Special relativity in Minkowski graph (II)

In the previous blog post we noted that the Lorentz transformation is the map from one inertial coordinate system to another. This means that the coordinates of a person standing on the platform can be mapped to the coordinates of the person sitting in the train using the transformation, which results from special relativity postulates, and quantitative analysis of the physical phenomenon can be done in both the frames.

We also note that these inertial frames (coordinate systems) are same in nature (that is the reason why we used the same constant c in both the coordinate systems). In other words, no inertial frame is to be given any sort of preference. If you like, you can think a 4 dimensional grid of space and time built from the 4 dimensional version of a unit cube. The 2 dimensional version of this grid is shown in the figure of coordinate chart below.

Now this grid must be exactly same for all the coordinate systems (henceforth, we will talk of inertial coordinate systems only). Therefore the grids in coordinate charts 1, 2 and 3 (of different inertial observers) are congruent. All that differs is the mapping of same spacetime events. For instance the green dots represent the events of aging of Bob. Now this set of spacetime events is mapped differently in different frames as shown.

More precisely, one meter “of” a coordinate system should be equal to one meter “of” any other coordinate system and so must be the case with one second (I am really not sure which preposition to use here, but I think “of” should convey the message). These units of measure which define the grid size, are same for all the frames and are termed as “proper length” and “proper time”. Although, a rod (a set of special spacetime events defined later) of one meter in one frame might not be one meter in other coordinate system. In fact Lorentz transformation makes sure that the length is contracted by a factor of \gamma = 1/\sqrt{1-\frac{v^2}{c^2}} (https://en.wikipedia.org/wiki/Length_contraction). We will also explain this phenomenon later in this post.

The German mathematician Hermann Minkowski utilized the property of coordinate charts (maps from manifold to \mathbb{R}^4) in making a useful geometrical tool called Minkowski graph. The idea is that on a graph paper we have two dimensions at our disposal. And, the Lorentz transformation shows that if relative velocity between two frames is along x-axis then the map of the coordinates y and z to another coordinate system is identity. Therefore we consider the map from the events in the spacetime manifold to \mathbb{R}^2 space (our graph paper) which includes only two coordinates x and t as shown in the figure.

chart

In this coordinate system (frame of reference), the event A has been assigned the coordinates (5,3). Also note that the time axis has been divided by the constant c and, here, we define one light second as the time taken by light to travel one meter (or inverse of c).

Now each point in this graph (coordinate system) represents an event in the spacetime manifold. As the consequence, the evolution of a particle in the spacetime is mapped to a trajectory in the graph (which is drawn by the observer at the origin). This trajectory is called the “world line” of the particle. On carefully examining the transformations, one can note that when the relative speed between two frames is greater than the constant c (which we now set to unity) then the factor \gamma becomes imaginary. Furthermore, the factor approaches infinity when the relative speed approaches unity. In relativistic dynamics the total energy of a free massive particle is directly proportional to \gamma. Hence it would require infinite amount of energy to accelerate a particle near the speed of light.

The slope of the trajectory in the Minkowski graph is the inverse of the speed of particle with respect to the observer (who is drawing the graph). Therefore the slope of the trajectory is always greater than unity in the graph because a massive particle cannot be accelerated to the speed of light. A light ray always follows the trajectory of straight line with unity slope. The Lorentz transformation makes sure that the trajectory of the light remains same in all the graphs, corresponding to various coordinate systems.

The Minkowski graph is quite helpful in comparing the physics from two frames. Consider two observers F and F’ moving with some relative speed.

Lorentz_transforms_2.svg-2

Observer F uses her coordinate system to draw a Minkowski graph as shown in the figure above.

Now consider a massive particle at the origin with respect to the observer F’ (i.e x^\prime =0) from time t^\prime = 0 to t^\prime = 1. So how will F draw the world line of the particle in her coordinate system (or Minkowski graph)? As you might have guessed, we will use the Lorentz transformation (because we have to map the information given in F’ to F). In other words we have to find the equation of the particle in coordinates of F subjected to the constraints \Delta x^\prime = 0 and \Delta t^\prime = 1 in the coordinates of F’.

