Title QM DPP Sheet 07.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: CONCEPT OF WAVE FUNCTION - II Daily Practice Problem (DPP) Sheet 7 PART - I: Multiple Choice Questions (MCQ) 1. 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    2. If the probability that x lies between x and x+dx is p x dx a    expax dx  , where 0     x , a 0, then the probability that x lies between 1 x and   2 2 1 x x  x is: (a)   ax1 ax2 e e    (b)   ax1 ax2 a e e    (c)   ax2 ax1 ax2 e e e     (d)   ax1 ax1 ax2 e e e     3. 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   4. 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
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 5. 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 6. A system in a normalized state 1 1 2 2      c c , with 1 and 2 representing two different eigenstates of the system, requires that the constants 1 2 c c and must satisfy the condition (a) 1 2 c c. 1  (b) 1 2 c c  1 (c)   2 1 2 c c  1 (d) 2 2 1 2 c c   1 PART II: Numerical Answer Type (NAT) Questions 7. The wave function of a particle is given by   2 2 ( ) x x Ce x         , where c and  are constants. The probability of finding the particle in the region 0    x is equal to _________________________ (Your answer should be upto ONE DECIMAL PLACES) 8. The wave function of a particle is given by     2 2 2 x x Cx e x         , where C and  are positive real constants. The probability of finding the particle in the region 0    x is equal to ______________ (Your answer should be upto ONE DECIMAL PLACES) 9. The wave function of the particle at a certain instant is given as   2 2 exp x x A ikx a           If P1 and P2 denote the probabilities of finding the particle in the range a to a+da and 2a to 2a+da respectively and the ratio P1 / P2 is m e , then m is equal to _______________________________ (Your answer should be AN INTEGER) 10. Consider a system whose state is given in terms of an orthonormal set of three vectors 1 2 3 3 2 2 3 3 3        Take an ensemble of 1800 identical system each one of them is in the state  . If measurements are done, the number of systems that will be in the state 2 is ___________________________ (Your answer should be AN INTEGER)
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 11. In a 1-D potential system, 0 1 2    , and represents ground state, 1st excited state & 2nd excited states respectively. Two states are defined as following: 0 1 2 0 1 2               2 4 , 2 3 The value of  for which   and are orthogonal to each other, is __________________________ (Your answer should be AN INTEGER) 12. The state of the system is given by 0 1 2        2 3 5 where 0 1 2    , and forms an orthonormal set. The probability of finding the particle in the state  2 , will be ______________________________________ (Your answer should be upto TWO DECIMAL PLACES) PART III: Multiple Select Questions (MSQ) 13. 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. 14. The wave function corresponding to a particle is given by   1 | | x a x e a    [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 ln 2 2 a . (d) The value of ‘b’ such that probability of finding the particle between    b x b is 0.5, is 2 ln 2 a .
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 15. The wave function of a particle at time t = 0 is given as   2 sin ; 0 1   0 ; otherwise n x x x                Which of the following statements is/are CORRECT? (a) Probability of finding the particle in the left half is 0.5 for n  1,2,3,...... (b) Probability of finding the particle in the left half is 0.5 for n  1,3,5,...... only. (c) Probability of finding the particle in the region 0 4 L  x is 0.25 for n  1,3,5,...... (d) Probability of finding the particle in the region 0 4 L  x is 0.25 for n  2, 4,6,...... Answer Key PART I: Multiple Choice Questions (MCQ) 1. (b) 2. (a) 3. (c) 4. (a) 5. (a) 6. (d) PART II: Numerical Answer Type (NAT) Questions 7. (0.5) 8. (0.5) 9. (6) 10. (800) 11. (7) 12. (0.68 to 0.71) PART III: Multiple Select Questions (MSQ) 13. (a,d) 14. (a,c) 15. (a,d)