Tiêu đề QM DPP Sheet 10.pdf ✅

1 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 IIT JAM PHYSICS 2024 Section: MODERN PHYSICS (Quantum Mechanics) Chapter: OPERATOR FORMALISM - III Daily Practice Problem (DPP) Sheet 10 1. A particle of mass ‘m’ is confined in a 1-D box from x = - L to x = L. The wave function of the particle in this state is   0 cos 2 x x L           The expectation value of p2 in this state is (a) 2 2 2 2L   (b) 2 2 2 4L   (c) 2 2 2 16L   (d) 2 2 2 32L   2. Let, wave function corresponding to two particles is given as following:   2 2 1/4 / 1 2 2 x a x e a           and   2 2 1/4 / 2 6 32 x a x xe a           then the value of 2 x 1  p  is (a) i a  (b) i a   (c) a   (d) a  3. An one-dimensional wave function of a particle is given as following:   2 2 x a/ ikx  x e    The expectation value of position and momentum of the particle is equal to (a) 0, k (b) a k ,  (c) a / 2, k (d) a / 2,k / 2 4. The wave function of a particle is   i ax bt    x Ae   , where A, a, b are positive real constants. The uncertainty in the momentum of the particle will be (a) a (b) a (c) 0 (d) a b   5. If the wave function of a quantum mechanical system is an eigenfunction of the operator associated with a physical observable A, then the expectation value of n A will be (a) A (b) n A (c) nA (d) n A
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 is given by 0 1 2 1 1 2 2 2 i        where 0 1 2    , , are the normalized normalized eigenfunctions with energies 0 1 2 E E E , , corresponding to the ground state, first excited state and second excited state, respectively. The expextation value of energy in the given state of the particle, is (a) 0 1 2 2 2 4 E E E   (b) 0 1 2 2 2 4 E E E   (c) 0 1 2 2 2 5 E E E   (d) 0 1 2 2 2 5 E E E   7. Consider a state 1 2 3 1 1 1 2 5 10        , which is given in terms of three orthonormal eigenstates of the operator Bˆ such that 2 ˆB n   n n  . The expectation value of Bˆ in the state  , is (a) 3/4 (b) 7/4 (c) 11/5 (d) 11/4 COMMON DATA FOR Q. 8 & Q.9 A particle of mass ‘m’ is in the state   2  , exp                   mx x t A a it where A and a are positive real constants. 8. The expectation value 2 x will be (a) 4  am (b) 2  am (c) 4 am (d) 8  am 9. The uncertainty in the momentum of the particle, will be (a) ma (b) ma (c) ma / 2 (d) ma / 4 COMMON DATA FOR Q. 10 & Q. 11 Consider a particle whose normalized wave function is given as   3/2 2 for 0 0 for 0           x xe x x x 10. The most probable position of the particle, is (a) 1  (b) 1 2 (c) 3 2 (d) 2 3 11. The uncertainty product  . x x p in the given state, will be (a) 2  (b) 2  (c) 3 2  (d) 3 4 
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 12. Consider a normalized Gaussian wave packet at time t = 0 as following:    2 0 0 2 1 exp 2 x x p  x i x               Calculate 2 2 , , , , , x x x x x p p x p   . 2 2 2 2 0 0 0 0 2 1 1 1 1 Answer: , , , , , 2 2 2 2 x x p p               13. The wave function of the particle at any instant of time is given as following:   2 2 N x x a    where N is the Normalization constant. What will be the average position & average linear momentum of the particle? [Answer: 0, 0] 14. A particle moving along x-axis has the following wave function:   for 0 1 0 otherwise ax x  x       The expectation value of the position of the particle is (a) 0.25 (b) 0.5 (c) 0.75 (d) 0 15. The wave function of the particle at time t = 0 is given as following: 1 2 1 3 2 2      i where 1 and 2 are normalized energy eigenfunctions with energy eigenvalues E1 and E2 respectively. The uncertainty in the energy of the particle at t = 0, will be (a) 1 2 1 4 E E (b) 1 2 3 4 E E (c) 1 2 3 2 E E (d) 1 2 1 2 E E Answer Key 1. (b) 2. ( ) 3. (a) 4. (c) 5. (d) 6. (c) 7. (d) 8. (a) 9. (b) 10. (a) 11. (c) 14. (c) 15. (b)