![HW12.2 (3 points) [1,1],[1,0),[1,-1) are the three eigenstates of Lạ& Ly with quantum number l = 1 and m = 1,0,-1, respective](http://img.homeworklib.com/questions/da491550-8e34-11eb-bcd3-a3d1adfc852b.png?x-oss-process=image/resize,w_560)
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HW12.2 (3 points) [1,1],[1,0),[1,-1) are the three eigenstates of Lạ& Ly with quantum number l =...
(a) Use the states with total angular momentum 1-1, |1,1), |1,0), and 3. |1,-1), as a basis. Express L as a 3 x 3 matrix. (Ans: (b) Find the eigenvalues of L. (Ans: h, 0, -h.) (c) Express the cor...
(a) Use the states with total angular momentum 1-1, |1,1), |1,0), and 3. |1,-1), as a basis. Express L as a 3 x 3 matrix. (Ans: (b) Find the eigenvalues of L. (Ans: h, 0, -h.) (c) Express the corresponding eigenstates of L in terms of the states 1,1), |1,0), and |1,-1). (Ans: For eigenvalue h, the eigenstate is (1, 1) + V21,0) + 1, -1))/2. For eigenvalue 0, the eigenstate is (1, 1)-|1,-1))/V2. For eigenvalue -h, the eigenstate is...
1. (20 points) Using the eigenstates of S, as the basis, (a) determine the eigenvalues and...
1. (20 points) Using the eigenstates of S, as the basis, (a) determine the eigenvalues and eigenstates of Sy; (b) determine the eigenvalues and eigenstates of S.ñ, where S is the spin-1/2 angular momentum, ñ is an unit vector. 2.(30 points) Consider a system with j = 1. (a) Explicitly write down <j = 1, m'J j = 1, m > in 3 x 3 matrix form. (b) Determine the eigenstate and eigenvectors of Jr. (c) Consider the eigenstate of...
4. If the general angular momentum quantum number j is 1 there is a triplet of |j, mj) states 1,1, 1,0), and 1,-1) In this case a matrix representation for the operators J, Jj and J, can be construct...
4. If the general angular momentum quantum number j is 1 there is a triplet of |j, mj) states 1,1, 1,0), and 1,-1) In this case a matrix representation for the operators J, Jj and J, can be constructed if we represent the lj,m,) triplet by three component column vectors as follows 0 0 0 0 0 Jz can then be represented by the matrix: 00 1 (a) Construct matrix representations for the raising and lowering operators, J and J...
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB....
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB. 1- Show that the commutator[L.L]is equal to zero. 2-Computethecommutator(Ly, Lx]. 3-Compute the commutator 3.Lx]. 4-Compute the commutator[LLY.Lx]. 5- Add the results from (2)-(4) to compute [LLyLx]. Lxercise 3: Matrix elements The angular momentum operators acting on the angular momentum eigenstates. Il determined by L-11.m) = (1 + 1) - m(m + 1)2.m +1) L_11.m) = /(1+1) - mm - 1)/1.m-1) L211,m) = hm|1,m) 1-...
2. (8 points) Consider an electron in a hydrogen atom with 1. There are three possible...
2. (8 points) Consider an electron in a hydrogen atom with 1. There are three possible eigen- states of the operator L2, given by 11, m) 11,1), 11,0), and 11,-1). (a) Recall that the raising/lower operators are given by L±-L, ±ill. Also recall the rela- tionship Use this relationship to determine the 3x3 matrix representation of Ly (using a basis com- prised of eigenstates of L). (b) What are the eigenvalues and associated eigenstates of the operator Ly? (c) If...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 4 = 2...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 4 = 2 with vi = and |_ G 12 = -2 with v2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: x(t) (50) = C1 + C2 e e B. In fundamental matrix form: (MCO) = I: C. As two equations: (write "c1" and "c2" for C1 and c2) x(t) = yt) =
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if...
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an...
2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an orthonormal basis for the linear space. Let us consider a real 3D space where a vector is denoted by a 3x1 column vector. Consider the symmetric matrix B-1 1 1 Show that the vectors 1,0, and1are eigenvectors of B, and find 0 their eigenvalues. Notice that these vectors are not orthogonal. (Of course they are not normalized but let's don't worry about it. You can...
Suppose that the matrix A A has the following eigenvalues and eigenvectors: (1 point) Suppose that...
