Here Ï is an independent variable. The solution is periodic in Ïwith period âÏ = 2Ï. Also, a, e, T are constants that determine theparameters of the orbit.

Figure 2 shows the relation between Ï and the spatialcoordinates. In the figure, the coordinates (Î¾,Î·) are defined by Î¾= x + ae and Î· = y a /b,

withb=a*1âe2 ]1/2.

(A) What kind of curve is the orbit?

(D) Determine Î¾(Ï) and Î·(Ï).

(E) Express r as a function Ï .

(F) The angular momentum is L = m r2dÏ/dt. Determine L in termsof {a,e,T} Hence verify that L is a constant of the motion.

(G) The energy isE = [1/2] m ( dr/dt )2+ [1/2] m r2(dÏ/dt)2â GMm/r.Determine E in terms of {a,e,T}.

(H) E must be a constant of the motion. Hence determine T and Efrom the equation that you obtained in (F).

(I) Write T in terms of the spatial orbit parameters and theforce constant GM.

(J) Now express L and E in terms of a, e, GM.

(K) In Figure 1, determine the coordinates of the points P, A,and R

(L) In Figure 1, determine the time t at the point R.

A Particle in a box
In order to answer the following questions you will need toconsideran infinite potential
well of length L, in which a particle has been confined.

1. For principal quantum number n=7, write code to calculatethenormalization constant.
Then, plot the probability density function for this state (usingL= 10).
2. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <x> and <x2> . Is <x>oneof the most probable positions? Explain
why.
3. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <p> and <p2> . Show the valueyouobtain for <p2> is equal to
2mE7 (where E7 is the energy of the n = 7 level). Remember that pisa differential
operator so you will have to use the diff command.
4. The uncertainty of a quantum measurement is defined on page222,Equation 6.41 (3rd
edition SMM). Using this equation calculate the uncertainties oftheposition and
momentum of a particle in the n = 5 state. Compute the productofthese uncertainties and
compare this with the prediction of the Heisenberguncertaintyprinciple.
Quantum oscillator
5. Repeat calculations in the previous question for the groundstatewave function of the
quantum oscillator. Comment on the obtained result.

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Physics 110,000

Show that the spontaneous emission rate (Equation 9.56 A=[w0^3|p|^2]/[3pieohbarc^3]) for a transition from n,l to n',l' inhydrogen is

9.97 [e^2 w^3 I^2]/[3pieohbarx^3] * { [l+1] /[2l+1] if l' =l+1
'' ' ' '' ' ' '' { l/[2l+1] if l'=l-1

Where I=Integra[r^3 Rnl Rn'l'] dr

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Physics: Principles with Applications

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Show that the spontaneous emission rate (Equation 9.56 A=[w0^3|p|^2]/[3*pi*eo*hbar*c^3]) for a transition from n,l to n',l' inhydrogen is9.97 [e^2 w^3 I^2]/[3*pi*eo*hbar*x^3] * { [l+1] /[2l+1] if l' =l+1 '' ' ' '' ' ' '' { l/[2l+1] if l'=l-1Where I=Integra[r^3 Rnl Rn'l'] dr

Related textbook solutions

Physics: Principles with Applications

Related questions

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Show that the spontaneous emission rate (Equation 9.56 A=[w0^3|p|^2]/[3pieohbarc^3]) for a transition from n,l to n',l' inhydrogen is

9.97 [e^2 w^3 I^2]/[3pieohbarx^3] * { [l+1] /[2l+1] if l' =l+1
'' ' ' '' ' ' '' { l/[2l+1] if l'=l-1

Where I=Integra[r^3 Rnl Rn'l'] dr