From the Lorentz transformation, the first constraint will give a trejectory equation \Delta t = \frac{\Delta x}{v} which F will interpret as the time axis of the observer F’ (because the line of constant x-coordinate giving t-axis and vice versa is a property of the orthogonal coordinate system in the \mathbb{R}^2 space). Therefore F will draw a straight line in her graph with slope as inverse of the relative speed. Let v=.6 meters per light second. The question now arises is that how will she mark the scale of that time axis in her coordinate system (or graph)? The second constraint gives a result that \Delta t = \frac{1}{\sqrt{1-v^2}} which means that a unit second in the coordinates of F’ is mapped t0 1.25 seconds in the coordinates of F. It means that the two events with a unit time separation at the origin in frame F’, are separated by 1.25 seconds in the frame F. Thus the graph of F will look like

minkowskitime

The violet dots correspond to the events with unit time intervals and zero space displacements in F’ (hence it is the time axis for F’) while green dots in vertical line correspond to the events with unit time intervals and zero space displacements in F. Note that in the graph drawn by the observer F, the 15 units of her time is equal to the 12 units of the time in F’. It is consistent with the fact that \Delta t = 1.25\Delta t^\prime. This is why we say that moving clocks are slower.

Similarly, we can draw the x-axis of F’ in the coordinates of F using the same technique. Consider a situation in which \Delta t^\prime = 0 (simultaneous events) and \Delta x^\prime = 1. Using the Lorentz transformation and first constraint we get \Delta t = v\Delta x. This equation gives the x-axis of the observer F’ in the frame of F. Second constraint gives the relation \Delta x = \frac{1}{\sqrt{1-v^2}}. This means that a unit length (between two simultaneous events) in F’ is mapped to 1.25 meters in F. The graph now looks like

minkowskispacetimeThe violet line with lesser slope represents all the simultaneous events of F’ (with t^\prime = 0) in the coordinates of F where they are having different coordinate t. Thus simultaneity is a relative concept and depends of the frame of reference.

In the end of this post, we will explain length contraction using the Minkowski graph. The length of a rigid rod is defined as the distance (in \mathbb{R}^n space) between its end points at the same time coordinate (convince yourself!). Consider a rod of 2 meters length at rest in the frame F’. Let its first end point be at (4, t^\prime). Therefore, according to the definition of length, the coordinates of other end point will be (6, t^\prime). Now this rod will trace a “world sheet” in the Minkowski graph as shown in the figure below. The red dots on the lower violet line (the x-axis of F’) are the coordinates of the endpoints of the rod which are, according to F’, (4,0) and (6,0). They evolve in the time t^\prime such that after unit second in F’, the coordinates are (4,1) and (6,1) (the couple of next red points on each red line) in F’. And it should be, because the 2 meter rod is at rest in F’.

minklencont

Now how much length will the observer F measure? Let us say that at t=8 light seconds she measures the length. Now according to the definition of the length, she will cut the world sheet of the rod with a line (to obtain simultaneous coordinates in her frame) and measure the distance between space coordinates. In this example the coordinates (pointed by the arrows) are 8 and 9.6. Thus the length of the rod is 1.6 meters in the frame F. This is called length contraction and this is why we say moving trains are shorter.

The main point to keep in mind are that in special relativity, an event in spacetime might not have same set of coordinates in different frames of reference. To map an event from one coordinate chart to another we must always use Lorentz transformation. If this is followed honestly, then all the paradoxes of special relativity can be resolved. One such paradox is “twin paradox” which I will explain in next post.

If you are wondering how I made these cool Minkowski graphs then head over here.

A modern viewpoint on special relativity (I)

In this series of blog posts, I will explain our current understanding of spacetime using the notions of relativity.

Special relativity originates from two simple principles but its predictions challenge our day to day experiences. For instance, according to the relativity, a person in the train station will see the the length of a moving train a little smaller than that of stationary train (of same type). Furthermore, the same person will notice that the clocks hanging in the train ticking  little slower than that on her wrist. These phenomena are termed as “length contraction” and “time dilation”. The magnitude of these contractions and dilations depends directly on the relative velocity of the train and the person standing on the platform. Now, commonly, the relative speed of train is so low that one doesn’t see any appreciable changes. Cosmic particles which travel at very high speeds are known to have longer half lives than their laboratory counterparts. But convincing the audience that relativity is a correct theory of the nature is not the purpose of this post. Assuming that relativity is a valid theory, the aim of this post is to describe and discuss the theory.

