Tiêu đề QM DPP Sheet 06.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 (Online Batch) Section: MODERN PHYSICS (Quantum Mechanics) Chapter: CONCEPT OF WAVE FUNCTION - I Daily Practice Problem (DPP) Sheet 6 1. The plot of a particular function is as shown: (x) x (a) This is not acceptable as a bound state wave function because it is not zero at the origin. (b) This is not acceptable as a bound state wave function because it is discontinuous. (c) This is acceptable as a bound state wave function. (d) This is not acceptable as a bound state wave function but acceptable as a scattering state wave function. 2. Which of the following functions represents acceptable realistic wave function of a bound state of the particle in the range     x  ? (A, B, D, E, F are positive real constants & C is negative real constant) (a)   x A   tan x (b)   x B   cos x (c)     2  x C exp D x/ (d)     2  x E x  exp Fx 3. Which of the following is an allowed wavefunction for a particle in a bound state? (where N is a constant and  ,  0) (a)   3 x e x N x     (b)   1  x x N e      (c)     2 2 2 x y z x x Ne e         (d)   non-zero constant if 0 if x R x x R        4. Which of the following wave function is an allowed wavefunction for a particle in a bound state for all values of x? (A is a real constant) (a)   x A   sec x (b)   x A   tan x (c)   2 x  x Ae  (d)   2 x  x Ae  5. An one dimensional wave function is given by   sin ; 0 2 2 0 ; otherwise x A x l x l                      The value of the normalization constant ‘A’ is (a) 1 l (b) 2 3l (c) 2 l (d) 1 3l
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. Consider the wave function of a bound state in the one dimensional potential shown below. –2a V0 2V0 V(x) –a a 2a (a)   2 | , | 1 a a  x t dx    (b)   2 | , |  x t dx   is time dependent (c)   2 2 2 | , | 1 a a  x t dx    (d)     2 2 2 2 | , | | , | 1 a a a a   x t dx x t dx       7. Which of the following wave function is an allowed 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         8. 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  9. 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  10. 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) 
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. 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 12. 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 13. 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   14. Check whether the following function represent a physically acceptable reallistic wavefunction for the bound state for the particle or NOT? (A and a are positive real constants) (i)   2 2 exp x x A ikx a           (ii)   2 2 exp x x A a         (iii)   | |/ x a  x Ae  (iv)   x a/  x Ae  (v)   x A ax   tan   (vi)   x A ax   sin   (vii)    x A x x     sin for 0 1   (viii)   2 A ax sin x x   (ix)   2 2 ax  x Ax e  (x)   0 for | | for | | x a x c x a           (xi)   2 ax  x Ax e  (xii)   x A ax   sec  (xiii)   2 for 0 ax  x Ax e x      (xiv)   2 ar  r Ar e  (xv)   A ikr r e r   [ANSWER: (i) Yes, (ii) Yes, (iii) Yes, (iv) No, (v) No, (vi) No, (vii) Yes, (viii) No, (ix) Yes, (x) No, (xi) No, (xii) No, (xiii) Yes, (xiv) Yes, (xv) No] 15. Check whether the following functions are normalizable or NOT? If YES, Find the normalization constant. (A and a are positive real constants) (i)   sin for 0 1   0 otherwise n x x x         (ii)   2 2 x a ikx /  x e    (iii)   2 2 2 a x  x x e   (iv)   2 cos for 0 otherwise x a x a x a                  (v)   r a/  r e   (vi)   ia e     Answer Key 1. (b) 2. (d) 3. (c) 4. (d) 5. (a) 6. (a) 7. (a,d) 8. (a) 9. (b) 10. (d) 11. (a) 12. (a) 13. (a)