and in the relativistic limit…

(c) … and compute the fractional error in using the non-relativistic limit.

[Hint: Recall the conditions that quantize the wavenumber, *k*, and hence the momentum, *p*,

and how this relates to the energy.]

*You construct the device. Aligning the axis of your DougAd collector with the hot star, you*

*begin… collecting… DougAds. Relieved that your calculations were good, you check on Dr.*

*G. “I thought the spectrum of that star looked familiar. It’s Rigel. It used to be part of*

*the constellation of Orion. Apparently it is now part of the Centaurus constellation. It was*

*deflected here through a gravity assist from a wandering black hole. From the direction that*

*Rigel had to travel, I have figured out the most likely location of the black hole and that its*

*mass is most likely *6 *⇥ *1031 *kg.”*

*“We’re going use an old trick from Thorne & Nolan (2014)**a **and use the DougAds to send a message*

*to Earth prior to the onset of this Orange Party. We need to do some more calculations if this*

*is going to work. Hope you remember your 3D quantum mechanics in the presence of a central*

*potential. The plan is to use the DougAds to encode a signal, send them into the black hole, and*

*have them tunnel out of the black hole in the year 2020. This will be right at the end of The Orange*

*One’s first term when we discovered DougAds. We’ll need to know the probability of tunneling so*

*that we know how many DougAds to use.”*

*a**http://www.imdb.com/title/tt0816692/*

3. You don’t seem to recall working out tunneling out of a black hole in class. But you do

remember Dr. G saying something about it. Well, no time like the present to figure it out.

You do remember working out the wavefunction, energies, and orbital angular momenta of

the Coulomb potential. So, those insights should be useful here. The radial wavefunctions in

the presence of the purely radial potential are given by the series sum:

*R*

*n,`*(*⇢*) = *⇢**` n*

*n**!**`**!*1

X *k*

=0

*a**k**⇢**k n**e**!**⇢**n**/*2*,*

where the coe”cients *a**k *are defined recursively as

*a**k*+1 =

*k *+ *` *+ 1 *” **n*

(*k *+ 1)(*k *+ 2*` *+ 2)*a**k**,*

*n *is the principal quantum number, *` *is the orbital angular momentum quantum number,

and *⇢**n *is a “normalized” radial coordinate which can be written *⇢**n *= *r/r**n*. For a Coulomb

potential (*U *= *“**kZe r *2 ), *r**n *= *na *2*Z **! *(Note: In this expression for the normalizing radius, *a**“*

is the Bohr radius, not to be confused with the coe”cient of the *k *= 0 term in the radial

wavefunction. The coe”cient of the zeroth term of the wavefunction is determine through

normalization.)

(a) To get *r**n *for the gravitational potential energy, write down the 3D, time-independent

radial wave equation using *U*(*r*) = *“**GMm r *, and follow the development in the book for

the Coulomb potential energy (section 8.5, equations 8.17–8.38). [The key di↵erence

is that you are replacing *kZe*2 for the Coulomb potential energy with *GMm *for the

gravitation potential energy. ] Use the radial wave equation with the *n *= 1 radial

wavefunction and ensure the resulting equation remains valid at *r *= 0 to determine *r*1,

and then by extension, *r**n*.

(b) Continuing the analogy of the gravitational potential with the Coulomb potential, show

that the permitted energy levels under a gravitational potential are:

*E**n*

= *“*

*G*2*M *2*m*3

2~2*n*2

[Hint: Start with the quantized energy for the Coulomb potential and recast it in terms

of the *r*2

*n *given above for the Coulomb potential energy and other constants that are

not specific to the Coulomb potential (e.g., the Bohr radius is specific to the Coulomb

potential energy as it depends on *ke*2, but ~ is not). Treat this as a more fundamental relation for central potentials and use your new gravity-specific *r**n *to arrive at its

quantized energy levels.]

(c) What minimum gravitational potential energy does a DougAd orbiting the singularity

need in order to reach the gravitational radius (i.e., where the classical escape speed is

the speed of light)?

(d) Determine the smallest principal quantum number that yields at least the energy from

the previous step.

(e) Normalize the wavefunction for the principal quantum number from the previous step,

and the orbital angular momentum quantum number that yields the largest fraction

of the wavefunction outside the gravitational radius. [Note: Use physical insights to

reduce the apparent complexity of the problem. The definition of the gamma function

may be useful here: #(*n*) = R0*1 **t**n**!*1*e**!**t**dt*, which also follows the recursive definition

#(*n *+ 1) = *n*#(*n*), and is therefore a more general form of a factorial. You may leave

your answer in terms of this function, but you must specify the argument where the

function is to be evaluated.]

(f) From the wavefunction, determine the probability that a DougAd can tunnel out of the

black hole. [Note: The definition of the incomplete gamma function may be useful here:

#(*n, x*) = R*x **1 **t**n**!*1*e**!**t**dt*. You may leave your answer in terms of this function, but you

must specify the arguments where the function is to be evaluated.]

*Something begins to perplex you about this plan. How, you wonder, are you going to control the*

*direction that the DougAds are going to tunnel out of the black hole? Will they, and the message*

*they will carry, even get to Earth? You ask Dr. G and he agrees. “You’re right… we don’t know*

*what direction they will travel. We’ll have to constrain the magnetic quantum number to narrow*

*down the plane in which the tunneling occurs. But we’ll need a continuous stream of DougAds to*

*maximize the likelihood of detection.” You look over at your DougAd collector. “We’re going to*

*need a bigger boat…”*

*Stay tuned for*

*The Salvation of Earth, Part 2*

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