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QM2 Concept Test 2.1 Which one of the following pictures represents the surface of constant π 2 for π 200 ? A. B. C. D. E.
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QM2 Concept Test 2.2 Choose all of the following statements that are true about the quantum number π. The quantum number π is determined by using separation of variables to solve for the angular wavefunction π π,π =Ξ(π)Ξ¦(π). Using separation of variables to solve the angular equation π π,π =Ξ π Ξ¦ π yields π 2 Ξ¦ π π 2 =β π 2 Ξ¦β Ξ¦ π ~ π πππ . The quantum number π must be an integer (π=0, Β±1, Β±2β¦) because when π advances by 2Ο, we return to the same point in space, i.e., Ξ¦ π+2π =Ξ¦(π). 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. All of the above
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QM2 Concept Test 2.3 Choose all of the following statements that are true related to the quantum number π. The associated Legendre function is π π π π₯ = 1β π₯ π π ππ₯ π π π π₯ and the πth Legendre polynomial is π π π₯ = 1 2 π π! π ππ₯ π π₯ 2 β1 π . We use separation of variables to solve for the angular wavefunction π π,π =Ξ(π)Ξ¦(π) which yields Ξ π ~ π π π ( cos π) . Due to the definition of π π π₯ , π must be a nonnegative integer. From the relation between π π π π₯ and π π π₯ , π π π π₯ =0 unless π β€π. So for a normalizable wavefunction, for any given π, π=βπ, β¦,β1, 0, 1, β¦,π. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. All of the above
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QM2 Concept Test 2.4 Choose all of the following statements that are true about the radial equation and radial wavefunction π
(π) for spherically symmetric potential energy: The radial wavefunction π
(π) cannot be found until a specific potential energy π(π) is provided. Solving for the radial wavefunction π
(π) determines the possible quantum number π and its relation to quantum number π Solving the radial equation shows that the degeneracy of the π π‘β energy level πΈ π for hydrogen atom is 2π. 1 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above
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QM2 Concept Test 2.5 A hydrogen atom is in the ground state π 100 π = 1 π π 3 π β π π , where π is the Bohr radius. What is the most probable value of π? π=0 m π= π 2 π=π π=2π None of the above
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QM2 Concept Test 2.6 A hydrogen atom is initially in the state ( π π 210 ) at time π‘=0. Choose all of the following statements that are correct about measurements performed on the atom at time π‘=0: If you make a measurement of the square of the magnitude of the angular momentum πΏ 2 immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ of the electron at time π‘=0, the measurement of πΏ 2 will yield 2 β 2 or zero with 50% probability. If you make a measurement of the π§-component of the orbital angular momentum πΏ π§ immediately following a measurement of the π₯-component of the orbital angular momentum πΏ π₯ of the electron at time π‘=0, the measurement of πΏ π§ will yield zero with 100% probability. If you make a measurement of the energy of the electron immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ at time π‘=0, the energy measurement will yield β13.6ππ 4 with 100% probability. A. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM2 Concept Test 2.7 A hydrogen atom is initially in the state ( π π 210 ) at time π‘=0. Choose all of the following statements that are correct about measurements performed on the atom after a long time π‘: If you make a measurement of the square of the magnitude of the angular momentum πΏ 2 immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ , the measurement of πΏ 2 will yield 2 β 2 or zero with 50% probability. If you make a measurement of the π§-component of the orbital angular momentum πΏ π§ immediately following a measurement of the π₯-component of the orbital angular momentum πΏ π₯ , the measurement of πΏ π§ will yield zero with 100% probability. If you make a measurement of the energy of the atom immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ at time, the energy measurement will yield β13.6ππ 4 with 100% probability. A. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM2 Concept Test 2.8 A hydrogen atom is initially in the state ( π πππ + π πππ ) at time π‘=0. Choose all of the following statements that are correct about measurements performed on the atom after a long time π‘: If you make a measurement of the square of the magnitude of the angular momentum πΏ 2 immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ , the measurement of πΏ 2 will yield 2 β 2 or zero with 50% probability. If you make a measurement of the π§-component of the orbital angular momentum πΏ π§ immediately following a measurement of the π₯-component of the orbital angular momentum πΏ π₯ , the measurement of πΏ π§ will yield zero with 100% probability. If you make a measurement of the energy of the electron immediately following a measurement of the z-component of the orbital angular momentum πΏ π§ , the energy measurement will yield β13.6ππ 4 with 100% probability. A. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM2 Concept Test 2.9 A hydrogen atom is initially in the state π 200 at time π‘=0. Choose all of the following statements that are correct. The expectation value of the distance of the electron from the nucleus π depends on time. The expectation value of the π₯-component of angular momentum of the electron πΏ π₯ depends on time. The expectation value of the π§-component of angular momentum of the electron πΏ π§ depends on time. A. 1 only B. 2 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM2 Concept Test 2.10 A hydrogen atom is initially in the state ( π πππ + π πππ ) at time π‘=0. Choose all of the following statements that are correct. The expectation value of the distance of the electron from the nucleus π depends on time. The expectation value of the π₯-component of angular momentum of the electron πΏ π₯ depends on time. The expectation value of the π§-component of angular momentum of the electron πΏ π§ depends on time. A. 1 only B. 2 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM2 Concept Test 2.11 A hydrogen atom is initially in the state ( π π 210 ) at time π‘=0. Choose all of the following statements that are correct. The expectation value of the distance of the electron from the nucleus π depends on time. The expectation value of the π₯-component of angular momentum of the electron πΏ π₯ depends on time. The expectation value of the π§-component of angular momentum of the electron πΏ π§ depends on time. A. 1 only B. 2 only C. 1 and 2 only D. 1 and 3 only E. None of the above
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QM1 Concept test 2.12 π» is the Hamiltonian for the electron in a Hydrogen atom with Coulomb potential energy only. Choose all of the following statements that are correct. π» , πΏ π§ =0 π» , πΏ 2 =0 In spherical polar coordinates, the potential energy V of the Hydrogen atom is independent of the angular variables π and π. A. 3 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. all of the above
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QM1 Concept test 2.13 The stationary states of a hydrogen atom are π πππ . Choose all of the following statements that are correct. πΏ 2 π πππ =π(π+1) β 2 π πππ πΏ π§ π πππ =πβ π πππ π» π πππ = β13.6ππ π π πππ , where π» is the Hamiltonian of the system. A. 1 only B. 2 only C. 1 and 2 only D. 2 and 3 only E. all of the above
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