![Problem # 2 Consider the operators A =-and B 흙 + ах a) Show that A, B] = 0. b) Show that f(x,y)- exty is an eigenfunction of](http://img.homeworklib.com/questions/a69cd850-d29a-11ea-a211-633269d80b0f.png?x-oss-process=image/resize,w_560)
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Problem # 2 Consider the operators A =-and B 흙 + ах a) Show that A,...
3. Consider the eigenvalue problem 1
The answer is given. Please show more detailed steps, thank
you.
3. Consider the eigenvalue problem 1<x<2 dx2 y(1)=0,y(2) = 0. dx iwrite it in the standard Sturm-Liouville form. ii) Show that 0 by the Rayleigh Quotient. dx p(x)-x, q(x) = 0, σ(x)-1 According the Raileigh Quotient Any eigenvalue is related to its eigenfunction φ(x) by - x p(x) dr Since the B.C. are ф(1)-0 and ф(2-0, so dx
3. Consider the eigenvalue problem 1
qm 09.4 4. The commutation relations defining the angular momentum operators can be written [Îx, Îy]...
qm 09.4
4. The commutation relations defining the angular momentum operators can be written [Îx, Îy] = iħẢz, with similar equations for cyclic permutations of x, y and z. Angular momentum raising and lowering operators can be defined as În = Îx ihy (i) Show that [Lz, L.] = +ħL. [6 marks] (ii) If øm is an eigenfunction of ł, with eigenvalue mħ, show that the state given by L+øm is also an eigenfunction of L, but with an eigenvalue...
I need some help on this problem. Thank you! 15. Consider the differential equation (*) х%3D Ах + vex where v is an eig...
I need some help on this problem.
Thank you!
15. Consider the differential equation (*) х%3D Ах + vex where v is an eigenvector of A with eigenvalue X. Suppose moreover, that A has linearly independent eigenvectors v', v2,... , v", with distinct eigenvalues Ар А, гespectively. (a) Show that (*) has no solution /(t) of the form V(t)= ae*. Hint: Write a= а у+.. +a, v. (b) Show that (*) has a solution /(t) of the form n 0)...
3. Consider the boundary value problem for y(x), -1 < x < 1: **) g” +...
3. Consider the boundary value problem for y(x), -1 < x < 1: **) g” + Ag 0, y(-1) 0, y(1) = 0 (a) Find all positive eigenvalues for (**). (b) For each positive eigenvalue In, find a correspoding eigenfunction yn(x).
1. Consider the following operators: a. T = ve dax2 b. p = Wild c. 8=...
1. Consider the following operators: a. T = ve dax2 b. p = Wild c. 8= x Are the following functions (i.-iv. eigenfunctions of operators a, b, or c? If so, provide the eigenvalue. i. 4 cos (4x) ii. e-7x iii. ei3x iv. x2-3
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form,...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determi...
#2 ONLY PLEASE
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
Solve part (d) 6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the proble...
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
1. [Total: 12 pts] Some practice with the ladder operators: a) (4 pts) Show that for...
1. [Total: 12 pts] Some practice with the ladder operators: a) (4 pts) Show that for the harmonic oscillator [À,ât] = = kwât b) [8 pts] Show that for any functions f(x) and g(x), that go to zero at infinity, i.e. f(+00) = g(+00) = 0: f*(x)(â+9(x))dx = | (âf(x))*g(x))dx (2) - In other words, â+ and â_ are Hermitian conjugate or adjoint to each other.
The answer is given. Please show more detailed steps, thank
you.
3. Consider the eigenvalue problem 1<x<2 dx2 y(1)=0,y(2) = 0. dx iwrite it in the standard Sturm-Liouville form. ii) Show that 0 by the Rayleigh Quotient. dx p(x)-x, q(x) = 0, σ(x)-1 According the Raileigh Quotient Any eigenvalue is related to its eigenfunction φ(x) by - x p(x) dr Since the B.C. are ф(1)-0 and ф(2-0, so dx
3. Consider the eigenvalue problem 1
qm 09.4
4. The commutation relations defining the angular momentum operators can be written [Îx, Îy] = iħẢz, with similar equations for cyclic permutations of x, y and z. Angular momentum raising and lowering operators can be defined as În = Îx ihy (i) Show that [Lz, L.] = +ħL. [6 marks] (ii) If øm is an eigenfunction of ł, with eigenvalue mħ, show that the state given by L+øm is also an eigenfunction of L, but with an eigenvalue...
I need some help on this problem.
Thank you!
15. Consider the differential equation (*) х%3D Ах + vex where v is an eigenvector of A with eigenvalue X. Suppose moreover, that A has linearly independent eigenvectors v', v2,... , v", with distinct eigenvalues Ар А, гespectively. (a) Show that (*) has no solution /(t) of the form V(t)= ae*. Hint: Write a= а у+.. +a, v. (b) Show that (*) has a solution /(t) of the form n 0)...
3. Consider the boundary value problem for y(x), -1 < x < 1: **) g” + Ag 0, y(-1) 0, y(1) = 0 (a) Find all positive eigenvalues for (**). (b) For each positive eigenvalue In, find a correspoding eigenfunction yn(x).
1. Consider the following operators: a. T = ve dax2 b. p = Wild c. 8= x Are the following functions (i.-iv. eigenfunctions of operators a, b, or c? If so, provide the eigenvalue. i. 4 cos (4x) ii. e-7x iii. ei3x iv. x2-3
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
#2 ONLY PLEASE
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
1. [Total: 12 pts] Some practice with the ladder operators: a) (4 pts) Show that for the harmonic oscillator [À,ât] = = kwât b) [8 pts] Show that for any functions f(x) and g(x), that go to zero at infinity, i.e. f(+00) = g(+00) = 0: f*(x)(â+9(x))dx = | (âf(x))*g(x))dx (2) - In other words, â+ and â_ are Hermitian conjugate or adjoint to each other.
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