Hilbert spaces And Quantum Mechanics

Question 1

Define a field? Give examples

Answer 1

Set on which addition subtraction multiplication are defined and behave as they should on that field

Rational numbers, Real numbers, Complex numbers

Question 2

Define a vector space V

Answer 2

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

Question 3

8 properties of vector space V

Answer 3

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

Question 4

For vector space V
1) associativity of +

Answer 4

v_1 + (v_2 + v_3) = (v_1 +v_2) +v_3

Question 5

For vector space V

2) comutativity of +

Answer 5

v_1 + v_2 = v_2 + v_1

Question 6

For vector space V
3) compatibility of scalar * with field *

Answer 6

a(bv) = (ab)v , (for all a,b€ K, for all v€V)

Question 7

For vector space V

4) distributivity of scalar * over vector +

Answer 7

a(v_1 + v_2) = av_1 +av_2

Question 8

For vector space V
5) distributivity of scalar * over scalar +

Answer 8

(a+b)v = av + bv

Question 9

For vector space V
6) existence of neutral 0 € V under vector addition

Answer 9

v+0 = 0+v =v , for all v€V

Question 10

For vector space V
7) existence of inverse -v for every v€V

Answer 10

v+(-v) = 0 , for all v€V

Question 11

For vector space V

8) triviality of * by 1

Answer 11

1v = v , for all v€V

Question 12

3 Implications of 8 axioms of vector space

Answer 12

Zero vector is unique
Multiplication by zero vector gives zero vector
Inverse to given vector is unique

Question 13

A vector space V is said to be finite dimensional if

Answer 13

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

Question 14

Define a linearly independent set of vectors

Answer 14

Question 15

Define vector space of all functions

Answer 15

Question 16

Define L^2 (R)

Answer 16

I

Question 17

Verify that a sum of 2 square-íntegrable functions is square integrable

Answer 17

Question 18

Binary operation

Answer 18

Function who’s input is 2 elements of the same same as it’s output

Question 19

Define a basis S for V

Answer 19

Any linearly independent set S is completed if #S = dimV

Question 20

Construct example infinite dimensional vector space

Answer 20

Question 21

Square integrable function

Answer 21

Question 22

For a vector space V on C, define a norm?
3 properties of norm

Answer 22

Way of measuring:
-size of vector
-Distance between 2 vectors

Question 23

4 properties of Hermitian Inner Product

Answer 23

Question 24

Define Hermitian Inner Product on Cⁿ

Answer 24

Question 25

Cauchy Schwartz inequality
(For Hermitian inner product on V)

Answer 25

Question 26

Prove Cauchy Schwartz inequality for Hermitian inner product on V

Answer 26

Question 27

Define Hermitian inner product on L²(R)

Answer 27

Question 28

Define Cauchy sequence

Answer 28

Question 29

A normed vector space is called complete if

Answer 29

Every Cauchy sequence in this space has a limit

Question 30

A Banach space is

Answer 30

A complete normed vector space

Question 31

A Hilbert space is

Answer 31

A complete Hermitian Inner Product Space (where the topology is derived from the norm defined by the inner product)

Question 32

Relate the set of equivalence classes of square integrable functions to Hilbert spaces

Answer 32

Question 33

Define a linear functional α on a complex vector space V

Answer 33

α: V -> C
Such that

Question 34

Dual space?

Answer 34

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)

Question 35

Relate H^* and H

Answer 35

There is a natural continuous 1 to 1 anti-linear map between them

Question 36

Riesz representation theorem

Answer 36

Any α€ Hilbert space can be written as α=α_u for some u

Question 37

Dirac notation for inner product

Answer 37

Question 38

Kronecker δ function

Answer 38

δ_i,k = 1 if i=k and 0 otherwise

Question 39

For an infinite dimensional Hilbert space H define an orthonormal set of vectors

Answer 39

Question 40

Given that |e_k > are orthonormal;

Answer 40

This is an infinite sum

Question 41

Prove the below converges to some limit in H

Answer 41

<=> if infinite sum of s_N < infinity

Question 42

What is a square summable sequence

Answer 42

Sequence such that

Question 43

Identity in a space of linear operators can be formed from

Answer 43

Where e_k form an orthonormal basis

Question 44

Basis in Hilbert space?

Answer 44

Question 45

How to use Dirac delta function

Answer 45

Only use

Question 46

Basis for Η = L^2 (R)

Answer 46

Question 47

Answer 47

2πδ(k)

Question 48

|Ψ> = ?

Answer 48

Question 49

Answer 49

Question 50

Answer 50

Question 51

Given |Ψ> how do we compute the function f(k)?

Answer 51

Question 52

Every |Ψ> € Η can be represented as

Answer 52

Question 53

Typical |e_k>

Answer 53

Question 54

Fourier transform of Ψ(x)

Answer 54

Question 55

Inverse Fourier transform

Answer 55

Recovers Ψ(x) from its Fourier transform <e_k|Ψ>

Question 56

What is a separable Hilbert space

Answer 56

Hilbert space that has a basis

Question 57

2πδ(k) =

Answer 57

Question 58

e_k (x)
=?

Answer 58

Question 59

Answer 59

e_k (x)

Question 60

What does normed vector space allow? What about Hermitian inner product space ?

Answer 60

Normed - measuring distance

Hermitian - measuring angle between 2 vectors (specifically orthogonality)

Question 61

Usual way to denote Fourier transform of Ψ(x)

Answer 61

Question 62

A set |e_1> , … , |e_n> is a basis of H if?

How to check?

Answer 62

All vectors in H can be given in form