A Hilbert space is an infinite-dimensional vector space consisting of sequences of real numbers that satisfy a convergence condition. It allows vector addition and scalar multiplication. In quantum mechanics, state vectors span a Hilbert space. For identical particles, boson state vectors are symmetric and fermion state vectors are antisymmetric. Linear algebra concepts like operators, eigenvectors, and superposition are used in Dirac's formulation of quantum mechanics postulates. Observables are represented by operators and eigenvectors correspond to eigenvalues. Any state can be written as a superposition of eigenvectors.

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Lecture 13
Hilbert space
Named after David Hilbert (1862-1943), a German mathematician. A Hilbert space is a
vector space, v consisting of say, an infinite sequence of real number (𝑎1, 𝑎2 … … … … … )
satisfying ∑∞ 𝑎𝑖
2 = 𝑎 2 + 𝑎 2 + … … … … … < ∞ that is the sum converges where 𝑎
𝑖=1 1 2 𝑖
forms the basis for the Hilbert space.
Addition and scalar multiplication are defined componentwise.
(𝑎1,𝑎2, … … … . ) + (𝑏1, 𝑏2, … … … ) = (𝑎1 + 𝑏1, 𝑎2 + 𝑏2, … … … )
𝛼(𝑎1, 𝑎2 … … . . ) = (𝛼𝑎1, 𝛼𝑎2, … … … … . . ).
It can also be shown that inner products converge as well, that is
and
∑∞ 𝑎𝑖 𝑏𝑖 = 𝑎 𝑏 + 𝑎 𝑏 + … … … …. converges. The proof can be given
𝑖=1 1 1 2 2 as in the
following-
|𝑎1𝑏1| + … … … … |𝑎𝑛𝑏𝑛| ≤ √∑𝑛 𝑎𝑖
2 √∑𝑛
𝑖=1 𝑖=1 𝑏𝑖
2
≤ √∑∞ 𝑎𝑖
2 √∑∞
𝑖=1 𝑖=1 𝑖
𝑏 2 ( By Cauchy-Schwarz inequality)
Thus the sequence of sums
𝑆𝑛 = |𝑎1𝑏1| + … … …… . |𝑎𝑛 𝑏𝑛 | is bounded for all n and also for very large n.
In Quantum Mechanics, the state vectors span an infinite vector space which is termed as
the Hilbert space. As an example, let us talk about the Hilbert space of fermions
and bosons and compare and contrast between them.
A system of (identical) bosons will always have symmetric state vectors and a
system (identical) fermions will always have antisymmetric state vectors. Let us call the
Hilbert spaces for bosons and fermions as 𝑣𝑆 and 𝑣𝐴 respectively. For the sake of
simplicity in
terms of representation, we consider two particles. At any given time, the state of
two bosons is an element of 𝑣𝑆 and that of two fermions an element of 𝑣𝐴. The
normalized
vectors which satisfy the criterion of being elements are,
1
|ΨS > = [|𝑥1𝑥2 > +|𝑥2𝑥1
>]
√2
|ΨA > = [|𝑥1𝑥2 > −|𝑥2𝑥1 >]
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1
√2

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where the vectors are denoted in the position space for the particle (say). In a
general sense, the form for the vectors are valid in any representation.
Application of Linear Algebra in Quantum Mechanics
The theory of linear vector spaces is utilized by P.A.M. Dirac in formulating the
fundamental postulates of Quantum Mechanics.
We consider a complex vector space whose dimensionality is specified according
to the nature of a physical system under consideration. For example in a Stern-Gerlach
(SG) experiment, the only quantum mechanical degree of freedom is the spin of an atom.
The dimensionality in this case is determined by the number of alternative paths
the atoms can follow, when subjected to a SG apparatus. Whereas for the position
and momentum of a particle, the number of alternatives is infinite, in which case the
vector space in question is called a Hilbert Space.
Suppose a state is represented by a ket |Φ >, multiplying it by a complex number c,
physically |Φ > and 𝑐|Φ > represent the same space. In other words, the direction in
the vector space is of significance.
An observable, such as momentum or spin components can be represented by an
operator, e.g. 𝐴̂. In general 𝐴̂ |𝛼 >≠ (not analying)|𝛼 >. However there are some kets
which obey the equality and they are known as eigenkets.
𝐴̂ |𝛼 > = 𝑎|𝛼 >
a : eigenvalue of the operator 𝐴̂.
We mentioned earlier the dimensionality of the vector space is determined by the number
of option the system can assume. Thus any arbitrary state |Ψ > of the system is written as
a superposition of many such options.
|Ψ > = ∑𝑛 𝑐𝑛|Φn >
where 𝑐𝑛’s are the complex coefficients.
Two kets are said to be orthogonal when
< Φi|Φj > = 0
and < Φi|Φi > = 1 where |Φi > are properly normalized. Operators
1. 𝐴̂ [𝑐1|Φ1 > + c2|Φ2 >] = 𝑐1𝐴̂|Φ1 > + c2 𝐴̂|Φ2 >
2. An operator A is said to be Hermitian when 𝐴̂† = 𝐴̂ where 𝐴̂† is the adjoint A,
i.e. complex conjugate and transposed.
Applications:
1. The eigenvalues of a Hermitian operator are real. Further the eigenkets of A
corresponding to different eigenvalues are orthogonal.
A|Φ > = 𝑎|Φ >
As A is Hermitian,
(13.1)

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< Φ|𝐴 = 𝑎∗ <Φ|
Where |Φ > is an eigenvalue of A
(13.2)
use < Φ| on (1) and |Φ > on (13.2)
< Φ|𝐴|Φ > = 𝑎
< Φ|𝐴|Φ > = 𝑎∗
Thus 𝑎 = 𝑎∗, Also < Φ| Φ > = 𝛿ΦΦ′
2.Given an arbitrary state (ket) in ket space spanned by the eigenkets of A,
we attempt to expand it as follows-
|Ψ > = ∑𝑛 𝑐𝑛|Φn >
𝑐𝑛 =< Φn|Ψ >
Thus, |Ψ > = ∑𝑛|Φn >< Φn|Ψ >
Thus, ∑𝑛|Φn >< Φn| = 1 This is called completeness or closure relation.
3.The operator multiplication is usually not commutative. That is
𝐴𝐵 ≠ 𝐵𝐴
The multiplication is however associative.
𝐴(𝐵𝐶) = (𝐴𝐵)𝐶 = 𝐴𝐵𝐶.
𝐴(𝐵|Φ >) = (𝐴𝐵)|Φ > = 𝐴𝐵|Φ >. (𝐴𝐵 )† = 𝐵†𝐴†
4. Outer product is |Φ1 >< Φ2|. It should be emphasized that
|Φ1 >< Φ2| is to be regarded as an opearator, whereas the inner product is just a
number.
If 𝑥 = |Φ1 >< Φ2|
𝑥† = |Φ2 >< Φ1|
For a Hermitian x we have,
< Φ1|𝑥|Φ2 > = < Φ2|𝑥|Φ1 >∗