Hilbert spaces And Quantum Mechanics Flashcards
Define a field? Give examples
Set on which addition subtraction multiplication are defined and behave as they should on that field
Rational numbers, Real numbers, Complex numbers
Define a vector space V
V is a non empty set over a field K
V has a binary operation that takes 2 vectors from V to produce another from V for all v€V
8 properties of vector space V
1) associativity of +
2) comutativity of +
3) compatibility of scalar * with field *
4) distributivity of scalar * over vector +
5) distributivity of scalar * over scalar +
6) existence of neutral 0 € V under vector addition
7) existence of inverse -v for every v€V
8) triviality of * by 1
For vector space V
1) associativity of +
v_1 + (v_2 + v_3) = (v_1 +v_2) +v_3
For vector space V
2) comutativity of +
v_1 + v_2 = v_2 + v_1
For vector space V
3) compatibility of scalar * with field *
a(bv) = (ab)v , (for all a,b€ K, for all v€V)
For vector space V
4) distributivity of scalar * over vector +
a(v_1 + v_2) = av_1 +av_2
For vector space V
5) distributivity of scalar * over scalar +
(a+b)v = av + bv
For vector space V
6) existence of neutral 0 € V under vector addition
v+0 = 0+v =v , for all v€V
For vector space V
7) existence of inverse -v for every v€V
v+(-v) = 0 , for all v€V
For vector space V
8) triviality of * by 1
1v = v , for all v€V
3 Implications of 8 axioms of vector space
Zero vector is unique
Multiplication by zero vector gives zero vector
Inverse to given vector is unique
A vector space V is said to be finite dimensional if
There exists a finite set of vectors e_1, e_2, …, e_n such that any vector v € V can be written as (below)
For some α_1,α_2, …,α_n € C such a set is called complete
Define a linearly independent set of vectors
Define vector space of all functions
Define L^2 (R)
I
Verify that a sum of 2 square-íntegrable functions is square integrable
Binary operation
Function who’s input is 2 elements of the same same as it’s output
Define a basis S for V
Any linearly independent set S is completed if #S = dimV
Construct example infinite dimensional vector space
Square integrable function
For a vector space V on C, define a norm?
3 properties of norm
Way of measuring:
-size of vector
-Distance between 2 vectors
4 properties of Hermitian Inner Product
Define Hermitian Inner Product on Cn
Cauchy Schwartz inequality
(For Hermitian inner product on V)
Prove Cauchy Schwartz inequality for Hermitian inner product on V
Define Hermitian inner product on L2(R)
Define Cauchy sequence
A normed vector space is called complete if
Every Cauchy sequence in this space has a limit
A Banach space is
A complete normed vector space
A Hilbert space is
A complete Hermitian Inner Product Space (where the topology is derived from the norm defined by the inner product)
Relate the set of equivalence classes of square integrable functions to Hilbert spaces
Define a linear functional α on a complex vector space V
α: V -> C
Such that
Dual space?
For a complex vector space V, V^* is the vector space of linear functionals on V
In context if Hilbet spaces, V^* is being used to denote the CONTINUOUS dual space to V
(Vector space consisting of all functional α which are continuous as functions V -> C)
Relate H* and H
There is a natural continuous 1 to 1 anti-linear map between them
Riesz representation theorem
Any α€ Hilbert space can be written as α=α_u for some u
Dirac notation for inner product
Kronecker δ function
δ_i,k = 1 if i=k and 0 otherwise
For an infinite dimensional Hilbert space H define an orthonormal set of vectors
Given that |e_k > are orthonormal;
This is an infinite sum
Prove the below converges to some limit in H
<=> if infinite sum of s_N < infinity
What is a square summable sequence
Sequence such that
Identity in a space of linear operators can be formed from
Where e_k form an orthonormal basis
Basis in Hilbert space?
How to use Dirac delta function
Only use
Basis for Η = L^2 (R)
2πδ(k)
|Ψ> = ?
Given |Ψ> how do we compute the function f(k)?
Every |Ψ> € Η can be represented as
Typical |e_k>
Fourier transform of Ψ(x)
Inverse Fourier transform
Recovers Ψ(x) from its Fourier transform <e_k|Ψ>
What is a separable Hilbert space
Hilbert space that has a basis
2πδ(k) =
e_k (x)
=?
e_k (x)
What does normed vector space allow? What about Hermitian inner product space ?
Normed - measuring distance
Hermitian - measuring angle between 2 vectors (specifically orthogonality)
Usual way to denote Fourier transform of Ψ(x)
A set |e_1> , … , |e_n> is a basis of H if?
How to check?
All vectors in H can be given in form
