UCD Fermionic Version of A Harmonic Oscillator Questions

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qnynbjnat

Mathematics

University of California Davis

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Explanation & Answer

I edited my answer a little bit, and upload the final answer again as attached below.

Problem 1:

 dt i − V ( , ) , where M is the 1-manifold of time such as

It is initially given that S =

M

t

(

)

, and V has the form of V  , = a + b + c + d . It is sufficient to prove that

i − V  ,  dt = i  −   dt , but this is true since





(

)

(

)

(

)

dt i − V  ,  = dt i − a + b + c + d 




 d

 dt i − V  ,  = dt i
− a + b + c + d 


 dt

 d

 dt i − V  ,  = dt i
−  


 dt



 dt i − V  ,  = dt i  −  





(

)

(

)

(

)

(

)



Hence, it follows that S =



 dt i − V ( , ) implies S =  dt i  −   , as desired

M

M

result.
Problem 2:


Since H =

1    
   −   is the Hamiltonian, the momentum  of  is calculated by using
2 


Lagrangian as follow:


H
=

1

= 
2 

   
  −  



1             
  = 
  + 
−
  − 

2  


 


       

1  

  =  −    −    −  −     −   −    

 
2 
  



   
 
1
  =  −2
2
  = −i

(

)

Similarly, the momentum  of  is calculated by using Lagrangian as follow:


H
=−

1 
=− 
2 

   
  −  



1             
  = − 
  +   −   −   
2  


 




1                 
  = −  −    +  −   − −    −  −  


 
2 
     



 

  

 
1
  = − i ( 2 )
2
  = −i
Hence, we can ignore the constant

 and conclude that the conjugate momentum to  is

 = −i , and conversely, the conjugate momentum to  is  = −i , as desired result.
Problem 3:

...

Npr_Ghgbe (5316)
University of Virginia

Anonymous
Really great stuff, couldn't ask for more.

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