Nitty-Gritty of Quantum Mechanics from a Rubberneck’s POV (Detour Chapter: Space)
(Prologue: In order to understand how mathematical formalism in Quantum Mechanics works, we have to get a good grasp on the Hilbert Space, and for that, we need to know what Space is in terms of mathematics before jumping onto the Hilbert Space. Although this is a detour chapter that mostly ignores the charming word ‘Quantum’, this will not fail to mesmerize you, for sure.)
Space is what we see around us. In a general context, space means some kind of distance; this can be verified when your fiance says, “I need some space. Don’t call or text me.” Jokes aside, space really means distance, at least from a layman’s perspective. But, mathematics does not concur with this notion. In mathematics, the term space is a set of objects which follows some ground rules and postulates or properties. So, space in mathematical jargon can be referred to as a mathematical construct.
A set of elements + Ground Rules = Space
[Apple, Orange, Guava…..] + Rules {interaction between any two and/or Certain Properties} = A Space composed of these elements [Apple, Orange, Guava…..]
Just think about a set of elements satisfying some properties whenever you come across the term space. Try to ignore the overly-used concepts of space that we use daily. By overly-used concepts, I meant our surroundings or a graph paper to draw geometric figures on. However, this 2D graph paper can also be linked to what Space is in mathematics; wait a moment!! How so? Let’s say this 2D graph paper has 21 by 21 lines (so, 21 lines on each dimension). So, we write the extremities as -10 for the leftmost position, 0 as the origin, and +10 for the right extremity.
For each dimension:
10 (lines to the left of origin)+ 1 (origin line)+ 10 (lines to the right of origin) =21 lines
We can draw lines, circles, different geometric shapes on this graph. Also, we can measure the distance between any two points using the distance formula. So, this graph can be denoted as a Space in mathematics which contains all these figures, lines, numbers. So, a set of all these numbers and figures can be called Space. This is known as 2D Euclidean Space, a subset of Space. A simple representation of 2D Euclidean Space that I have drawn using Python is as follows:
We can connect a line using any 2 points in this Euclidean space, draw a circle or square, evaluate its area, determine the distance between two points, and shift a line from one place to another by just multiplying the slope with a value of our interest (we can use constants to increase the y-intercept as well). So, these figures abide by some rules and regulations for which the set containing all these elements is a Space in mathematical jargon.
2D/3D Euclidean Space⊂ Space
Space can be classified into two basic types: linear space (vector space) and topological space.
Linear Space:
It is a space that is full of vectors and scalars, which are subject to addition and multiplication properties. So, there are some ground rules for these vector spaces. Members of this space can be added through vector addition rules and also, can be multiplied by constant numbers through scalar multiplication.
In the GIF above, we can see that vectors can be shifted and rotated. This is done following some rules. Also, we can multiply these vectors to scale them up or down. In the GIF below, the vector is multiplied by a positive constant and hence, it gets enlarged. So, these are the ground rules in vector spaces.
Now, hold on to your horses because I am going to make a statement, a big one. Here it goes: in linear spaces, vectors are NOT necessarily just a line segment with an arrow pointing towards a direction, exactly like in the above GIFs, rather, vectors encompass many other situations as well. So, to form a vector space, a set of vectors has to satisfy some properties, i.e. vector addition, scalar multiplication, associative, and a few more. That means, anything, literally speaking, that satisfies those axioms or properties can form a vector space. To understand this, we need to look at the properties first.
Vector Properties:
If A and B are two vectors, then
(1) Addition: A+B will also be a vector
(2) Commutativity: A + B = B + A
(3) Zero Vector: 0 (doesn’t signify origin of vector space, just a dummy vector)
(4) Identity Element: 0 + A = A + 0 = A
(5) Inverses: A + (-A) = (-A) + A = 0
(6) Associativity: If C is also a vector, then (A + B) + C = A + (B + C)Scalar Properties:
If A and B are two vectors and x, y are two scalars, then
(7) Distributivity: x(A + B) = xA + xB, (x + y)B = xB + yB
(8) Associativity: x (yA) = (xy)A
Matrices as Vectors:
So, if a set of matrices follow these rules, then we can conclude that they form a vector space, and the elements (matrices) of that set are vectors. We can add, subtract, multiply two matrices which satisfy (1), (2), (3), (4), (5) [null matrix after subtracting from itself], (6), and also, (7) and (8) since we can take out or multiply a constant from or into a matrix. So, matrices can be vectors, in a broader sense. And, they can form a vector space.