In my opinion, special relativity is all about the properties of spacetime in fairly small region and how various observers record the events occurring in that spacetime. The words in italics have precise definitions and meanings in physics so let us spend some time in understanding them. When you read the word spacetime you might think of it as a three dimensional space with time t as separate entity running in the background. This is totally a wrong picture of the spacetime. If you are a physics major, and have had a course in relativity, you might think of it as a strange mixed space of \mathbb{R}^3 and t with 4 coordinates in which the invariant path length is given ds^2 = dx^2+dy^2+dz^2-c^2dt^2, where c is the speed of light.

The problem with the last definition is that it does’t capture the essence of spacetime as such. It is like explaining an alien that the person in the red shirt is a boy. Just like the word “boy” is assigned to a specific creature having certain characteristics and is independent of the shirt he wears, the word “spacetime” is assigned to a specific physical entity and is independent of the coordinates assigned to it by the observer (yes, an observer is someone who assigns the coordinates such that her coordinates are fixed  in the coordinate system). Please note that this is not a standard definition of the observer but it is quite helpful in explaining the subject. In relativity, spacetime is a special set, with many interesting properties, which mathematicians call “Manifold”. And the coordinate system is a map from the spacetime manifold (say the set \mathcal{M}) to the set \mathbb{R}^4. The coordinate chart of a N dimensional manifold is shown in the figure below (courtesy Carroll’s notes)

Manifold

Thus, when an observer is assigning coordinates x, y, z and t to an event, he/she is basically mapping the events in \mathcal{M} to a \mathbb{R}^4 space. When an observer assigns a coordinate system, he/she defines a frame of reference. So when we say “the frame of reference of the observer A”, we mean the coordinates assigned by A.

In physics we postulate that all the physical phenomena, comprising of events, happen in this spacetime manifold. In order to quantitatively analyze them, we require an observer who can assign the coordinates  to these events. First postulate of relativity says that the physics doesn’t depend on the coordinate systems of the observers moving with constant relative velocities with respect to each other. We call all such observers as “inertial observers”. It simply means that if we do the physics calculations using the coordinates of any inertial observer we will find the same results. This shouldn’t be hard to accept as the real physical phenomena take place in the spacetime manifold and no matter which map (coordinate system) we use for quantitative analysis, we must find the same physical results. Now be careful! this postulate talks about the maps which are inertial. So we have to be careful when working with accelerated bodies.

Vectors are mathematical  objects (maps, to be precise) which, by definition, remain unchanged under the coordinate transformations. In relativity, we have 4 dimensional vectors living in the spacetime manifold which remain same in all the inertial coordinate systems. We call such objects as invariants. In fact there is a class of invariants called “Tensors” and we assign all sorts of physical objects (energy-momentum, electromagnetic field etc) to the tensors as they, too, don’t depend on the frame of reference. Now events, which are also physical objects, are assigned to 4 vectors. The distance between two events is called the “path length” and is a very good example of an invariant (actually it is a scalar which is a tensor of rank 0). The second postulate gives us a prescription to make scalars, in terms of what is known as metric. According to this prescription the squared path length is mapped to dx^2+dy^2+dz^2-c^2dt^2 in a coordinate system constructed from x, y, z  and ct . Here c is a constant introduced to match the dimensions of the length squared when multiplied by dt^2 . Simple analysis shows that it must have the dimension of \frac{\text{length}}{\text{time}}. In another coordinate system with the coordinates x^\prime, y^\prime, z^\prime and ct^\prime the squared path length is mapped to dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}-c^2dt^{\prime 2}. Note the same constant c has been used to maintain equality of the frames. Now, due to the invariance of path length, dx^2+dy^2+dz^2-c^2dt^2=dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}-c^2dt^{\prime 2}. This is the starting point of the derivation of the Lorentz transformation in the standard relativity textbooks (although they arrive at this expression using light rays which is misleading sometimes). The Lorentz transformation is a relation between the coordinate systems of two inertial observers, assigned to the same events in the spacetime manifold \mathcal{M}.

