5. a A quantum harmonic oscillator (QHO) with a fundamental vibrational frequency v is in superposition state described by a wave function, where Φ_v(x)'s are normalized eigenfunctions of the QHO (the subscripts "v" denote the vibrational quantum numbers) and c₁ is the expansion coefficient for Φ₁(x). What is the average energy when a very large number of identically prepared QHO's φin (x) state are measured?7,
(A)
(B) 2hv
(C)
(D)
(E) 4hv

1.Given that the ionization energy of hydrogen-atom ground state is 13.6 eV, how much energy in eV is required to remove an electron from the 3d2 orbital in a Be3+ ion? (A) 6.8 eV (B) 18.1 eV (C) 24.2 eV (D) 54.4 eV (E) 61.2 eV

2. Consider a quantum particle of mass m trapped in a 3-dimensional cubic box. The potential energy is zero inside the box and infinity outside the box. The length on each side of the box is a. What is the degeneracy of the energy level of E= ? (h = Planck constant) 8ma² (A) 1 (B) 2 (C) 4 (D) 6 (E) 8

3. In quantum mechanics, the measurements of two physical quantities, a and b, are represented by operators and , respectively. If the two operators commute, which of the following statements is NOT true? (A) It is possible to measure a and b precisely and simultaneously. (B) The two operators possess a common set of eigenfunctions. (C) The two operators possess a common set of eigenvalues. (D), where f is a well-behaved function. (E)

4. Consider a quantum particle confined in a 1-D symmetric finite potential well defined asWhich of the following schematic diagrams of the bound eigenstate energy distribution and probability density functions is most qualitatively correct in all aspects? (A) (B)(C) (D) (E)

6. The first excited state of helium (He) atom is described by electron configuration 1s¹2s1. Which of the followings is the correct form of spin-orbital functions for He atom first excited-state wave function satisfying the Pauli Principle? (Note: 1S(i) and 2S(1) are the spatial orbitals; α(i) and B(i) are the spin functions; i=1, 2 denotes the electron coordinates; N is the normalization constant) (A) N[IS(1)2S(2) -1S(2)2S(1)][a(1)β(2)] (B) N[IS(1)2S(2)+1S(2)2S(1)][a(1)a(2)] (C) N[1S(1)a(1)][2S(2)β(2)] (D) N[IS(1)2S(2)+1S(2)2S(1)][a(1)β(2) +a(2)β(1([ (E) N[IS(1)2S(2)-1S(2)2S(1)][a(1)β(2)+a(2)β(1)]

7. Consider the variational method for the ground state of a two-electron atom. The simplest trial function is the Hartree product of the form ), where isthe hydrogen-like satomic orbitalwith anefective nuclear changes， ζ , as the variational parameter. The variational integral can be evaluated analytically to give, where Z is the true nuclear charge and Eh=27.2 eV is the Hartree energy.What is the ground state energy of Li+ ion at this level of approximation? (A) -77.5 eV (B) -196.5 eV (C) -54.4 eV (D) -108.8 eV (E) -98.2 eV

8. The hydrogen atomic wave functions can be expressed as , whereAn electron is in the 2p0 orbital of a H atom. What is the most probable radius (the radius where the radial distribution function is at maximum) of finding that electron? (A) 5a0(B) 4a0(C) 3a0(D) 2a0(Ε) a0

9. The complex form of H-atom wave functions for n=3, 1=2, m=±1 states are,where R3d(r) is the radial function and (sinθcosθ)e±iφ are spherical harmonic functions with 1=2 and m=±1. A real form function (3dxz orbital) can be constructed with the linear combination of the two complex wave functions as. Which of the following statements about the 3dxz orbital is NOT correct? (A) 3dxz wave function is an eigenfunction ofH-atom (Hamiltonian operator). (B)3dxz wave function is an eigenfunction of H-atom(operator for square of orbital angular momentum). (C) 3dxz wave function is an eigenfunction of H-atom (operator for angular momentum along the z axis). (D) The magnitude of orbital angular momentum is not zero. (E) the average values of Lz is zero.

10. Allyl radical (CH2-CH-CH2) is a planar molecular radical in which the three a electrons are delocalized. Hückel theory is a simplified molecular orbital (MO) theory describing the MOs of the n-electron system as linear combinations of the 2pz atomic orbitals at each carbon atom involved in the delocalized system. The basic assumptions of Hückel theory are: All overlap integrals are set equal to zero; All resonance integrals between non-neighboring atoms are set equal to zero; All coulomb integrals are set equal to a; All resonance integrals between neighboring atoms are set equal to β. What is the total energy of the three πelectrons predicted by Hückel MO theory? (A)(B) (C)(D) 2a +4β (E) 4α +4β

11. The first excited electron configuration of HF is (1σ)2(2σ)2²(3σ)2(1π)3(4σ*)1. What is (are) molecular term(s) that can arise from this excited configuration? (A) 1Σ and 3Σ (B)1Π and 3Π (C) 3Π and 1Σ (D)3Π (Ε) 3Δ

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