Caveat:
Let’s consider two matrices, A (2x3), and B (3x4). The inner product is 3x3, thus matrix multiplication is possible. So do matrix addition and other operations. The dimension of this resultant matrix, AB, will be 2x4.
Now, for BA, matrix multiplication and addition are not possible since their dimensions do not match. Thus, a set of different-sized matrices DO NOT form a vector space.
Matrices of same size → Vector Space
Matrices of different sizes → Not Vector Space
Also, if all elements of a set have 2 in their 2nd row, then adding any two of those elements will result in an element that will have 4 (2 + 2) in the 2nd row. But, the set, originally, does not have an element that has 4 in the 2nd row, thus, these two elements cannot form a vector space despite being vectors.
Polynomials as Vectors:
If we have two 3rd order polynomials, we can add/subtract those, multiply them with a number. The result will always be a 3rd order polynomial or less, but not more than order 3. We can factor out a number from a polynomial or factorize it; thus satisfying (7) and (8) axioms. So, these two polynomials form a vector space as well. Likewise, any number of functions can form a vector space if they satisfy the 8 axioms.
“Vectors are not just pointy things that we see or use, they can be made up of anything that follows definite properties”— Mathematics
Topological Space:
In linear spaces, there is no way to measure angles for which continuity and/or surfaces cannot be described by the definition of linear space. Topological spaces can be defined as such sets which include points and have a definite structure within them. Let’s de-confuse this.
Let’s consider we have a set X full of objects in it. These objects are called Points; not that they look like points, but it’s just a term to define them. So, this set or space has no structure but some points up until now. To operate certain geometry on these points, we need to define a structure for this space. The connections between points and their surroundings should be defined through this structure so as to there will be a concept of continuity or surface. So, that means between two points, there are some connections and these can be represented by subsets of X.
Connection/continuity between points = Subsets of Space X
A topological space is a pair (X, S) where X is a set and S is the accumulation of all subsets of X. S has the following properties:
1. The empty set { } and X belong to S
2. Any arbitrary union of members of S belongs to S
3. The intersection of any finite members of S belongs to S
Example:
Let’s consider a set X = {1, 3, 6}. So, using the first property, we can say that the collection of all subsets, S = {{ }, {1, 3, 6}}, encompasses X and the empty set { }.
Also, S = {{ }, {1}, {3}, {6}, {1, 6}, {3, 6}, {1, 3}, {1, 3, 6}} can form a topology and they belong to S ({1, 3} or {3, 6}… they all belong to S).
Lastly, {1, 3} ∩ {3, 6} = {3} belongs to S = {{ }, {1}, {3}, {6}, {1, 6}, {3, 6}, {1, 3}, {1, 3, 6}} as well.
Subsets of X = Open sets
All other subsets in space other than X’s subsets = Closed sets
How to select open sets to form a topology?
Defining a topology is equivalent to “choosing” which sets we want to consider as open sets. The only requirement is that those sets, taken together, satisfy the three properties above.
Let’s choose { }, {1}, {1, 3}, and {1, 3, 6} as our open sets for X = {1, 3, 6}.
Hence, S = {{ }, {1}, {1, 3}, {1, 3, 6}}.
Check 1: { } and X ={1, 3, 6} both belong to S. (DONE)
Check 2: {1} ∪ {1, 3} = {1, 3} which belongs to S or { } ∪ {1} = {1} which belongs to S. (DONE)
Check 3: { } ∩ {1, 3, 6} = { } which belongs to S or {1} ∩ {1, 3} = {1} which also belongs to S or {1, 3} ∩ {1, 3, 6} = {1, 3} which belongs to S. (DONE)
Hence, the collection of all 4 open sets, S = {{ }, {1}, {1, 3}, {1, 3, 6}} forms a topology on X.
Members of S = Open Sets
S = A Topology of X
X = Topological Space
So, in the upcoming sections, we will try to reflect through these spaces more intuitively and thoroughly as we are coming close to landing on the Hilbert Space. I planned to include this topic in my Quantum Mechanics series as I found very few intuitive articles on Spaces and I had almost zero knowledge about these spaces before writing this article. Also, I am new to making animated GIFs using Python, so please let me know if you have any queries or difficulties understanding any of those GIFs. Also, I am sharing my source code if anyone is interested, but a heads-up for them- I always mess up while declaring variable names and I never change them afterward.
Thank you for your patience. Ping me if anything needs to be corrected.
Source Code: https://drive.google.com/drive/folders/1mm3eEx5Wuwks9Il3TtnXNre0leUFbtOx?usp=sharing