In the end of this post I would like to give the celebrated Lorentz transformations. Consider two observers moving with a constant relative speed along one particular direction (say along x axis), then the relation between the coordinates of the observers is (image courtesy Wikepedia)

          Lorentz_transforms_2.svg-2

Here \gamma = 1/\sqrt{1-\frac{v^2}{c^2}}. When physicists did the experiments they found that the value of the constant c is nearly 3\times 10^8meters per second which is the speed of light. The derivation of these transformations can be found in any standard textbook on relativity or in Wikipedia (https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations).

Spacetime odyssey

All physical phenomena we see around us take place on something that we call spacetime. Therefore it becomes important to get acquainted with spacetime and understand how it affects the physical phenomena or gets affected by them. I will chronologically introduce the notions associated with spacetime because, usually, everyone gets to learn it in the exact same fashion.

Until 20th century, space and time were thought as different entities. It was thought that there exists a three dimensional euclidean space which acts as a stage for the physical phenomena. This stage was supposed to be rigid, static and flat (rigorous definition will be given later). The space was a three dimensional extension of Euclid’s five axioms for plane geometry which were

  1. Between two points in the space, only one line can be drawn.
  2. A finite line in a space can be extended indefinitely.
  3. A circle with arbitrary radius and center can be drawn in the space
  4. All right angles are equal to one another.
  5. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

These axioms were based on the human experience (at the time of Euclid) and considered to be true in every part of the universe.  Even now, based on high-school experience, one doesn’t need enough convincing about the validity of these postulates.

Let us now define our three-dimensional Euclidean space in formal way. It is a set of points with well defined distance between any two points of the set. If you give me two points say P_1(x_1,y_1,z_1) and P_2(x_2,y_2,z_2) belonging to Euclidean space. I will give you the distance between them by computing  d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.

A couple of interesting points can be noted. First, the definition of distance between two points is static i.e the way to find distance doesn’t change with time. We can represent the definition of distance between two points using matrices. Let us represent the points by column matrices P_1 = \left(\begin{array}{c}x_1\\y_1\\z_1\end{array}\right) and P_2 = \left(\begin{array}{c}x_2\\y_2\\z_2\end{array}\right) and define a 3\times 3 matrix \eta = \left(\begin{array}{ccc}  1 & 0 & 0\\  0 & 1 & 0\\  0 & 0 & 1 \end{array}\right). The distance between two points can now be given by d^2 = (P_1-P_2)^T.\eta .(P_1-P_2) (here ‘.’ represents matrix multiplication). Note that the definition of distance depends on the matrix \eta. Physicists and mathematicians call \eta metric of the space. Now we can say that it is the metric which defines the distance between two points of a space. In fact the metric defines the notion of a ‘dot product’ in a space. Take any vector, write its components in column matrix and used the matrix multiplication defined above to find the length of the vector.

The metric of Euclidean space is time independent (build from constant numbers). Although I have used matrix to represent a metric, generally it should be thought as a set of numbers given by \eta^{\i\j} where i,j\in 1,2,3.

Next we note that the metric (of Euclidean space) is independent of space itself. In fact this independence is a direct consequence of Euclid’s axiom 5. This independence implies that the space is flat. I will demonstrate this and provide a formal definition of flatness in a later post.

 

Particle in a box (infinite square well).

Yes, I am aware of the ennui caused by this topic. We write schrodinger’s time independent equation, solve it, apply the boundary conditions, find the energy eigen states, normalize them and everybody is happy. If you are a physics major, then you must have done this countless times. But I always had a problem with this system because of its highly un-physical nature (I will come back to this later). That makes it a mathematical curiosity and if it is, then why not explore it to the full extent? Please note that in this post I will use rigorous mathematics and will define terms in proper mathematical language (I hope this post will serve as my notes for future reference). Here one can also appreciate the aesthetics of Linear Algebra.

