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ALL INDIA TEST SERIES, FOR, CSIR - JRF (PHYSICS) A, , Quantum Mechanics, , PHYSICAL SCIENCES, TIME: 3 HOURS MAXIMUM MARKS - 220

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wot NET/IRF, G,.. -, 1 e-sernee, ovee~- and ye, , , , Ql., , Q2., , Q3., , Ifa particle is represented by the normalized wave function, , vi5{a w(x)=) 4a*?, 0 otherwise, , , , for-a<x<a, , the uncertainty AX in its position is, , 2a 2a, ax =-—h b) AX =. ax =<4 d) AX =, (@) aX =F (b) aX => (c) 7 (@ ax =, , x, If probability density of a one dimensional system is ew(-=), x20 then the, probability that system has value *= 4 is, , 1 e-1, , 1 I, (a) CO) Fay (> @ Fa, , The Hermitian operator A has two normalized eigen state |I) and |2) with eigen values, , 1 and 2 respectively .if any state |y} is defined as |y) = cos¢|1)+sing|2). The value of, , wlAlw wl the value of ¢ is given by, 4, , x, , x x 3x, a) = 2) = 5 =, (a) 4 (b) 6 (c) 3 (d) a, The wave function of a tree particle in one dimension is given by, w (x)= Asinx+ Bsin3x, Then w(x) is an eigenstate of, , (a) the position operator (b) the Hamiltonian, , (c) the momentum operator (d) the parity operator

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fiziks, , Institute for NET/JRF, GATE, IIT-JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics, , fiziks, LISIK2, , , , Q’., , A particle of mass mm is in infinite square well of width @ centered at <. The probability, , will be equally likely in left half of box and zero in right half of box, What will be, , , , probability that measurement of energy is i E :, 2mae, 1 a 4, (a4 ) |, © 4 a) 4, 2 r x r, , A particle of mass m is confined to a one dimension region OSxSa at ¢=0 its, , normalised wave function is given by y(x,t=0)= A, , 2 1+coo{ =) uin( =) what is, a L a a, , , , average energy of the system., , , , , , , , , , (a) pae () Beet : @) ee, Sma Sma 2ma 2ma, , A particle in one-dimension is in the potential, , w if x<0, V(x)=i-V, ff OSxs 2!, oO if x>2, , If there is at least one bound state, the minimum depth of potential is, hx? Wx? hx? x?, , @ ar (b) () —, @ oor, , , , , , , , , , l6ml* 32mi°, , , , H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-1 10016, Phone: 01 1-26865455/+91-9871145498, Website: www physicsbyfiziks.com | Email: [email protected], Revised Edition 2019

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fiziks, , Institute for NET/JRF, GATE, IIT-JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics, , fiziks, LISIK2, , , , Quo., , ao x<0, The potential is given by V(x)=j1 , ,, = moy x” x>@, , 2, , interacted by potential then what is the expectation value of energy on state, , ; if a particle of mass m is, , 2 1, =,\—¢, +=, w= 5% B bs, where ¢, is ground state and @., are first excited?, , dhe lhe She l3he, o> (b) ; (c) 2 (d) a, , , , The energy eigenvalues of a particle in the potential V (7) = 2mer*x? + ax are, , , , , , , , , , rE a Pe a, (a) E, -[n+} ]ano— (b) E, = [tty forse, ‘ 2 f \ 2, (c) £,= [neo ho = = (d) E,=[n+4 2ho+—* =, 2) ma \ ay 8meo™, , If the potential of two dimensional harmonic oscillator is V (x, y) = sme + 2mar y*, 5, , Statement I: The first excited is xno and doubly degencrate, , Statement II: the second excited state is The it is non degenerate., , (a) Statement | is correct, (b) Statement II is correct, (c) Both statements I and I] are correct, , (d) Neither statement I nor statement I] is correct, , , , H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-1 10016, Phone: 01 1-26865455/+91-9871145498, Website: www physicsbyfiziks.com | Email: [email protected], Revised Edition 2019

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fiziks, , Institute for NET/JRF, GATE, IIT-JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics, , fiziks, LISIK2, , , , Qu., , Qi2., , Q13., , If H is Hamiltonian for harmonic oscillator H -[«a+ 5 po where a and a’ are, lowering and raising operator respectively then commutation [H a” | is equivalent to, , (a) 0 (b) ua (c) nea (d) Awa, , The energy difference of the first excited state £,, to that of the ground state &,, to that, , of a particle in a three-dimensional rectangular box of side 1, and <, is, , a 627? 9x h* rr, , 3,, (a) {b) (c) (d) await, , , , , , , , , , ai, , 2m? 2m? 2m?, , w (0,4) is written in basis of spherical harmonics Y,;", , As w(0.¢)=—Ly!(a,4)+-L¥#(0,6)+-Ly, '(0,)., v6 3 v2, , Statement |-y (0,@) is cigen state of 1? operator, Statement 2- If L. is measured on y(@,¢) measurement is f,0, and —h, , Statement 3- (a, , ~~, i, , +(2)-4(2) on state yw (4,¢), , (a) Statement 1 and statement 2 are correct, (b) Statement | and statement 3 arc correct, (c) Statement 2 and statement 3 are correct, , (d) All statements are correct, , , , H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-1 10016, Phone: 01 1-26865455/+91-9871145498, Website: www physicsbyfiziks.com | Email: [email protected], Revised Edition 2019