Exam_3-2020_Fall-01.pdf - 2020 Fall BCH 463 | Examination...

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Unformatted text preview: 2020 Fall BCH 463 | Examination 3 (Nov 2nd, 2020) Name: ID: Please write down how you derive and solve the equations to the answers for the calculation problems. The credits will not be given if only the answers are provided. 1. Which is the wave function of the Schrödinger equation for the hydrogen (C is the constant that can be determined by the quantum numbers) (5 pts) |m| im l (A) (r, ✓, ) = Cn,l,m Pl (cos ✓)e r Rn,l (r)r e l na0 (B) (r, ✓, ) = Cn,l,m Rn,l (r)r e |m| im (C) (r, ✓, ) = Cn,l,m Pl (cos ✓)e r na0 (D) The Schrödinger equation for the hydrogen does not have an exact solution. 2. Which of the quantum numbers is NOT determined from the Schrödinger equation? (5 pts) (A) The principal quantum number n (B) The angular momentum quantum number l (C) The magnetic quantum number ml (D) The electron spin quantum number ms 3. Given the wave function of the 1s orbital of the hydrogen as <latexit sha1_base64="WFMN0vMG29z/5u1JvwzixXVdTzA=">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</latexit> 100 1 1 1 = p ( )2 e ⇡ a0 r a0 , where a0 is the Bohr radius. (a) Write the equation for the expectation value of the kinetic energy to the orbital boundary for the ground state. Do not solve the equation. (5 pts) <latexit sha1_base64="frUydUGW8sKUKL4b+ycp/0So32k=">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</latexit> r2 = 1 @ 2 @ 1 @ @ 1 @2 (r ) + (sin ✓ ) + ( ) r2 @r @r r2 sin ✓ @✓ @✓ r2 sin2 ✓ @ 2 . Hints: Laplacian operator (b) Calculate the total energy of an electron in the 1s orbital of the hydrogen atom. (5 pts) (c) Calculate the average position of the electron in 1s orbital. (10 pts) 2020 Fall BCH 463 | Examination 3 (Nov 2nd, 2020) 4. The average value of the squared radius of the 2s orbital can be calculated by (d is the Z volume element r2sindr d dɸ) (5 pts) 1 (A) Z0 (B) Z0 (C) Z0 1 1 (D) 0 1 ⇤ 200 r 200 d⌧ ⇤ 200 ⇤ 2 200 r 200 d⌧ 200 d⌧ ⇤ 200 r 200 d⌧ 5. (a) Write the Schrödinger equation for a beryllium (Be) atom. (Hint: A beryllium atom has a nucleus with 4+ charge.) (10 pts) (b) Write the wave function of the beryllium atom using the Hartree-Fock approximation and Slater determinant. Please specify the function and variable you use. (10 pts) 6. For a hydrogen atom, the eigenvalue (the energy term) of the Hamiltonian operator predicts that (5 pts) (A) the 2s and 2p orbitals have the same energy; (B) the 2s has a larger energy than 2p orbital; (C) the 2p has a larger energy than 2s orbital; (D) the 1s and 2s have the same energy level. 2020 Fall BCH 463 | Examination 3 (Nov 2nd, 2020) 7. How many orbitals are there in a shell with n = 6? (5 pts) (A) 18 (B) 54 (C) 91 (D) 108. 8. For the angular momentum operators, the spherical harmonics are (5 pts) %# ; (A) the eigenfunctions of the %! , %" , and %# and the values are integral multiples of ℎ⁄2; (B) only the eigenfunctions of the (C) not the eigenfunctions of any components; (D) associated with the principal quantum number n. 9. (1) Given an orbital shape in the following figure, write down the possible combinations of all the quantum numbers. (5 pts) (2) Which angular component is associated with this orbital or spherical harmonics? (5 pts) 10. The ionization energies of the 2s and 1s of the lithium are 520 and 7,298 kJ/mol, respectively, and that of the hydrogen is 1,312 kJ/mol. Calculate the nuclear charge parameter for lithium. (5 pts) 2020 Fall BCH 463 | Examination 3 (Nov 2nd, 2020) 11. Given the wave function of the 3s orbital of the hydrogen atom as <latexit sha1_base64="j+ueDJ4KkIvGkC/MqdJNKBPdqNk=">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</latexit> 300 = 1 1 3 p ( ) 2 (27 81 3⇡ a0 18 r r2 + 2 2 )e a0 a0 r 3a0 . (a) Explain and write the equation for finding the most probable position for an electron in the 3s orbital. Do not solve it. (5 pts) (b) Where are the node positions? Use the Bohr radius a0 as a unit in yourpanswer. <latexit sha1_base64="8NlwQsyN6fHB3PYP/uPip/hCtHg=">AAACD3icbVC7SgNBFJ31GeNr1dJmMCg2CbshYhohYGMZwTwgu4bZyWwyZGZ2nZkVw7J/YOOv2FgoYmtr5984eRSaeODC4Zx7ufeeIGZUacf5tpaWV1bX1nMb+c2t7Z1de2+/qaJEYtLAEYtkO0CKMCpIQ1PNSDuWBPGAkVYwvBz7rXsiFY3EjR7FxOeoL2hIMdJG6tonD/ACeqFEOC0G0Is59NSd1GlwW4ZFWEE4y9Iyyrp2wSk5E8BF4s5IAcxQ79pfXi/CCSdCY4aU6rhOrP0USU0xI1neSxSJER6iPukYKhAnyk8n/2Tw2Cg9GEbSlNBwov6eSBFXasQD08mRHqh5byz+53USHVb9lIo40UTg6aIwYVBHcBwO7FFJsGYjQxCW1NwK8QCZcLSJMG9CcOdfXiTNcsk9KznXlUKtOosjBw7BETgFLjgHNXAF6qABMHgEz+AVvFlP1ov1bn1MW5es2cwB+APr8weFDpsO</latexit> 2 Hint: The solutions for a quadratic equation of ax + bx + c = 0 are (10 pts) x= b± b2 2a 4ac . ...
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