Page 231 - Principles and Applications of NanoMEMS Physics

A. QUANTUM MECHANICS PRIMER 221

Other commutation relations are obtained as follows. For bosons, taking
the hermitian conjugate of (A.31) yields,

a α a = a β a . (A.33)
β
α
In addition, the facts that a α 0 = 0 and a α a + β 0 = δ αβ 0 , mean that we
+ +
can write a α a β 0 − a β a α 0 = δ αβ 0 , since the second term is zero, so
that we also have the commutation relation,
+
a α a + β − a β a α = δ αβ . (A.34)
The operator whose eigenvalue measures the number of particles in a
given state, say, state α, is the number operator, given by,
+
N = a α a , (A.35)
α
α
whereas the total number of particles, including all the distinct states, is
given by,

N = ¦ N . (A.36)
Total α
α
To measure (count) the number of particles in a given state, the operator N is
applied to that state’s eigenvector. The eigenvector of a state populated by n
particles is described by the application of the creation operator n times, i.e.,
+ n + + +
...a
a
( ) 0a = a 0 , (A.37)

n
so, measuring its occupation is effected by,
+ n + + + + + + +
N ( ) 0a = N a ...a 0 = a a a ⋅ a ...a 0 . (A.38)
a

n n
Now, using the commutation relation (A.34), we can substitute
+ +
aa →1 + a a in (A.38), so it becomes,
+ n + + + + + + + +
⋅
a
a
N ( ) 0a = N a ...a 0 = a 1 ( + a a ) a ...a 0

n n − 1
+
= a + a + 2 ( + N )a + ... a + 0 = ... . (A.39)
a

n − 2
= a + + + (n + N ) 0
a
...a
n
= ( ) (na + n + a + ) a 0 = n ( ) 0a + n