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BCS Quasiparticle Transformation and Normalization in Physics 610B, Assignments of Quantum Mechanics

San Diego State University (SDSU)Quantum Mechanics

A part of the homework assignment for physics 610b, fall 2007. It covers the quasiparticle transformation, the definition of the bcs quasiparticle vacuum, and the normalization of a specific eigenstate of the seniority hamiltonian for k = 1. The document also includes problems for students to solve, such as showing the normalization of the eigenstate for k = 1 and proving the commutation relation between the pairing operator and the number operator.

Typology: Assignments

Pre 2010

Uploaded on 03/28/2010

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Physics 610B Fall 2007
HW #6 Due Thursday, Nov 8, 2007
In HW 5 we introduced the quasiparticle (or Боглюбов) transformation,) transformation,
mmmmmmmmmmmmmmmmmmmm
cvcuacvcuacvcuacvcua ˆˆˆ
,
ˆˆˆ
,
ˆˆˆ
,
ˆˆˆ
and found the
inverse transform,
mmmmmmmmmm
avaucavauc ˆˆ
,
ˆˆ
, etc., where we assumed the
scalar numbers um, vm are real and (important) that um2+vm2 = 1. We also showed that the
quasiparticle operators obey the usual fermion anticommutation relations:
0
ˆ
,
ˆˆ
,
ˆˆ
,
ˆ
mmmmmm
cccccc
, etc, and
1
ˆ
,
ˆˆ
,
ˆ
mmmm
cccc
. Thus, we can
maniupulate quasiparticles like regular fermions.
We now define the BCS quasiparticle “vacuum” to be
0
ˆˆ
0
~
0
m
mmmm aavu
.
Problem 6.1 Show that
00
~
ˆ
0
~
ˆ
mm
cc
. Note this automatically implies the adjoint,
0
ˆ
0
~
ˆ
0
~
mm
cc
.
Problem 6.2 Consider the pairing operator
0
ˆˆ
1
ˆ
m
mm
aaS
. (NB: there are Ω values of
m > 0 (and hence a total of 2Ω single-particle states all together). I showed in class that
the state
0
ˆ
k
S
is an eigenstate (the ground state, actually) of the seniority
Hamiltonian. Note that this state has 2k particles in it. In general, however, this state is
unnormalized.
(a) Show that this state is normalized for k = 1.
(b) Prove the commutation relation
, where
0
ˆˆˆˆ
ˆ
m
mmmm
aaaaN
is the
number operator.
(c) Find the normalization for arbitrary k. Hint: Define the overlap
0
ˆˆ
0)(
kk
SSkO
. Assume you know O(k) for some k. Next you want to
compute O(k+1) from O(k). You can do this using the relation from part (b), and the fact
that
0
ˆ
k
S
has 2k particles.

Partial preview of the text

Download BCS Quasiparticle Transformation and Normalization in Physics 610B and more Assignments Quantum Mechanics in PDF only on Docsity!

Physics 610B Fall 2007

HW #6 Due Thursday, Nov 8, 2007

In HW 5 we introduced the quasiparticle (or Боглюбов) transformation,) transformation,

m m m m m m m m m m m m m m m m m m m m

a ˆ  u c ˆ  v c ˆ , a ˆ  u c ˆ  v c ˆ , a ˆ  u c ˆ  v c ˆ , a ˆ  u c ˆ  v c ˆ

     

and found the

inverse transform,

      m m m m m m m m m m

c ˆ u a ˆ v a , c ˆ u a ˆ v a , etc., where we assumed the

scalar numbers um, vm are real and (important) that u m

2

+ v m

2

= 1. We also showed that the

quasiparticle operators obey the usual fermion anticommutation relations:

 ˆ^ ,ˆ   ˆ ,ˆ   ˆ ,ˆ   0

  

m m m m m m

c c c c c c , etc, and  ˆ^ ,ˆ   ˆ ,ˆ   1

 

m m m m

c c c c. Thus, we can

maniupulate quasiparticles like regular fermions.

We now define the BCS quasiparticle “vacuum” to be

0

   

m

m m m m

u v a a

Problem 6.1 Show that

0 0

~ 0 ˆ

~ ˆ (^)   m m c c

. Note this automatically implies the adjoint,

0 ˆ 0

~ 0 ˆ

~  

  m m c c

Problem 6.2 Consider the pairing operator 

 

0

m

m m

S a a

. (NB: there are Ω values of

m > 0 (and hence a total of 2Ω single-particle states all together). I showed in class that

the state   0

ˆ

k

S

is an eigenstate (the ground state, actually) of the seniority

Hamiltonian. Note that this state has 2k particles in it. In general, however, this state is

unnormalized.

(a) Show that this state is normalized for k = 1.

(b) Prove the commutation relation  

   

N S S

ˆ

1

ˆ ,

ˆ

, where 

   

0

m

m m m m

N a a a a

is the

number operator.

(c) Find the normalization for arbitrary k. Hint: Define the overlap

ˆ ˆ ( ) 0

k k

O k S S  

. Assume you know O (k) for some k. Next you want to

compute O(k +1) from O(k). You can do this using the relation from part (b), and the fact

that   0

ˆ

k

S

has 2k particles.