Momentum Operator - Quantum Mechanics - Solved Past Exam, Exams for Quantum Mechanics. Ambedkar University, Delhi
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Momentum Operator - Quantum Mechanics - Solved Past Exam, Exams for Quantum Mechanics. Ambedkar University, Delhi

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This is the Solved Past Exam of Quantum Mechanics which includes Initial State of System, Eigenvalue of Matrix, Eigenvalue and Eigenvector, Expectation Value, Energy of System, Conserve Quantity, Odd Function, Result of...
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Solutions to PC2130 AY0506 Sem 1 Paper Part I:

1. 22 2

2 1

2 Xm

m PH 

 









dxaaaaaaV

aaaaaa m

aaaa m

X

XmP m

H

nn ][4

)( 2

))(( 2

2 1

2 1

22*

222

222

 



Hnnn

dxanna

dxaaaaaa

nnnnn

nn

2 1)

2 1(

2 1]1[

4

])1([ 4

][ 4

2 2

2*

22*



















(shown) 2 1

2 1

2 1

2 1

2 1

n

2

2

22

 HmP

P m

H

HP m

VP m

H

n



2. xi

P  

 

)(cos2sin2)( 11 xAL x

LLiL x

Lxi xP  

 

  

  

   

  

 

 

Since when momentum operator operates on stationary state )(1 x does not give back )(1 x , hence )(1 x is not an eigenstate of momentum operator, the particle does not have a well-defined momentum.

3. Average total energy H

  

    

  

  

    

  t

H t

HH t

H dt d

dt Hd

Using Schrodinger’s equation,  H t

i    H

t i    

 

t

H i HH

t HH

i H

dt Hd

 

  

  

Assuming H does not depend explicitly on time, i.e. 0  

t H

0  

t

H dt Hd

(shown)

National University of Singapore Physics Society 2009

4. )0,()exp(),( xHittx   

Time-dependent Schrodinger’s equation: ),(),( txHtx t

i    

When )0,()exp(),( xHittx    ,

LHS: ),()0,()exp()0,()exp( txHxHitHiixHit t

i    

 

 

The wavefunction satisfies the time-dependent Schrodinger equation, hence it’s a solution to Schrodinger equation.

5. )()( 011 xaAx  

  

  

 

  

  

  

 

  

  

  

  

   

  

 

  

  

  

 

     

     



2 4/1

1

2 4/1

1

2 4/1

1

2 4/1

1

2 exp2

2 exp)2(

2

2 exp

2

2 exp

2

xmxmmA

xmxmm m

A

xmxmxmm m

A

xmmxm xm

A





 



 

  

  

  

  

1

1 2 12

1 exp2

1)(

1

2/32/1 2

1

22 2/1

2 1

2 1

  

  

  

  

  

  

 

  

 



A m

mmA

dxxmxmmA

dxx

 

 

  

 



 



   

  

 

     

  

 

   dxxmmxmxmmdxxx 2

4/1 2

4/1

01 2 exp

2 exp2)(*)(

 

 

 

0exp2 2 2/1

  

  

 

   

 dxxmxmm

 

 

Since 0)(*)( 01  

 dxxx  , )(1 x and )(0 x are orthogonal. (shown)

6. )()exp()( xixqx  

 

  

  

  

xxm

ixj  * *

2 )( 

)exp()(

)exp()()exp(

* ixqx x

ixqxixqiq x

  

  





National University of Singapore Physics Society 2009

 

  

   

  

  

 

  

  

 

  

x xiqx

x xiqx

m ixj

x ixqxixqiq

x 



)()()()( 2

)(

)exp()()exp( *

 

2

2

))((

))((2 2

x m q

xiq m

i



The physical meaning of q is wave-number vector. Part II:

1(a)  

   

  

0 )(

2 1)0,(

n n

n

xAx 

Using normalization condition, 1)( 2  

 x

2 1

1 2

11 1

1 2 1

1)()( 2

1 2

1

1)0,(

2

0

2

0 0

*2

2

 

  

  

  

     

  







A

A

A

dxxxA

dxx

n

n

m n nm

nm



1(b)  



   

  

 

  

0

) 2 1(

0

/

2 1

2 1

2 1

2 1),(

n

tni

n

n

n

tiE n

n

eetx n

 

1(c)  

  

     

  

0 0

)(**2

2 1

2 1

2 1),(),(),(

n m

tnmi nm

nm

etxtxtx 

The above function is periodic as there is a term tnmie )(  present in the expression.

