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Physics 102 Extra Credit Legendre Polynomials Problem The following problem is worth 5 ertra credit points!...
The expression Φ(x, h)-(1-2xh + h2)-1/2 where |hl < 1 is the generating function for Legendre polynomials. φ(x, h) can be expressed as a sum of Legendre polynomials The function (x, h) = Po(x) + h...
The expression Φ(x, h)-(1-2xh + h2)-1/2 where |hl < 1 is the generating function for Legendre polynomials. φ(x, h) can be expressed as a sum of Legendre polynomials The function (x, h) = Po(x) + hA(x) + h2Pg(x) + hn (x) The generating function of the Legendre polynomials has some applications in Physics, such as expressing the electric potential at point P due to a charge q. The location of the charge is r with respect to the origin O...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But no...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation...
6. (10 points Extra Credit) Electrodynamics is not the only subject that utilizes Gauss' Law. We can also use it to study Newtonian gravity. The acceleration due to gravity (9can be written as, w...
6. (10 points Extra Credit) Electrodynamics is not the only subject that utilizes Gauss' Law. We can also use it to study Newtonian gravity. The acceleration due to gravity (9can be written as, where G is Newton's gravitational constant and ρ is the m ass density. This leads us to the usual formulation of Newton's universal law of gravity,或刃--GM(f/r, as expected (if we assume V xğ-0). This "irrotational" condition allows us write (in analogy to the electric field), --Vo and...
Please explain the solution and write clearly for nu, ber 25. Thanks. 25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-...
Please explain the solution and write clearly for nu, ber 25.
Thanks.
25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
ANSWER ALL PARTS Problem 1. (30 points) Consider a line with uniform charge density io along...
ANSWER ALL PARTS
Problem 1. (30 points) Consider a line with uniform charge density io along a z-axis that extends from in the diagram below a to- u, as seen 2a As found in the previous exam, the algebraic form of the electric potential along the z axis is (a) we want to expand this electric potential function as an infinite series for the case when a. However, the logarithm function is not amenable to a binomial expansion, so we...
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show...
Quantum Physics
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show, that if the wave function takes the form 9(r,6, o) . R (r) (6)$(o) that the SchrodinDer equation can be separated in three equations d. (sin ) +1(1+1)sin2@62 ㎡Θ, and b) Show that imposing the boundary condition ф (ф)-ф (ф 2x) feguires that m-0, 1, 2, 3, ' . . dThe hrst few Legendre polymomials are given by 0-63-15 The "associated Legendre functions"...
Recall that an energy eigenfunction of any central potential V (r) may be writtren as ψn`m(r,...
Recall that an energy eigenfunction of any central potential V
(r) may be writtren as ψn`m(r, θ, φ) = Rn`(r)Y`m(θ, φ). This
problem explores the behavior of ψ in the vicinity of the origin r
= 0. Recall that the function u(r) = rRn`(r) satisfies the
equation
− ~ 2 2m d 2u dr2 + ~ 2 `(` + 1) 2mr2 + V (r) u = Eu, (1)
where E is the energy eigenvalue. Note that Eq. (1) has the...
2nd attached picture is problem 1 from HW 2 1. (10 Points Exam Extra Credit): Let's...
2nd attached picture is problem 1 from HW 2
1. (10 Points Exam Extra Credit): Let's revisit the problem of how to compute derivatives of basis vectors, which we did in Problem 1 of HWW2 (note: you will need to refer back to this HW at to do this problem). Consider the Laplacian operator, V2, in spherical coordinates. It looks like this, where the scalar (say V) goes into the 2) 10.2001 8801 VO - por l" or ) +...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 regio...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
The expression Φ(x, h)-(1-2xh + h2)-1/2 where |hl < 1 is the generating function for Legendre polynomials. φ(x, h) can be expressed as a sum of Legendre polynomials The function (x, h) = Po(x) + hA(x) + h2Pg(x) + hn (x) The generating function of the Legendre polynomials has some applications in Physics, such as expressing the electric potential at point P due to a charge q. The location of the charge is r with respect to the origin O...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation...
6. (10 points Extra Credit) Electrodynamics is not the only subject that utilizes Gauss' Law. We can also use it to study Newtonian gravity. The acceleration due to gravity (9can be written as, where G is Newton's gravitational constant and ρ is the m ass density. This leads us to the usual formulation of Newton's universal law of gravity,或刃--GM(f/r, as expected (if we assume V xğ-0). This "irrotational" condition allows us write (in analogy to the electric field), --Vo and...
Please explain the solution and write clearly for nu, ber 25.
Thanks.
25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
ANSWER ALL PARTS
Problem 1. (30 points) Consider a line with uniform charge density io along a z-axis that extends from in the diagram below a to- u, as seen 2a As found in the previous exam, the algebraic form of the electric potential along the z axis is (a) we want to expand this electric potential function as an infinite series for the case when a. However, the logarithm function is not amenable to a binomial expansion, so we...
Quantum Physics
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show, that if the wave function takes the form 9(r,6, o) . R (r) (6)$(o) that the SchrodinDer equation can be separated in three equations d. (sin ) +1(1+1)sin2@62 ㎡Θ, and b) Show that imposing the boundary condition ф (ф)-ф (ф 2x) feguires that m-0, 1, 2, 3, ' . . dThe hrst few Legendre polymomials are given by 0-63-15 The "associated Legendre functions"...
Recall that an energy eigenfunction of any central potential V
(r) may be writtren as ψn`m(r, θ, φ) = Rn`(r)Y`m(θ, φ). This
problem explores the behavior of ψ in the vicinity of the origin r
= 0. Recall that the function u(r) = rRn`(r) satisfies the
equation
− ~ 2 2m d 2u dr2 + ~ 2 `(` + 1) 2mr2 + V (r) u = Eu, (1)
where E is the energy eigenvalue. Note that Eq. (1) has the...
2nd attached picture is problem 1 from HW 2
1. (10 Points Exam Extra Credit): Let's revisit the problem of how to compute derivatives of basis vectors, which we did in Problem 1 of HWW2 (note: you will need to refer back to this HW at to do this problem). Consider the Laplacian operator, V2, in spherical coordinates. It looks like this, where the scalar (say V) goes into the 2) 10.2001 8801 VO - por l" or ) +...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
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