Consider the interaction Hamiltonian Aint = Eint2 ® 2. Show that Û(t) = e-ifint/hSa. (.,yields a unitary transformation of the form Û(0) =() î – i sin (4) 2 2.= COSsb.) What is the value of 0 in terms of Eint, t and h?

Given the interaction Hamiltonian Hint=Eint Z⊗Z. We need to show that Ut=e-iHintth yeilds a unitary…

Answered: a. Consider the interaction Hamiltonian…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6.40. Find the matrix representing each linear transformation T on R³ relative to the usual basis of R': (a) T(x,y,z) = (x, y, 0). (c) T(x,y,z) = (2x – 7y – 4z, 3x+y+4z, 6x – 8y + z). (b) T(x,y,z) = (z, y+z, x+y+ z).
11. If the matrix of a linear transformation T ::: F. (C) with respect to the ordered basis B = {(1, 0), (0, 1)} is i %3D i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1)} %3D
9. A rotation is represented by the ZYZ Euler angles øi = 5. 02 = 0, y o3 = (a) Find the equivalent rotation matrix R.
1 Let T : R² → R² be the transformation that reflect points across y = x. B is a basis in R². Find the matrix of T relative to B. 2 Find the 3- matrix [T]å of the transformation T : x → Ax, where (=123)₁ (2) B = {b₁,b2}, A = b₁ = " b2 = (1¹). = {(1).(+)}
7. Find the Jacobian matrix of the following. 1 a) x=(u+v), y=÷(u-v), z= 2uvw b) x=rcos0, y=r sin 0,z = z
1.2 If R₁ = 2i-j+k, R₂ = i +3j - 2k, R3 = -2i+j3k and R4 = 3i+2j+5k. Determine scalars x, y, and z such that R4 = xR₁ +yR₂ + zR3.
IV. Let L be the operator given by L[y] = y"" – 2y" + 2u" i. Consider L as a linear operator T : P3 → P3. Find [T], where a is the standard basis of P3 ii. Determine the kernel of L from part i. iii. Now, consider L as a transformation L: Co (R) → C (R). Find the kernel of L.
1. Let A=MB (f) be the matrix associated with the linear map f(x,y,z) = (3x-y + 4z,y+z-x,x+y+z) relative to the bases B={u=(4, 18,6), v= (17.1.0), w (17,0,-5) } and Then the first B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0) } column of A is: O A. 2 "B 1 3 B. - - 16 On () 13 1 C. D. E. 7 -5 3 4 17 17 3 -
easy functional analysis
5.1
q3
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (3x−y+4z,y+z−x,x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,−5)} and B'={u'=(2,2,1); v'=(1,0,0); w'=(3,2,0) } Then the first column of A is: OA.-16 O B. 13 1 4 17 17 °C) O D. (2 213 O E. "G) ņ m

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a. Consider the interaction Hamiltonian Aint = Eint2 & 2. Show that Û(t) = e-it/ yields a unitary transformation of the form Û(0) = cos (2) Î – i sin (2) 2 2. b.) What is the value of 0 in terms of Eint, t and ħ?