212HomeWork8sol - Physics 212A Homework #8 Name: 1....

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Unformatted text preview: Physics 212A Homework #8 Name: 1. Relations between special functions and hypergeometric functions. a) Make a table for n=1,6 to show that the Legendre functions P n (1-2x) can be expressed in terms of 2F1[- n,n+1,1,x] . See Mathematical Functions-> Hypergeometric Related in the Help browser for more information on Mathematica 's commands for these functions b) Show that the Laguerre polynomials ( important for radial wave functions, often denoted by L n m ) can be expressed in terms of the confluent hypergeometric function U[-n, m+1, x] . Find the constant of proportionality. In[99]:= Table @8 HermiteH @ n, x D , HypergeometricU @- n 2, 1 2, x ^ 2 D , Simplify @ HermiteH @ n, x D HypergeometricU @- n 2, 1 2, x ^ 2 DD< , 8 n, 1, 6 <D PowerExpand Expand TableForm Out[99]//TableForm= 2 x x 2- 2 + 4 x 2- 1 2 + x 2 4- 12 x + 8 x 3- 3 x 2 + x 3 8 12- 48 x 2 + 16 x 4 3 4- 3 x 2 + x 4 16 120 x- 160 x 3 + 32 x 5 15 x 4- 5 x 3 + x 5 32- 120 + 720 x 2- 480 x 4 + 64 x 6- 15 8 + 45 x 2 4- 15 x 4 2 + x 6 64 ratio is 2^n : In[100]:= Table @8 HermiteH @ n, x D , 2 ^ n HypergeometricU @- n 2, 1 2, x ^ 2 D< , 8 n, 1, 6 <D PowerExpand Expand TableForm Out[100]//TableForm= 2 x 2 x- 2 + 4 x 2- 2 + 4 x 2- 12 x + 8 x 3- 12 x + 8 x 3 12- 48 x 2 + 16 x 4 12- 48 x 2 + 16 x 4 120 x- 160 x 3 + 32 x 5 120 x- 160 x 3 + 32 x 5- 120 + 720 x 2- 480 x 4 + 64 x 6- 120 + 720 x 2- 480 x 4 + 64 x 6 2. A particle of mass m is coupled to a quartic potential V0 x 4 . The Schrodinger equation for its stationary eigenstates is - 2 2 m '' @ x D + V0 x 4 @ x D = En @ x D . a) Find a characteristic length and energy that are independent of En, and put the equation in dimensionless form. Load the dimensions and DE packages Needs @ "Units`" D Needs @ "PhysicalConstants` " D Needs @ "MMHTools`DimTools` " D The symbols 8 c, kB, , g, G, me, amu, Me, Ms, Na, 0, 0, e, , Rgas < have been assigned SI unit specifiers Needs @ "MMHTools`DETools` " D The equation as written with dimensions is de4 =- 2 2 m '' @ x D + V0 x ^ 4 @ x D- En @ x D- En @ x D + V0 x 4 @ x D- 2 @ x D 2 m The relevant parameters of the problem are params = 8 m Kilogram, V0 Joule Meter ^ 4, , En Joule < ToSymbolsUnits ReduceUnits : Kilogram m, Kilogram V0 Meter 2 Second 2 , Kilogram Meter 2 Second , En Kilogram Meter 2 Second 2 > Use dimanal to find a combination that gives a length dimanal @ params, 8 Kilogram, Meter, Second < , Meter D : 1 3 m 1 6 V0 1 6 , : En 3 m 2 V0 4 >> and another combination that gives an energy dimanal @ params, 8 Kilogram, Meter, Second < , Joule ReduceUnits D : V0 1 3 4 3 m 2 3 , : En 3 m 2 V0 4 >> Now change the x variable in the DE to x = (characteristic lenth ) xr, where xr is dimensionless ChangeVariable B de4, x fi xr 1 3 m 1 6 V0 1 6 , , , x, xr F Simplify- En + V0 1 3 xr 4 4 3 m 2 3 @ xr D- V0 1 3 4 3 @ xr D 2 m 2 3 The combination of factors that multiplies and '' we recognize as the characteristic energy. Divide through by it:'' we recognize as the characteristic energy....
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212HomeWork8sol - Physics 212A Homework #8 Name: 1....

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