The question I had been asking myself was, “Why do we always work in position representation? What is wrong in working in momentum representation (for this particular problem)?”. The curiosity to find the answers compelled me to plunge into the depths that are generally not touched and mentioned in physics literature.

The conventional Quantum Mechanics (for this problem) is performed on the Hilbert Space L^2 ([0,1], dx) which I will denote with \mathcal{H}. Here L^2 (read L two) represents a functional space of square integrable functions and [0,1] represents the range of parameter x (position). I am interested in working with L^2 (??, dp) and want to obtain energy eigen states in terms of the wavefunction \varphi(p). I dont know the range of the parameter p (momentum). I can’t take it as whole real line (because of the bounded nature of position). Parseval-Plancherel theorem in not applicable here. I must first find the nature of momentum operator. Then maybe its spectrum might give me the idea about the range of momentum.

Just like a function has a domain, the linear operator on Hilbert space has a domain. The formal definition of the operator is (this definition is for general Hilbert Spaces and not restricted to L two spaces)

An operator on the Hilbert space is a linear map A : \mathcal{D}(A) \mapsto \mathcal{H} such that \psi \mapsto A \psi (where \psi \in \mathcal{D}(A)).

\mathcal{D}(A) represents the linear subspace of \mathcal{H}. This subspace is the domain of the operator. Strictly speaking, Hilbert space operator is the pair (A, \mathcal{D}(A)) which is the specification of the operation with the domain on which the operation is defined. Now two operators are said to be equal if and only if

A \varphi = B \varphi for all \varphi \in \mathcal{D}(A) = \mathcal{D}(B).

Let us consider position operator denoted by Q. The operation is defined as Q \psi (x) = x \psi (x). But given the boundary conditions, the domain is defined as

\mathcal{D}(Q) = \{\psi \in \mathcal{H} \mid Q\psi \in \mathcal{H} and \psi(0) = \psi(1) = 0 \}.

As mentioned earlier, the Quantum Mechanics is performed on the Hilbert Space of square integrable functions (denoted by \mathcal{H}), we certainly would not want a mapping which blows up the square integrability. Hence it is important to put the restriction that the output be Q\psi \in \mathcal{H} which translates into ||Q\psi||^2 = \int_{0}^{1} dx x^2|\psi (x)|^2< \infty. So, in words, the domain of Q is the set of all the square integrable functions such that

  • they satisfy boundary conditions
  • the output of linear map should be finitely square integrable

Now let us define momentum operator P. Quantum Mechanics defines the operation of this operator (on a function) as

P\psi = \frac{\hbar}{\iota} \frac{\partial \psi}{\partial x}

Certainly, we would like to have a set of those functions which besides satisfying boundary condition, also have square integrable derivatives. In formal language

 \mathcal{D}(P) = \{\psi \in \mathcal{H} \mid \psi^{\prime} \in \mathcal{H} and \psi(0) = \psi(1) = 0 \}.

Now let us see the definition of the adjoint of an operator A which means we have to define the operation and domain of the adjoint of operator (let’s denote it by A^{\dagger} ).

\mathcal{D}(A^{\dagger}) = \{\phi \in \mathcal{H} \mid \exists \tilde{\phi} \in \mathcal{H} such that \langle \phi , A\psi\rangle = \langle \tilde{\phi}, \psi\rangle \forall \psi \in \mathcal{D}(A)\}

This definition says that the domain of \mathcal{D}(A^{\dagger}) is set of \phi (square integrable)  such that there exists \tilde{\phi} (again square integrable) which makes the equation \langle \phi , A\psi\rangle = \langle \tilde{\phi}, \psi\rangle true for all the \psi belonging to domain of operator A.

For a given \psi , \tilde{\phi} depends on A and \phi . The operation of  A^{\dagger} is defined as

A^{\dagger} \phi = \tilde{\phi}.

So we, now, have a full working definition of the adjoint of a operator. Let’s find the adjoint of P. The inner product equation \langle \phi , P\psi\rangle = \langle \tilde{\phi}, \psi\rangle must hold. So

\int_{0}^{1} \overline{P^{\dagger}\phi}(x) \psi (x) dx = \int_{0}^{1} \overline{\phi}(x) \frac{\hbar}{\iota} \frac{\partial \psi}{\partial x}(x) dx .