Period 

 )(

2 nm

Period is longest when 1 nm ,  2

T

1(d) Expectation value of energy 

  dxxHx )0,()0,( * 

  



   

   

   

  

  

   dxxHx

n n

n

n n

n

00

* )( 2

1 2

1)( 2

1 2

1 

National University of Singapore Physics Society 2009

Expectation value   





  

   

   

  

  

   dxxHx

n n

n

n n

n

0

1

0

* 1

)( 2

1)( 2

1 

3

2

0

1

0

1

*

0

1

0

1

0

1

0

* 1

)12( 2

12 1

)12( 1

12 1

2 1

2 1

2 1

2 11

2 1

)()() 2 1(

2 1

2 1

)( 2

1)( 2

1

 

 

  

 

 

  

   

   

  

 

  

  

  

 

  

   

 

 

  

  

   

   

  

  

   

   

  

  

  





 























n

n

n

n

nn n

n

n

n

n nn

n

n n

n

n

dxxxn

dxxEx

2

3(a) Schrodinger’s equation:   

  

 

  )(

2

2

xV m

p t

i

* 22

* *

*

* 2*

2 **

2

* *

*

)( 2

1)( 2

1

)( 2

1

)( 2

1

)( 2

1

),(













 

  

 

  

 

 

  

 

  

 

 

 

  

 

 

 

  

 

 

  

 

  

 





xV m

p i

xV m

p itt

xV m

p it

xV m

p it

xV m

p it

dx ttt

dxtx



 

  

  

  

 

  

  

 

  

  

 

  

  



xxxm i

xm i

xm i

xmixmi *

* 2

*2

2

2 *

2

*22

2

22 *

222

2 1

2 1







 

 

Define  

  

  

  

xxm

itxj  * *

2 ),( 

0   j

t  (Continuity equation)

National University of Singapore Physics Society 2009

3(b)(i)   

    )exp()()exp()(

2 1),( 2211 

itExitExtx 

  

  

  

 

  

  

   

  

   

  

 

  

 

  

  

  

)exp()exp( 2

1

)exp()()exp()( 2

1

)exp()exp( 2

1

2 ),(

2211 *

2 2

1 1

*

2211

* *







itE x

itE xx

itExitEx

itE x

itE xx

xxm itxj









  

   

 

 

 

  

  

   ))(exp())(exp(

2 1 211

2 122

1 2

2 1

1

*

 EEit

x EEit

xxxx  (1)

  

   

 

 

 

  

  

   ))(exp())(exp(

2 1 121

2 212

1 2

2 1

1 *

 EEit

x EEit

xxxx  (2)

(1) – (2):

  

   

 

  

   

 

 

  

 

 

 

  

 

  

)sin(2)sin(2 21121221

)()( 1

2

)()( 2

1

12212112

tEEi x

tEEi x

ee x

ee x

EEitEEitEEitEEit









 

  

  

  

xxm

itxj  * *

2 ),( 

 

  

   

      

  

 

  

 

  

   

   

 

  

   

 



 

  

   

   

 

  

   

 



 

  

   

   

 

  

   

 

)sin(

)sin()sin(

)sin()sin(

)sin(2)sin(2 2

122 1

1 2

121 2

122 1

211 2

122 1

211 2

122 1

tEE xxm

tEE x

tEE xm

tEE x

tEE xm

tEEi x

tEEi xm

i

 

 

 

 









3(b)(ii)  *),( tx

 

  

  

  

  

  

 

  

     

   



)cos(4 2 1

2 2 1

)exp()()exp()()exp()()exp()( 2 1

12 21

2 2

2 1

)()(

21 2

2 2

1

2 2

1 1

2 2

1 1

1221

tEE

ee

itExitExitExitEx

EEitEEit











National University of Singapore Physics Society 2009

3(b)(iii) )sin(2)cos(4 2 1 12

21 2112

21 2

2 2

1 t EEEEtEE

tt  

 

  

  

  

 

  

   

     

  

  

  

 

  

   

      

  

 

  

 

   )sin()sin( 122

2 2

12 1

2

2 122

1 1

2 t EE

xxm tEE

xxmxx j

 

  

Using Schrodinger’s equation,  EH

))((2

))((2

)( 2

)( 2

)( 2

)( 2

22222 2

2

11122 1

2

1112 1

22

1112

22

2

222









ExVm x

ExVm x

xVE xm

ExV xm

xV xm

xV m

pH

  

  

  

 

  

 

 

  



 

  

   

     

  

  

  

   )sin( 122

2 2

12 1

2

2 t EE

xxmx j

  

t

tEEEE

tEEEE

tEEExVmExVm m

  

  

   

  

   

 

  

   

      

   







)sin()(2

)sin()(2

)sin()])((2[)])((2[

12 2121

12 1221

12 2222111122





 

0   

 

x j

t  (verified)

National University of Singapore Physics Society 2009

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