Consider the earth–sun system as a gravitational analog to the hydrogen atom.

(a) What is the potential energy function (replacing Equation 4.52)? (Let be the mass of the earth, and M the mass of the sun.)

V(r)=-e24π00,1r

(b) What is the “Bohr radius,”ag,for this system? Work out the actual number.

(c) Write down the gravitational “Bohr formula,” and, by equating Ento the classical energy of a planet in a circular orbit of radius r0, show that n=r0/ag.From this, estimate the quantum number n of the earth.

(d) Suppose the earth made a transition to the next lower level(n-1) . How much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, gravitation) be? (Express your answer in light years-is the remarkable answer a coincidence?).

Short Answer

Expert verified

(a)The potential energy function isV(r)=-GMmr

(b)The actual number isa=2.34×10-138)m

(c)By the quantum number of the earthn=r0ag

(d) The wave length emitted by photonsλ=1ly

Step by step solution

01

Given:

A potential function is a function of the position of an object. It can be defined only for conservative forces. A forces is conservative if the work it does on n object only on the initial and final position of the object and net on the path. The gravitational force is a conservative force.

The potential is given by:

V(r)=-e24πϵ01r

02

The potential energy function

Consdier the Earth-sun system as a gravitational analog to the hydrogen atom. The potential is:

V(r)=-GMmrm/r.

Where G is the gravitational constant, G=6.673×10-11m3kg-1s-2

translates hydrogen results to the gravitational analogs.

03

(b) The “Bohr radius” is

The Bohr radius for the Earth can be found by replacinge2 bymM and1/4πε0 by G ,so we get:

ag=2GMm2

Substitute with the numerical values to get:

ag=1.0546×10-34Js26.6726×10-11m/kg.s21.9892×1030kg5.98×1024kg2ag=2.34×10-138m

04

Step 4:(c) Finding the formula

Equation 4.70 (the allowed energies) is given by:

En=-m2ħ2(GMm)21n2

There are many items that could be altered to test the recreation of another. These changing quantities are called variables. A variables is any factor, trait, or condition that can exist in differing amounts or types. An experiment usually amounts or types, An experiment usually has three kinds of variables: independent, dependent, and controlled.

Ec=12mv2-GMmroGMmro2=mv2ro12mv2=GMm2ro

So,

cEc=-GMm2ro

=-m22(GMm)21n2n2=GMm22ro=roag

n=roag.

ro=earth-sundistance=1.496×1011mn=1.496×10112.34×10-138=2.53×1074r0=earth-sundistance=1m.

05

(d) Wavelength of the emitted photon

The wavelength of the emitted photon

E=-G2M2m32ħ21n+12-1n2.1n+12=1n21+1/n21n21-2n

So,

role="math" localid="1658389122151" 1(n+1)2-1n21n21-2n-1=-2n3;ΔE=G2M2m32n3E=6.67×10-1121.99×103025.98×102431.055×10-3422.53×743=2.09×10-41J

Ep=E=hv=hcλλ=3×1086.63×10-34/2.09×10-41=9.52×1015m

But1ly=9.46×1015m.thatλ1ly,n2=GMm2ro/2λ=chE=c2πħħ2n3G2M2m3=c2πħ2G2M2m3GMm2r0ħ23/2=c2πr03GM

But(from(c))

v=GM/ro=2πro, where T is the period of the orbit (in this case one year),

so T=2πro3/GMand hence λ=cT(one light year).

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