3, using L × L = ihL, show that the commutation relations take the form: 3, using L × L = ihL, show that the commutation relations take the form:
Show that the following commutation relations follow from the definitions in the previous question: [a+,a−] = −1 [a−,a+] = 1 [H, a+] = ̄hωa+[H, a−] = −̄hωa−
(L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B C]l-IA,BC] [A,CB] The commutator [..] is an important operation in quantum physics. If the elements A.B.C... satisfy the following conditions, they are said to form a Lie-algebra (ab,c are real or complex numbers): ii) [A,Bl -IB.Al iii) The Jacobi identity [AJB CII + [BJCA]] + [CJA,BII :0 - BA Prove the Jacobi identity for [A,B] AB
(L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B...
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...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
Classically, orbital angular momentum is given by L = r times p, where p is the linear momentum. To go from classical mechanics to quantum mechanics, replace p by the operator -i nabla (Section 14.6). Show that the quantum mechanical angular momentum operator has Cartesian components L_x = -i (y partial differential/partial differential z - z partial differential/partial differential y L_y = -i(z partial differential/partial differential x - x partial differential/partial differential z L_z = -i (x partial differential/partial differential...
2, Explicitly construct the three 3 × 3 matrices that represent (a) Lx, Ly, and Lz in the space of 1 1 functions: (Li/m , m' s(1-1, ml Lill = 1,m') 1m where i = x, y, z. (b) Show by explicit calculation that these three matrices obey the commutation relations of angular momentum (c) Find the matrices that represent L.+, L, and L2
(3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t)
(3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t)
3. By producing suitable examples of relations, show that it is not possible to deduce any one of the properties of being reflexive, symmetric or transitive from the other two.
1. Map the below ERD into a set of relations in at least Third
Normal Form (3NF). You must ensure that your relations meet 3NF.
Show your working.
2. List all the functional dependencies present in the
relation
i d leac ne Cowse Amociate Student Collec ri Maru ch ug
i d leac ne Cowse Amociate Student Collec ri Maru ch ug
3. (a) Given that e2i4 sin20 32m show by direct differentiation using the raising operator L+ that 1,12,-2(9,0) 0 (b) Also for e 32T sin 2,-2(0,9) show using the raising operator L. that
3. (a) Given that e2i4 sin20 32m show by direct differentiation using the raising operator L+ that 1,12,-2(9,0) 0 (b) Also for e 32T sin 2,-2(0,9) show using the raising operator L. that