Tiêu đề IIT-JAM QM DPP Sheet 06.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. If 2 exp   ax   an eigenfunction of the operator 2 2 2 d x dx         , then the constant B and cirresponding eigen- value are respectively (a) 2 B a a    2 , 2 (b) 2 B a a     4 , 2 (c) 2 B a a    4 , 2 (d) 2 B a a     2 , 2 7. The operator d d x x dx dx               is equivalent to (a) 2 2 2 1 d d x dx dx   (b) 2 2 2 1 d d x dx dx   (c) 2 2 2 2 1 d d x x dx dx    (d) 2 2 2 1 d x dx   8. A particle is subjected to the potential energy   1 2 2 V x kx  . At a particular time, it has the wave function   2 2 x a/  x Axe  . If the particle has a definite total mechanical energy, then the possible value of ‘a’ is (a) 1/4 2 4 mk        (b) 1/2 2 4 mk        (c) 1/2 4 mk        (d) 1/4 4 mk        9. Consider a one-dimensional particle which moves along the x-axis and it’s Hamiltonian is given as 2 2 2 ˆ      16 ˆ d H x dx where  is a real constant having the dimensions of energy. If the particle is in an energy eigenstate having wave function   2 2  2   x x xe , then the energy of the particle will be (a) 2 (b) 4 (c) 8 (d) 12 10. Consider a physical sysytem whose Hamiltonian H and initial state 0 are given by 0 0 0 1 1 0 0 ; 1 5 0 0 1                               i i H i i where  has the dimension of energy. If we measure the energy of the system, then the probability of getting the energy  is (a) 0.4 (b) 0.2 (c) 0.6 (d) 0 11. If the function   1 2 exp 2 f x x        is an eigenfunction of the operator 2 2 2 ˆ d A x dx   , then the corresponding eigenvalue will be (a) 1 (b) -1 (c) 2 (d) -2 12. Suppose a particle of mass m is confined within the region 0  x L and moving freely in it. Which of the following can possibly be the eigenstate of the Hamiltonian operator for the system? (n is a positive integer) (a) exp n x L        (b) exp n x L         (c) sin n x L        (d) cos n x L       
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 value of the commutator bracket 2 2 ˆ ˆ , x   x p   (where xˆ and ˆ x p are position and linear momentum operators respectively) is (a) 2i xp p x   ˆˆ ˆ ˆ x x   (b) 2i xp p x   ˆˆ ˆ ˆ x x   (c) 4 ˆˆ x i xp  (d) 4 ˆ ˆ x i p x  14. For a quantum mechanical system in 1-D, the commutator bracket ˆ x x H ˆ ˆ , ,         (where xˆ and Hˆ are position and Hamiltonian operators respectively) is (a) 2 i m  (b) 2 i m   (c) 2 m   (d) 2 m  15. The value of the commutator bracket   ˆ , F x x         is (a)   F x ˆ (b) ˆ I (c) - ˆ I (d) Fˆ x   16. Let xˆ and ˆ x p denote the coordinate and momentum operators satisfying the canonical commutation relation  x p i ˆ ˆ , x   in natural units  1. Then the commutator  x p ˆ ˆ ,sin x  is (a) cos ˆ x i p (b) cos ˆ x i p (c) sin ˆ x i p (d) sin ˆ x i p 17. The value of the commutator brackets  xp p x p ˆˆ ˆ ˆ ˆ x x x  ,  and  xp p x x ˆˆ ˆ ˆ ˆ x x  ,  (where xˆ and ˆ x p are position and linear momentum operators respectively) are (a) 2 , 2 ˆ ˆ x i p i x    (b) 2 , 2 ˆ ˆ x i p i x   (c) 2 , 2 ˆ ˆ x  i p i x   (d) 2 , 2 ˆ ˆ x   i p i x   18. The value of the commutator bracket 2 2 ˆ xˆ, x         is (a) ˆ 2 x    (b) ˆ 2 x   (c) ˆ x    (d) ˆ x   19. A and B ˆ ˆ are two quantum mechanical operators. If ˆ ˆ   A B,   stands for the commutator of Aˆ and Bˆ , then ˆ ˆ ˆ ˆ       A B B A , , ,       is equal to (a) ABAB BABA ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ  (b)     A AB BA B BA AB ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ    (c) zero (d)   2 ˆ ˆ   A B,   20. x and p are two operators which satisfy  x p i ,   . the operators X and Y are defined as X x p Y x p      cos sin and sin cos     where  is a real parameter. Then  X Y,  equals to (a) 1 (b) –1 (c) i (d) –i