Suppose that the matrix A A has the following eigenvalues and
eigenvectors:
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 2 = 2i with v1 = 2 - 5i and - 12 = -2i with v2 = (2+1) 2 + 5i Write the general real solution for the linear system r' = Ar, in the following forms: A. In eigenvalue/eigenvector form: 0 4 0 t MODE = C1 sin(2t) cos(2) 5 2 4 0 0...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: A1 = 4...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: A1 = 4 with = and [2] [i] Az = 3 with Ū2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: t (10) -- + C2 e e B. In fundamental matrix form: (39) - g(t). C. As two equations: (write "c1" and "c2" for C and C2) X(t) = g(t) = Note: if you are...
(a) Use the states with total angular momentum 1-1, |1,1), |1,0), and 3. |1,-1), as a basis. Express L as a 3 x 3 matrix. (Ans: (b) Find the eigenvalues of L. (Ans: h, 0, -h.) (c) Express the corresponding eigenstates of L in terms of the states 1,1), |1,0), and |1,-1). (Ans: For eigenvalue h, the eigenstate is (1, 1) + V21,0) + 1, -1))/2. For eigenvalue 0, the eigenstate is (1, 1)-|1,-1))/V2. For eigenvalue -h, the eigenstate is...
1. (20 points) Using the eigenstates of S, as the basis, (a) determine the eigenvalues and eigenstates of Sy; (b) determine the eigenvalues and eigenstates of S.ñ, where S is the spin-1/2 angular momentum, ñ is an unit vector. 2.(30 points) Consider a system with j = 1. (a) Explicitly write down <j = 1, m'J j = 1, m > in 3 x 3 matrix form. (b) Determine the eigenstate and eigenvectors of Jr. (c) Consider the eigenstate of...
4. If the general angular momentum quantum number j is 1 there is a triplet of |j, mj) states 1,1, 1,0), and 1,-1) In this case a matrix representation for the operators J, Jj and J, can be constructed if we represent the lj,m,) triplet by three component column vectors as follows 0 0 0 0 0 Jz can then be represented by the matrix: 00 1 (a) Construct matrix representations for the raising and lowering operators, J and J...
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB. 1- Show that the commutator[L.L]is equal to zero. 2-Computethecommutator(Ly, Lx]. 3-Compute the commutator 3.Lx]. 4-Compute the commutator[LLY.Lx]. 5- Add the results from (2)-(4) to compute [LLyLx]. Lxercise 3: Matrix elements The angular momentum operators acting on the angular momentum eigenstates. Il determined by L-11.m) = (1 + 1) - m(m + 1)2.m +1) L_11.m) = /(1+1) - mm - 1)/1.m-1) L211,m) = hm|1,m) 1-...
2. (8 points) Consider an electron in a hydrogen atom with 1. There are three possible eigen- states of the operator L2, given by 11, m) 11,1), 11,0), and 11,-1). (a) Recall that the raising/lower operators are given by L±-L, ±ill. Also recall the rela- tionship Use this relationship to determine the 3x3 matrix representation of Ly (using a basis com- prised of eigenstates of L). (b) What are the eigenvalues and associated eigenstates of the operator Ly? (c) If...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 4 = 2 with vi = and |_ G 12 = -2 with v2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: x(t) (50) = C1 + C2 e e B. In fundamental matrix form: (MCO) = I: C. As two equations: (write "c1" and "c2" for C1 and c2) x(t) = yt) =
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an orthonormal basis for the linear space. Let us consider a real 3D space where a vector is denoted by a 3x1 column vector. Consider the symmetric matrix B-1 1 1 Show that the vectors 1,0, and1are eigenvectors of B, and find 0 their eigenvalues. Notice that these vectors are not orthogonal. (Of course they are not normalized but let's don't worry about it. You can...
Suppose that the matrix A A has the following eigenvalues and
eigenvectors:
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 2 = 2i with v1 = 2 - 5i and - 12 = -2i with v2 = (2+1) 2 + 5i Write the general real solution for the linear system r' = Ar, in the following forms: A. In eigenvalue/eigenvector form: 0 4 0 t MODE = C1 sin(2t) cos(2) 5 2 4 0 0...
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: A1 = 4 with = and [2] [i] Az = 3 with Ū2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: t (10) -- + C2 e e B. In fundamental matrix form: (39) - g(t). C. As two equations: (write "c1" and "c2" for C and C2) X(t) = g(t) = Note: if you are...
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