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csir quantum mechanics by brij
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Brijmohan
Banaras Hindu University (BHU), Varanasi
PhD Condensed matter Physics
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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES QUANTUM MECHANICS SOLUTIONS Q1. Ans: NET/JRF (JUNE-2011) 1 The wavefunction of a particle is given by 0 i 1 , where 0 and 1 are the 2 normalized eigenfunctions with energies E0 and E1 corresponding to the ground state and first excited state, respectively. The expectation value of the Hamiltonian in the state is E E E 2 E1 E 2 E1 (a) 0 E1 (b) 0 E1 (c) 0 (d) 0 2 2 3 3 (d) Solution: Q2. 1 0 i 1 and 2 H H E 0 2 E1 3 The energy levels of the non-relativistic electron in a hydrogen atom (i.e. in a Coulomb potential V r 1 / r ) are given by E nlm 1 / n 2 , where n is the principal quantum number, and the corresponding wave functions are given by nlm , where l is the orbital angular momentum quantum number and m is the magnetic quantum number. The spin of the electron is not considered. Which of the following is a correct statement? (a) There are exactly (2l + 1) different wave functions nlm , for each Enlm. (b) There are l(l + 1) different wave functions nlm , for each Enlm. (c) Enlm does not depend on l and m for the Coulomb potential. (d) There is a unique wave function nlm for each Enlm. Ans: Q3. (c) The Hamiltonian of an electron in a constant magnetic field B is given by H B . where is a positive constant and 1 , 2 , 3 denotes the Pauli matrices. Let B / and I be the 2 2 unit matrix. Then the operator e i H t / simplifies to t i B t i B sin sin t (a) I cos (b) I cos t 2 B 2 B i B i B cos t cos 2 t (c) I sin t (d) I sin 2 t B B Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 1
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