The angular momentum raising and lowering operators are defined by on9 Although the angular momen...
(2.) Consider the orbital angular momentum operator defined in terms of the position and momentum operators as p. Define the angular momentum raising and lowering operators as L± = LztiLy. Use the commutation relations for the position and m omentum operators and find the commutators for: (a.) Lx, Lz and Ly, Lz; (b.) L2, Lz; (c.) L+,L
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...
Problem #7 The quantum harmonic oscillator Hamiltonian, expressed in terms of raising and lowering operators, is We also know that Using these two statements, show that if then both Yn and ah have well-defined energies. Give these energies in terms of μη.
Problem #7 The quantum harmonic oscillator Hamiltonian, expressed in terms of raising and lowering operators, is We also know that Using these two statements, show that if then both Yn and ah have well-defined energies. Give these energies...
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...
Using the properties of the raising and lowering operators for the 1 dimensional simple harmonic oscillator to compute where is an integer and
3. Between the Two Methods, I Prefer the Ladder Using the raising and lowering operators, and their properties as discussed in class, work out the explicit form for the second excited state of the harmonic oscillator, that is, derive the normalized wave function pa (x) for the SHO. Show explicitly that the function you derived is orthogonal to the ground state wave function ho(x).
Problem 5 The spin raising and lowering operators are define as Su = Sx+iS, and S. = Szy, blon that the operator S2 is diagonalized in the basis of eigenvectors of S. (15)
2. (a) From your knowledge of the raising and lowering operators and their relation to X and P, calculate for each of the following states(X), ΔΧ, 〈P), and ΔΡ. Comment on the Heisenberg uncertainty relation in each case. i. 10) ii. In) (b) For the following cases, calculate(X) and A. ii. (11) 13))/V2
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...
Show that the angular momentum operator, Îz = (ħ/i) d/dφ, is hermitian. Hint: consider the wavefunction ψ(φ), where φ varies from 0 to 2π