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ij
h | i
and the formula (5.17) for the expansion coefficients becomes
α = ix . (5.21)
i
h | i
If the ketvector x is representedconcretely by a column vector,the corre-
| i
sponding dual bra vector is represented by a row vector, the elements of which
are complex conjugates of the elements of the ket vector, or
α
1
α
 2
|x i= . ⇒hx |=(α∗1,α∗2,...,α∗n). (5.22)
.
.
 
α 
 n
In this notationwe have to decide how to treatthe equation x,y = y,x ∗
h i h i
of the definition of the inner product. Referring to equation (5.18) we see that
taking the complex conjugate we get
x,y =(x y +x y +...+x y )
i∗ ∗1 1 ∗2 2 ∗n n ∗
h
=y x +y x +...+y x = y,x
1∗ 1 2∗ 2 n∗ n
h i
making the identification natural
xy ∗ = y x (5.23)
h | i h | i
where
y = y and x = x . (5.24)
† †
h | | i | i h |
Fromnowon,the Diracnotationwillbe usedalmostexclusively. Theterms
state, vector, bra and ket will be used interchangeably in the following. No
confusion can arise.
TheDiracsystemofnotationcapturesadeeppropertyofquantummechan-
ics,namely athat aphysicalsystemcanbe representedinmanydifferent ways.
The explicit representation that we use carries very little significance and can
be chosen for convenience.
5.1.4 Tensor products
Using tensor products new (larger) vector spaces can be built out of existing
vector spaces. This is a common practice in quantum mechanics where for
examplethe jointstates ψ oftwoindependentsystemscanbebuiltfromthe
12
| i
states of the constituents ψ and ψ .
1 2
| i | i
To make this notion precise, let V and W be vector spaces of dimension n
and m, and let v and w denote generic vectors in these spaces respectively.
| i | i
102
The tensor product V W is defined as the nm-dimensional vector space con-
⊗
sisting of all linear combinations of ’tensor products’ of vectors v w . This
| i⊗| i
is an abstract product, which by definition satisfies
α(v w )=(αv ) w = v (αw ) (5.25)
| i⊗| i | i ⊗| i | i⊗ | i
(v + v ) w = v w + v w (5.26)
1 2 1 2
| i | i ⊗| i | i⊗| i | i⊗| i
v (w + w )= v w + v w . (5.27)
1 2 1 2
| i⊗ | i | i | i⊗| i | i| i
Theseconditionson v w sufficestoprovethatV W isindeedalinear
| i⊗| i ⊗
vector space, i.e, the conditions (1.1) are satisfied.
For an example of a concrete realizationof the tensor product, consider two
vectors x and y
| i | i
α β
x = 1 and y = 1
| i (cid:18)α 2(cid:19) | i (cid:18)β 2(cid:19)
in a 2-dimensional vector space. The tensor product of the vectors becomes
α β
1 1
α β
x y = 1 2.
| i⊗| i α 2β 1
α β 
 2 2
5.2 Operators and dynamical variables
Theintuitionbehind(abstract)quantummechanicsisthatifvectorsinaHilbert
space are used to describe states of a physical system, then it is natural to use
linear operators acting on the space to describe the dynamics, i.e changes of
state. A linear operator acting on a state yields a new state. In the following
sectionwe will make the basis for this intuition exact. The dynamicalvariables
of classical physics, like position, momentum and energy, will be mapped to
linear operators in quantum mechanics.
5.2.1 Linear operators
A linear operator A on a vector space V is a linear map from V to itself, that
is A: V V. The image A(x ) of a vector x is written Ax . The linearity
→ | i | i | i
requirement is expressed as