Title Assignment-2._Schrodinger Equation.pdf ✅

Assignment 2: Wave function & General formalism of Quantum mechanics 3 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 GATE Previous Years’ Questions Common data for Q. 1 and Q. 2 The wavefunction of a particle moving in free space is given by, ikx ikx e 2e    1. The energy of the particle is [GATE 2012] (a) 2 2 5 k 2m  (b) 2 2 3 k 4m  (c) 2 2 k 2m  (d) 2 2 k m  2. The probability current density for the real part of the wavefunction is (a) 1 (b) k m  (c) k 2m  (d) 0 3. Consider the wave function 0 ( / ) ikr Ae r r , where A is the normalization constant. For 0 r r  2 , the magnitude of probability current density up to two decimal places, in units of 2 ( / ) A k m  , is _____ [GATE 2013] 4. The state of a system is given by [GATE 2016]        1 2 2 2 3 where, 1 2 2    , and form an orthonormal set. The probability of finding the system in the state 2 is ______________.(Give your answer upto two decimal places) 5. A two-state quantum system has energy eigenvalues  corresponding to the normalized states  . At time t  0, the system is in quantum state 1 2         . The probability that the system will be in the same state at (6 ) h t   is ________ (up to two decimal places). [GATE 2018] 6. Let 1 2 1 0 , 0 1                 represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of 1 and  2 is given by a density matrix that is defined as 1 2 1 2 2 2        c c . If 1 c  0.4 and 2 c  0.6 , the matrix element 22 (rounded off to one decimal place) is __________________ [GATE 2019] 7. The Hamiltonian operator for a two-level quantum system is 1 2 0 0 E H E        . If the state of the system at t  0 is given by   1 1 0 2 1         then     2   0 | t at a later time t is [GATE 2019] (a)     / 1 2 1 1 2 E E t e     (b)     / 1 2 1 1 2 E E t e     (c)   1 2   1 1 cos / 2     E E t    (d)   1 2   1 1 cos / 2     E E t   
Assignment 2: Wave function & General formalism of Quantum mechanics 4 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 8. The wavefunction of a particle in one dimension is given by: [GATE 2023] ; ( ) 0 ; otherwise M a x a  x        Here M and a are positive constants. If ( ) p is the corresponding momentum space wavefunction, which one of the following plots best represents 2 ( ) p ? (a) p | ( )|  p 2 (b) | ( )|  p 2 p (c) | ( )|  p 2 p (d) | ( )|  p 2 p ANSWER KEY 1. (c) 2. (d) 3. (0.25 ) 4. (0.27 to 0.29 ) 5. (0.25) 6. (0.6) 7. (c) 8. (c)
Assignment 2: Wave function & General formalism of Quantum mechanics 5 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 CSIR-UGC-NET Previous Years’ Questions 1. If a particle is represented by the normalized wave function     2 2 5 2 15 a x for a x a x 4a 0 otherwise            the uncertainty p in its momentum is [CSIR Dec 2012] (a) 2 5a  (b) 5 2a  (c) 10 a  (d) 5 2 a  2. The energies in the ground state and first excited state of a particle of mass 1 m 2  in a potential V(x) are –4 and –1, respectively, (in units in which  1). If the corresponding wavefunctions are related by    1 0  x x sinh x    , then the ground state eigenfunction is [CSIR Dec 2012] (a)   0  x sec hx  (b)   0  x sechx  (c)   2 0   x sec h x (d)   3 0   x sec h x 3. If 4    ( ) exp( ) x A x is the eigenfunction of a one dimensional Hamiltonian with eigenvalue E = 0, the potential V(x) (in units where    2 1 m ) is [CSIR Dec 2013] (a) 12x 2 (b) 16x 6 (c) 16x 6 + 12x 2 (d) 16x 6 – 12x 2 4. Suppose the Hamiltonian of a conservative system in classical mechanics is H xp   , where  is a constant and x and p are the position and momentum respectively. The corresponding Hamiltonian in quantum mechanics, in the coordinate representation, is [CSIR Dec 2014] (a) 1 2 i x x             (b) 1 2 i x x             (c) i x x      (d) 2 i x x      5. Let 1 and 2 denote, the normalized eigenstates of a particle with energy eigenvalues E1 and E2 respectively, with E E 2 1  . At time t  0 the particle is prepared in a state 1 2 1 ( 0) ( ) 2       t . The shortest time T at which   ( ) t T will be orthogonal to   ( 0) t is [CSIR Dec 2014] (a) 2 1 2 ( ) E E    (b) 2 1 ( ) E E    (c) 2 1 2( ) E E    (d) 2 1 4( ) E E    6. A Hermitian operator Oˆ has two normalised eigenstates 1 and 2 with eigenvalues 1 and 2, respectively. The two states u     cos 1 sin 2 and v     cos 1 sin 2 are such that ˆ v O v | | 7 / 4  and u v| 0  . Which of the following are possible values of  and  ? [CSIR Dec. 2015] (a) 6     and 3    (b) 6    and 3    (c) 4     and 4    (d) 3    and 6    
Assignment 2: Wave function & General formalism of Quantum mechanics 6 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 7. The eigenstates corresponding to eigenvalues 1 2 E E and of a time-independent Hamiltonian are 1 and 2 respectively. If at t  0, the system is in a state    ( 0) sin 1 cos 2 t    the value of   ( ) | ( ) t t at time t will be [CSIR June 2016] (a) 1 (b) 2 2 1 2 2 2 1 2 ( sin cos ) E E E E     (c) 1 2 / / sin cos iE t iE t e e      (d) 1 2 / / 2 2 sin cos iE t iE t e e        8. Consider the two lowest normalized energy eigenfunctions 0  ( ) x and 1  ( ) x of a one dimensional system. They satisfy 0 0 0 1 ( ) ( ) and ( ) , where d x x x dx          is a real constant. The expectation value of the momentum operator in the state 1 is [CSIR Dec. 2016] (a) 2    (b) 0 (c) 2   (d) 2 2   9. The normalized wavefunction in the momentum space of a particle in one dimension is 2 2 ( )      p p , where   and are real constants. The uncertainty x in measuring its position is [CSIR Dec. 2017] (a) 2     (b) 3     (c) 2  (d)      10. The Hamiltonian of a two-level quantum system is 1 1 1 2 1 1         H  . A possible initial state in which the probability of the system being in that quantum state does not change with time, is [CSIR Dec. 2017] (a) 4 4 cos sin           (b) 8 8 cos sin           (c) 2 2 cos sin           (d) 6 6 cos sin           11. The wavefunction of a particle of mass m, in a potential V(x) in one dimension is   , where 0    x x Ae    and A are constants. If this wavefunction is an energy eigenfunction, then a possible form of the potential V(x) is [CSIR Dec. 2019] (a) 2  m x  (b) 2   m x  (c)   2  x m   (d)   2   x m   12. Let the normalized eigenstates of the Hamiltonian , 2 1 0 1 2 0 0 0 2 H            be , and    1 2 3 . The expectation value   H and the variance of H in the state  1 2 3  1 3        i are [CSIR Dec. 2019] (a) 4 1 and 3 3 (b) 4 2 and 3 3 (c) 2 2 and 3 (d) 2 and 1