text stringlengths 0 8.13M |
|---|
| i | i |
belongs inthe interface, being effectively a specificationof a functionality to be |
provided by the implementation. The implementation then, could be in terms |
ofwavefunctions orin terms ofmatrices,or perhaps interms ofsome other for |
the purpose suitable (mathematical) constructs. In the next section on angular |
momentum and spin, we will see a concrete example of this. |
86 |
Derivation of the normalization conditions |
This section is somewhat technical, and do involve certain concept not yet dis- |
cussed. The purpose is to derive the normalization coefficients ξ(n) and η(n). |
The reader might want to skip it for now and return after reading chapter 5. |
The coefficients ξ and η are subject to some consistency conditions. First, |
since [a,a ]=1, we have |
† |
[a,a ]n = n |
† |
| i | i⇒ |
ξ(n)η(n+1) ξ(n 1)η(n)=1. (4.59) |
− − |
Furthermore, the states n are subject to a orthonormalitycondition, anal- |
| i |
ogous to (??) |
nm =δ , |
nm |
h | i |
and in particular |
nn =1. |
h | i |
Thequestionarises,whatis n? Adetailedexplanationofthiswillbegiven |
h | |
i chapter 5. Here we can think of n as a form of conjugate to n . As regards |
h | | i |
the equation (4.52), this conjugation, denoted by a dagger , works as follows |
† |
(a n ) =(ξ(n)n+1 ) |
† † † |
| i | i ⇒ |
na= n+1ξ(n) , (4.60) |
∗ |
h | h | |
and as regards equation (4.53) |
(an )† =(η(n)n 1 )† |
| i | − i ⇒ |
na = n 1η(n) (4.61) |
† ∗ |
h | h − | |
Enforcingthecondition nn =1onthestate n+1 ,thatis n+1n+1 =1 |
h | i | i h | i |
and using (4.52) and (4.53) as well as (4.60) and (4.61) yields |
1 |
n+1n+1 = naa† n = |
h | i ξ(n)ξ(n) h | | i |
∗ |
1 1 |
n[a,a ]+a an = n1+η(n)η(n) n = |
† † ∗ |
ξ(n)ξ(n) h | | i ξ(n)ξ(n) h | | i |
∗ ∗ |
1 |
1+η(n)η(n)∗ , |
ξ(n)ξ(n) |
∗(cid:0) (cid:1) |
where the common rewriting trick aa =[a,a ]+a a has been used. |
† † † |
Concluding, we get |
1 |
1+η(n)η(n) =1, |
∗ |
ξ(n)ξ(n) |
∗(cid:0) (cid:1) |
87 |
or more succinctly |
ξ(n)ξ(n) η(n)η(n) =1 (4.62) |
∗ ∗ |
− |
Before trying to solve equations (4.59) and (4.62) we will make the simpli- |
fying assumption that ξ and η are real. This assumption can be justified after |
the fact. So equation (4.62) becomes |
ξ(n)2 η(n)2 =1. (4.63) |
− |
The solution to (4.59) and (4.63) can now be constructed in an inductive |
way. Starting from a0 =0 which implies |
| i |
η(0)=0 |
we find that (4.59) implies |
ξ(0)=1, |
which in its turn using (4.63) implies |
η(1)=1. |
Goingoninthis way,usingequations(4.59)and(4.63), wegetthe sequence |
of equations |
ξ(1)=√2 |
η(2)=√2 |
ξ(2)=√3 |
η(3)=√3 |
. |
. |
. |
η(n)=√n, |
or in general |
ξ(n)=√n+1 |
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