(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators: [Lz, y] = -ihx, [L,, z) = 0, [L2, x] = ihy, [L,, p.) = ihpy, [L, Py] = -ihp, [Lz, Pa] = 0, [4.122] (b) Use these results to obtain [Lz, Lx] = ihLy directly from Equation 4.96. (c) Evaluate the commutators [Lz, r²] and [Lz, p²] (where, of course, r2 = x² + y? + z² and p² = p+ p; + p?). %3D
(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators: [Lz, y] = -ihx, [L,, z) = 0, [L2, x] = ihy, [L,, p.) = ihpy, [L, Py] = -ihp, [Lz, Pa] = 0, [4.122] (b) Use these results to obtain [Lz, Lx] = ihLy directly from Equation 4.96. (c) Evaluate the commutators [Lz, r²] and [Lz, p²] (where, of course, r2 = x² + y? + z² and p² = p+ p; + p?). %3D
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![*Problem 4.19
(a) Starting with the canonical commutation relations for position and momentum
(Equation 4.10), work out the following commutators:
[L2, x] = ihy,
[Lz, y] = -iħx,
[Lz, Px] = ih py, [L2, Py] = -ihp;, [L, Pe] = 0.
[L2, z) = 0,
[4.122]
(b) Use these results to obtain [Lz, L;] = iħLy directly from Equation 4.96.
(c) Evaluate the commutators [Lz, r*] and [L, p²] (where, of course, r2
x² + y? +z? and p² = p;+ p;+ p?).
(p²/2m) + V commutes with all three
(d) Show that the Hamiltonian H
components of L, provided that V depends only on r. (Thus H, L², and L2
are mutually compatible observables.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12f2c931-076b-42ac-bfa4-9253ef314b65%2Ffb8789db-977a-4b1b-979e-66eca19d196a%2F2xuh9gn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:*Problem 4.19
(a) Starting with the canonical commutation relations for position and momentum
(Equation 4.10), work out the following commutators:
[L2, x] = ihy,
[Lz, y] = -iħx,
[Lz, Px] = ih py, [L2, Py] = -ihp;, [L, Pe] = 0.
[L2, z) = 0,
[4.122]
(b) Use these results to obtain [Lz, L;] = iħLy directly from Equation 4.96.
(c) Evaluate the commutators [Lz, r*] and [L, p²] (where, of course, r2
x² + y? +z? and p² = p;+ p;+ p?).
(p²/2m) + V commutes with all three
(d) Show that the Hamiltonian H
components of L, provided that V depends only on r. (Thus H, L², and L2
are mutually compatible observables.)
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