(1) Prove the identity,


  dxe
 ax 2 
 a
.
Hint: I ( a )
 


 dxe
 ax 2
,
I a 2 
[
 
 dxe
 ax
2
][
 
 dye
 ay
2
]
  dxdye

(
2  y
2
) .
Then use polar coordinate to evaluate the integral.
(2) Extending the previous result to show that


  dxe
 ax 2
 bx 
 b 2 a e 4 a .
Hint: complete the square first,
 ax
2  bx
  a ( x
2  b a x
 b
2
4 a
2
)
 b
2
4 a
.
(3) Calculate the moment n
  
 dx 2 n
 ax
2 x e =?
One of the most important questions in quantum dynamics is the calculation of the transition amplitude between two given states. For instance, for a free particle,
 p
2
, we are interested in the following
2 m question: if at the initial time t
 t
1
, the particle is in an eigenstate of the position operator q
ˆ
, with eigenvalue x , that is, q t
1 x t
1
 
H
| ,
1

H
.

What is the transition amplitude that at the final time t
 t
2
, the particle will be observed to be at position y, that is, q t
2 y t
2
 
H
| ,
2

H
?
(1) The quantity we need to calculate is
H

,
2
| ,
1
 
H
|
ˆ
( , ) | x

, where U
ˆ
( t
2
, t
1
) is the evolution operator as discussed in the class.
What is the relation between H
 t
 t
2
 t
1
, and U
ˆ
( t
2
, t
1
) ?
(2) By inserting two complete sets of momentum eigenstates,
 dk
2

| k k 1,
 d 
2

|    |

1,

|
ˆ
( , ) | x
   d
2

 y |

|
ˆ
( , ) | k

|

.
Remember that we have derived in the lectures that
 y |
 e i y ,
  x k
  e
 ikx .
Show that the matrix element of U
ˆ
( t
2
, t
1
) can be written as

|
ˆ
( , ) | k
 
( k
 f k , what is f( k ) ?
(3) Put everything together, you should get an integral over k,
 y | U
ˆ
( t
2
, t
1
) | x

   d
2

 y |

|
ˆ
( , ) | k

|

   d
2
 e i y

( k
 f k e
 ikx
  dk

2 e
(

) ─ ( ＊ ) f(k) should look like a Gaussian distribution and you can now evaluate the integral by applying the formula you have derived in problem (I),
 dxe
 ax 2
 bx 
 b 2 a e 4 a . Identity what is a & b in formula ( ＊ ) and write

down the final answer for the transition amplitude
 y | U
ˆ
( t
2
, t
1
) | x

.
(4) If we take the equal time limit t
2
 t
1
, the transition amplitude lim t
2
 t
1
H

,
2
| ,
1

H will become the orthonormal condition for position eigenstates

|
 
( x
 y ) . Check that this is true.
(5) Show that, by directly applying the time derivative i

 t
2
on
U
ˆ
( y , t
2
; x , t
1
) , we can check that U
ˆ
( y , t
2
; x , t
1
) satisfies the Schrodinger equation, i

 t
2
ˆ ( , ; , )
2 1
2
 
2 m

2
 y 2
U y t x t
2 1
.