Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
k k | |k k
x 0 with equality only if x =0. (5.11)
k k≥ k k
x+y x + y , (5.12)
k k≤k k k k
ensuringthat x satisfiesallrequirementsforapropernormonavectorspace.
k k
The last property, (c), is the triangle inequality, which can be shown from the
Schwarz inequality.
A vector x is said to be normalized if x =1.
k k
Quantum states and rays
Quantum states are represented by normalized vectors Ψ. Actually, if the nor-
malized states Ψ and Ψ are related by the equation Ψ = αΨ where α = 1,
′ ′
| |
then Ψ = Ψ = 1, and they are thought of as representing the same state.
′
k k k k
This is refereed to as saying that the two vectors belong to the same ray in the
space. Thus it is more correct to say that quantum states are represented by
equivalence classes of states.
1Amorecorrectwaytoexpressthisis: thereisanisomorphismi:V Cn.
→
98
Linear independence and basis sets of vectors
A finite set of vectors x ,x ,...x is linearly dependent it there exists some
1 2 k
{ }
set of complex numbers α ,α ,...α (not all of them zero) such that
1 2 k
{ }
k
α x =0
i i
Xi=1
.
If there is no such set of complex numbers, then the set of vectors is said
to be linearly independent. This gives a precise way of defining the dimension
of a finite dimensional vector space. A vector space is n-dimensional if it con-
tains a subset of n linearly independent vectors but no subset of n+1 linearly
independent vectors.
A subset e ,e ,...e of an n-dimensionalvector space V is a basis for V
1 2 n
{ }
if any vector x in V can be expanded as
n
x= α e . (5.13)
i i
Xi=1
The numbers α are called expansion coeffients. They are unique. This
i
follows from the linear independence of the basis vectors.
Orthonormal basis vectors
Apairofvectorsxandyaresaidtoorthonormalpairiftheyarebothnormalized
andorthogonaltoeachother. Anorthonormalbasisforann-dimensionalvector
spaceis a setofbasis vectors e ,e ,...e ,allof whicharenormalizedandin
1 2 n
{ }
which every pair is orthogonal. Or more concisely
e ,e =δ for i,j =1,2,...,n, (5.14)
i j ij
h i
where δ is the Kronecker delta which is equal to 1 for i=j and 0 otherwise.
ij
It can be shown that every finite-dimensional vector space has an orthonor-
mal basis. A constructive proof of this is given by the inductive Gram-Schmidt
procedure. Suppose v ,v ,...v isabasisforthe vectorspaceunderconsid-
1 2 n
{ }
eration. For the base case, define
v
1
e = . (5.15)
1
v
1
k k
Then define inductively for 1 k n 1
≤ ≤ −
v k e ,v e
e = k − i=1h i k+1 i i . (5.16)
k+1 v Pk e ,v e
k k − i=1h i k+1 i i k
P
It is straightforward to verify by direct calculation that this yields an or-
thonormal basis e ,e ,...e .
1 2 n
{ }
99
Orthonormal basis vectors are useful as it is possible to compute the ex-
pansion coefficients explicitly. Suppose we have a vector x with (unknown)
expansion
n
x= α e .
i i
Xi=1
Taking inner-product on both sides of the equation with an arbitrary basis
vector e yields the following short calculation
j