Solutions to PC2130 AY0708 Sem 2: Quantum Mechanics Problems, Exams of Quantum Mechanics

Download Solutions to PC2130 AY0708 Sem 2: Quantum Mechanics Problems and more Exams Quantum Mechanics in PDF only on Docsity! Solutions to PC2130 AY0708 Sem 2 1(i)            E E E H 300 020 00 ,            010 101 010 2 1 xS 1(ii)   0det  xSI 1 ,0 0)1( 0 2 1 2 1 0 210 2121 021 2 3              Possible outcomes 1 ,0  1(iii) Let            c b a )0( , )0()0(  xS )( 2 1 , 2 1 2 1 010 101 010 2 1 cabbca c b a b ca b c b a c b a                                                       Let 1 ,2  cab ,            1 2 1 )0( Normalizing the state )0( , we have            1 2 1 2 1 )0( 1(iv)                                               /3 /2 / /3 /2 / / 2 2 1 1 2 1 2 1 00 00 00 )0()( iEt iEt iEt iEt iEt iEt iHt e e e e e e et  1(v) Average value )()( tSt x                                /3 /2 / /3/2/ 2 010 101 010 2 1 2 4 1 iEt iEt iEt iEtiEtiEt e e e eee National University of Singapore Physics Society 2009 1(v) Average value        Et tSt x cos)()(  2(i) Infinitely many 2(ii)        0 1 z ,        1 0 z                    1 2 5 1 1 0 5 1 0 1 5 4 5 1 5 4 zz Those state vectors that are orthogonal to  will be perfectly distinguishable from single shot measurement. Let the state vector be  ,         2 1 5 1 2(iii)                12 24 5 1 12 5 1 1 2 5 1P 2(iv) Density matrix,                10 04 5 1 2(v) 2211 5 1 5 4 }{ AAAtrA         22211211 2221 1211 2221 1211 224 5 1 2 2 12 5 1 1 2 12 5 1 AAAA AA AA AA AA AA                         Since  AA  for any A in general, hence  and  do not represent the same state. 3(i) Schrodinger equation: 0)())(( 2 22 2        xxVE m dx d E For 00  EV , 0)( 0  EVxVE as E is negative and smaller than 0V in magnitude. Let )( 2 02 2 1 EV m k   , xikxikE BeAex 11)(  , the states are bound. For 0E , )( 2 02 2 2 EV m k   , xikxikE DeCex 22)(  , the states are not bound. 3(ii) Between ax 0 , 0)( VxV  , )( 2 021 VE m k   National University of Singapore Physics Society 2009