Linear vectors and Quantum Mechanics. (I)

i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=\hat{H}\left|\Psi(t)\right>

This is called Schrondinger’s equation and it is often used in Quantum Mechanics. To understand this equation one should understand the mathematics on which Quantum Mechanics is based.

Note \left|\Psi(t)\right> appearing on both the sides of the equation. Mathematicians call it “linear vector”.

In mathematics linear vectors are recognised by the following property:

On adding two linear vectors the entity obtained is again a linear vector.

The mathematical examples of linear vectors are numbers, integers, matrices and solutions of linear differential equations (there are more, think!). You can see that if you add two entities belonging to a type (say integer), then you end up with same type of entity again (integer) and not some other entity (matrix or irrational numbers). On the other hand solutions of non-linear differential equations are not linear vectors. If you add two solutions, the resultant function will not satisfy the equation!

In Quantum Mechanics we use linear vectors to represent the state of a physical system at certain time ‘t’. In physics, state of a system means the information of the various variables (in form of numbers) related to a system. If I know these numbers, I know the physical system through and through. And if I know these numbers as function of time, I know the dynamics of the system. One natural question which may arise is

If we know these numbers as function of time then does it mean we know the future of that physical system?

According to Classical Physics answer is yes (in principle), but in Quantum Mechanics we can only estimate the odds, so the answer is no (even in principle).

So Quantum Mechanics says that I can associate \left|\Psi(t)\right> with any physical system (starting from electrons and molecules (even myself!) upto galaxy and beyond). All the information (set of numbers) associated with the physical system is buried inside this state vector. It is the job of physicist to extract the information and use it.

Quantum Mechanics is telling us that the properties of physical systems are linear in nature. If I add two physical states then the resulting state will also be a physical state!

Let us consider a physical example which explicitly shows this property of linear vectors. Then I will come back to explain what I said in last paragraph. Consider a city which is inhabited by only 8 people. Now each person has different characteristics which make him/her unique. For simplicity, let us consider that they have only three characteristics with two possible choices.

  1. The hand they use to do work (lefty/righty)
  2. The language they know (hindi/english)
  3. Habit (lazy/active)

Let us consider a person ‘A’ having three characteristics as

  1. Righty
  2. Hindi
  3. Active

and denote this person with linear vector \left|A\right>. In the city if we search for set of characteristics mentioned above, it means we are searching for person A and thus we should use \left|A\right> for these set of characteristics which describes the state.

Similarly, let there be person ‘B’ having characteristics

  1. Lefty
  2. English
  3. Lazy

and denote this set or state with \left|B\right>. Analyse the table and take its snapshot in your mind.

Person A or \left|A\right> Person B or \left|B\right> Phase (\varPhi)
Work Righty Lefty 0
Language Hindi English \frac{2\pi}{3}
Habit Active Lazy \frac{4\pi}{3}

Now my aim is to show that one can represent other 6 persons using these two persons or in other words I can represent 6 other states using these two states (\left|A\right> and \left|B\right>). For the given scenario, the only possible way is to choose one characteristic from any one person and rest two from other. This way we can build 6 other set of characteristics (6 more states and hence 6 more people). Now we set the coefficients of \left|A\right> and \left|B\right> as the square root of number of characteristics picked from that particular state (I will explain the reason little later).

Consider a case: Select 1 characteristic of person A, say work, and rest from characteristics of person B, language and habit. The state thus formed is {Righty, English and Lazy} which neither represents person A nor person B. It represents new state or a new person denoted by \left|C\right> = \sqrt{\frac{1}{3}}\left|A\right> + \sqrt{\frac{2}{3}}\left|B\right>. Here coefficient of \left|A\right> denotes the square root of number of characteristics picked from person A and similarly coefficient of  \left|B\right> denotes the square root of number of characteristics picked from person B. I have divided whole linear vector by \sqrt{3} in order to normalize it. We can represent new person in terms of two special persons (call them basis).

