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3. Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other arguments) (a) \( \langle n = 2, l = 1, m = 0 \vert x \vert n = 2, l = 0, m = 0 \rangle \) (b) \( \langle n = 2, l = 1, m = 0 \vert p_z \vert n = 2, l = 0, m = 0 \rangle \) (c) \( \langle L_z \rangle \) for an electron in a central field with \( j = 9/2, m = 7/2, l = 4 \). (d) \( \langle \text{singlet}, m_s = 0 \vert S_z^{(e)} - S_z^{(p)} \vert \text{triplet}, m_s = 0 \rangle \) for an s-state positronium (e) \( \langle \mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)} \rangle \) for the ground state of a hydrogen molecule.

          3. Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why
it vanishes using simple symmetry (or other arguments)
(a) \( \langle n = 2, l = 1, m = 0 \vert x \vert n = 2, l = 0, m = 0 \rangle \)
(b) \( \langle n = 2, l = 1, m = 0 \vert p_z \vert n = 2, l = 0, m = 0 \rangle \)
(c) \( \langle L_z \rangle \) for an electron in a central field with \( j = 9/2, m = 7/2, l = 4 \).
(d) \( \langle \text{singlet}, m_s = 0 \vert S_z^{(e)} - S_z^{(p)} \vert \text{triplet}, m_s = 0 \rangle \) for an s-state positronium
(e) \( \langle \mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)} \rangle \) for the ground state of a hydrogen molecule.

3. Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why
it vanishes using simple symmetry (or other arguments)
(a) ⟨ n = 2, l = 1, m = 0 | x | n = 2, l = 0, m = 0 ⟩
(b) ⟨ n = 2, l = 1, m = 0 | pz | n = 2, l = 0, m = 0 ⟩
(c) ⟨ Lz ⟩ for an electron in a central field with j = 9/2, m = 7/2, l = 4.
(d) ⟨singlet, ms = 0 | Sz^(e) - Sz^(p)|triplet, ms = 0 ⟩ for an s-state positronium
(e) ⟨𝐒^(1)·𝐒^(2)⟩ for the ground state of a hydrogen molecule.

Added by David G.

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