Tiêu đề IIT-JAM QM DPP Sheet 05.pdf ✅

2 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 6. The wave function of a particle, constraint to move along x axis at certain instant of time is given as   2 2 exp x x A ibx a           [a,b are positive real constants] The normalization constant ‘A’ is (a) 2 a  (b) 2 2 a  (c) 2 2 a (d) 2 2 a  7. The wave function of a particle at a certain time is given as following:   2 2 A ikx x e x a    [a,k are positive real constants] The real value of A such that   x is normalized, is (a) a  (b) a  (c) 2a  (d) 2 a  8. In an one-dimensional system, the normalized wave function is given by        x N x exp is positive real constant     where N is normalization constant. What is the value of N? (a) 2 / (b) 2 (c) 1/ (d)  9. The normalized wave function of the particle is given as following:     2 1 for 1 1 0 otherwise C x x  x         The value of the constant C is (a) 15 / 4 (b) 15 / 2 (c) 15 / 8 (d) 2 / 3 10. The wave function  r  of a particle moving in three-dimensional space has the physical dimensions of (a)   3 Length 2  (b)  3 Length 2 (c)   1 Length  (d) Length 11. The wave function  p  of a particle in two dimensional momentum space has the physical dimensions of (a)   1 1 MLT   (b)   1 2 2 2 M L T   (c) 3/2 3/2 3/2 M L T  (d)   1 1/2 1/2 1/2 M L T   12. A one dimensional wave function corresponding to a particle is given by   a x| |  x ae   [a are positive real constant] The probability of finding the particle between x = 1/a to x = 2/a is equal to (a) 2 4 e e    (b) 2 4 2 e e    (c) 2 4 4 e e    (d) 2 4 4 e e   
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 13. The wave function of a particle at time t = 0 is given as   2 3 sin ; 0 0 ; otherwise x x L x L L                      The probability of finding the particle in the range 3 4 4 L L   x is (a) 1 2 (b) 1 1 2 3  (c) 1 1 2 3  (d) 1 1 2   14. A particle moving along x-axis, has the following wave function:   sin for 0 2 2 0 otherwise n x x a x a                 The probability of finding the particle between x = 0 and x = 2a/n, will be (a) 1/n (b) 2/n (c) 1/2n (d) n/2 15. Let    , , represents three ortho-normalized wave functions and two state is defined: 1   2 c i 2 1 c i                        The real, positive values of c1 and c2 for which   and are normalized are (a) 1 1 , 7 3 (b) 1 1 , 3 7 (c) 1 1 , 5 7 (d) 1 1 , 5 3 16. Consider the following position space wave function   op x i  x Ae   where A and 0 p are positive real constants. The momentum space wavefunction   p will be (a)   op x i Apo  x e    (b)   op x i Apo  x e     (c)     x A p p    2   0  (d)     0 2 A   x p p     17. The wave function of a particle is given as   2 sin ; (0 ) 0 ; ( ) x x a x a a otherwise                      The probability of its momentum being in the range 2 2 50 h h h to a a a  is (a) 2 8 5625 (b) 2 16 5625 (c) 2 32 5625 (d) 2 4 5625
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 18. If the position space wave function is a Gaussian function peaked at x = 0, then the most probable momentum of the particle will be (Symbols have their usual meaning) (a) 0 (b) k (c) k (d) k / 2 PART II: Multiple Select Questions (MSQ) 19. Which of the following wave function is/are physically admissible wavefunction for a particle in a bound state for all values of x? (A is a real constant) (a)   exp x x A ikx a           (b)   2 2 exp x x A ikx a          (c)   2 sin kx x A x   (d)   exp x x A a         20. Which of the following wave function is/are well-behaved wavefunction for a particle in a bound state? (Symbols have their usual meaning) (a)   2 ax  x Ax e  (b)   2 ar  r Ar e  (c)   0 for | | for | | x a x c x a           (d)   sin for 0 1 0 otherwise A x x x         21. The normalized ground state wave function of an electron in a hydrogen atom is given by   1/2 3 0 1 o r a r e a           where 0 a is the first Bohr radius. Which of the following statements is/are CORRECT? (a) Probability of finding the electron at a distance greater than 0 a from the nucleus is 0.68. (b) Probability of finding the electron at a distance greater than 0 a from the nucleus is 0.32. (c) Probability of finding the electron at a distance less than 0 a from the nucleus is 0.68. (d) Probability of finding the electron at a distance less than 0 a from the nucleus is 0.32. 22. The wave function corresponding to a particle is given by   1 | | x a x e a    where a is positive real constant. Which of the following statements is/are CORRECT? (a) The probability of finding the particle in the region    a x a is   2 1 1/  e . (b) The probability of finding the particle in the region    a x a is 2 1/ e . (c) The value of b such that probability of finding the particle between    b x b is 0.5, is a. ln 2 / 2  . (d) The value of b such that probability of finding the particle between    b x b is 0.5, is 2 / ln 2 a .