Let us pause here and think why linear vectors are suitable to describe persons. We defined a person with a set of characteristics. And we know that if we add characteristics (in any proportion) we get another set of characteristics which describes another person (and not some animal!). This idea is important and this is what allowed us to use linear vectors for persons.

If you are wondering why I chose to take the square root of number of characteristics picked as coefficients, then now is the time to explain. We can say that the (|coefficient|)^2 shows us the similarity to the person (or state) it is written with. Thus person C is 1/3 like person A and 2/3 like person B. Also it is obvious by the choices I made.

But this is not a correct way and there is a fallacy. If I select language of person A and rest two characteristic from person B then the state I get is {Lefty, Hindi and Lazy} which represents different person altogether (who is certainly not person C). So let us call him person D (if you did think this on your own then you are really paying attention). Hence I need one more variable to store one more piece of information. The information being what kind of choice is made first. The previously mentioned coefficients will tell me how many characteristics were picked from person A and B. So I need to concentrate on the person from which only one characteristic was chosen (because this choice will determine other two choices from other person). Hence I have introduced a term (call it phase) \varphi . Take a look in the table. The fourth column shows the phase associated with each characteristic (later I will explain the reason for this particular allotment of phase).

If I define the linear vector associated with person C as \left|C\right> = \sqrt{\frac{1}{3}}\left|A\right> + e^{i*0}\sqrt{\frac{2}{3}}\left|B\right> and person D as \left|D\right> = \sqrt{\frac{1}{3}}\left|A\right> + e^{\frac{2*i*\pi}{3}}\sqrt{\frac{2}{3}}\left|B\right> then everything will fall in right place. Not only I can see the mathematical difference between person C and D, but I am also able to preserve the definition of “likeliness” in the way that both the person C and D are 1/3 like person A and 2/3 like person B and still they are different!

No you might ask why did I assign phase like this. I considered the number of ways in which one characteristic is chosen from one person and two from another (for this case it is 3). I, then, divided 2*\pi by that number and allotted the phases accordingly.

Another question might rise at this point. What if we have a set of more than three characteristics? Will this method work? Certainly, the answer is yes. In that case you can consider all sorts of permutation and assign a phase to each permutation like I did in this case.

But why to divide the number 2\pi? Well it clearly means that I want to divide a circle in parts. The number of parts being equal to number of permutations calculated. Also I know that coefficients should vary from 0 to 1. So I can use sine and cosine of a parameter instead of these coefficients. Let that parameter be \theta/2 where \theta varies from 0 to \pi .  And we also know that the phase \varphi is varying from 0 to 2\pi (in parts). What do you make out of it? Have you seen these limits elsewhere? Yes, of course, these are the polar and azimuthal angles of a spherical-polar coordinate system where the length (defined as (|coefficient 1|)^2 + (|coefficient 2|)^2) is constant as \theta and \varphi  are varied. The surface traced this way will be, no doubt, a sphere. Discreet points on the sphere represent the states (decided by the \theta and \varphi of that point).

In general state \left|\Psi\right> representing a person can be written as

\left|\Psi\right> = \cos(\frac{\theta}{2})\left|A\right> + e^{i*\varphi}\sin(\frac{\theta}{2})\left|B\right>

This sphere has special name Bloch’s Sphere. The only difference is that on our sphere we have discrete points where as on bloch sphere there are uncountable points and, hence, uncountably infinite states. \theta and \varphi vary continuously.

Let us stop here conclude this post with following statements:

  1. Anything (mathematical or physical) that is linear can be represented by linear vectors.
  2. Quantum Mechanics says that all the physical systems are linear.
  3. To represent the set of linear entities (in this example we had a set of 8 persons) we can find a subset of those entities (persons A and B), known as basis, to represent all the other entities.

In next post I will demonstrate how linear vectors are used in Quantum Mechanics.

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