Angular momentum, spin, hydrogen atom, identical particles, perturbation theory and Dirac notation.
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- What is the commutation relation [L_x, L_y]?
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[L_x, L_y] = i*hbar*L_z- What are the eigenvalues of the angular momentum operator L_z?
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m*hbar, where m = -l, -l+1, ..., l-1, l- What does the raising operator L+ do to the eigenstate |l,m>?
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L+ |l,m> = hbar*sqrt(l(l+1) - m(m+1)) |l,m+1>- What does the lowering operator L- do to the eigenstate |l,m>?
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L- |l,m> = hbar*sqrt(l(l+1) - m(m-1)) |l,m-1>- What are the eigenvalues of the total angular momentum squared operator L^2?
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l(l+1)*hbar^2, where l = 0, 1, 2, ...- What is the spin quantum number s for an electron?
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s = 1/2- What are the eigenvalues of S^2 for a spin-1/2 particle?
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s(s+1)*hbar^2 = (3/4)*hbar^2- How many independent basis states exist for a spin-1/2 system?
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Two: spin-up |up> and spin-down |down>, or |+> and |->- What is the Pauli matrix sigma_z?
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[[1, 0], [0, -1]]- What commutation relation do the Pauli matrices satisfy?
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[sigma_i, sigma_j] = 2i*epsilon_ijk*sigma_k- What anticommutation relation do the Pauli matrices satisfy?
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{sigma_i, sigma_j} = 2*delta_ij- What commutation relation do spin operators S_x and S_y satisfy?
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[S_x, S_y] = i*hbar*S_z- In the Stern-Gerlach experiment with a spin-1/2 particle, how many beams result?
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Two beams: spin-up and spin-down- What is the ground state energy of the hydrogen atom?
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E_1 = -13.6 eV- What is the Bohr radius a_0?
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a_0 = 0.529 Angstroms, or hbar^2 / (m_e*e^2*k_e)
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