On integrating by parts, the RHS

\int_{0}^{1} \overline{P^{\dagger}\phi}(x) \psi (x) dx = \frac{\hbar}{\iota}\left[_{0}^{1} \overline{\phi}(x) \psi (x)\right] -\frac{\hbar}{\iota} \int_{0}^{1}\overline{\frac{\partial \phi}{\partial x}}(x) \psi (x) dx.

For this equation to be true we must have

\overline{P^{\dagger}\phi}(x) = -\frac{\hbar}{\iota}\overline{\frac{\partial \phi}{\partial x}}(x)

or

P^{\dagger}\phi(x) = \frac{\hbar}{\iota}\frac{\partial \phi}{\partial x}(x).

The most curious and interesting point is that the function \phi(x) does’t need to satisfy boundary condition (of infinite square well) since \left[_{0}^{1} \overline{\phi}(x) \psi (x)\right] = \left[ \overline{\phi}(1) \psi (1) - \overline{\phi}(0) \psi (0) \right] is zero as \psi(x) already satisfies the boundary condition. Clearly the \phi(x) has no restriction save square integrability which shows that the set \mathcal{D}(P^{\dagger}) has more elements than set \mathcal{D}(P). The formal definition of \mathcal{D}(P^{\dagger}) is

\mathcal{D}(P^{\dagger}) = \{\phi \in \mathcal{H} \mid \phi^{\prime} \in \mathcal{H}\}.

It can be seen that  \mathcal{D}(P) \subset \mathcal{D}(P^{\dagger}). Thus P^{\dagger} \neq P i.e the momentum operator is not self-adjoint. On the other hand it is quiet simple to show that Q^{\dagger} = Q i.e the position operator is self-adjoint.

Now let us see the definition of Hermitian operators.

The operator A on \mathcal{H} is Hermitian if  \langle \phi , A\psi\rangle = \langle A\phi,\psi\rangle for all \phi, \psi \in \mathcal{D}(A)

Since A operates on both the functions, needless to say, they must belong to the domain of the operator. Let us use the definition of adjoint operators. Operator A^{\dagger} is such that \langle \phi , A\psi\rangle = \langle A^{\dagger}\phi,\psi\rangle which implies \langle A\phi,\psi\rangle = \langle A^{\dagger}\phi,\psi\rangle . Thus if the operation of the operator is same as that of its adjoint, then it is Hermitian operator. The domain of both the operators need not be same.

We have seen that the specification of both P and P^{\dagger} is same \frac{\hbar}{\iota}\frac{\partial}{\partial x}. The momentum operator is Hermitian for this case but not self-adjoint. It is easy to note that all the self-adjoint operators are Hermitian but vice-versa is not true.

Now the spectral theorem of linear operators states that

If the Hilbert space operator A is self-adjoint, then its spectrum is real and the eigenvectors associated to different eigenvalues are mutually orthogonal; moreover, the eigenvectors together with the generalized eigenvectors yield a complete system of (generalized) vectors of the Hilbert space.

This property does not hold for only Hermitian operators.

As the momentum operator, here, is only Hermitian its eigen vectors won’t form complete basis. Thus there is no use in solving this problem in momentum space. Moreover, L^2 (??, dp) space wont have continuous momentum (because of quantization imparted by boundary conditions).

Normally one assumes that the eigen vectors of an observable forms a complete set of basis. This means that observables must be represented by self-adjoint operators. From above we note that momentum operator is not an observable for this problem! 

Absurd, isn’t it? Well, the boundary condition itself gives us the hint of the highly non-physical nature of the problem. The “non self-adjointness” of momentum operator further shows us how the infinite potential square well is a mathematical curiosity with tenuous nexus to real physical systems.

Linear vectors and Quantum Mechanics. (II)

So it has been more than a week since I wrote my last blog post. Although, time to time, I was tempted to write the sequel, but I had to distance myself from that temptation. I have been studying David Griffiths’s “Introduction to Elementary Particles” and much of my time has been devoted to it. Hopefully one day I will know Particle Physics well enough to blog about it.

So by now I am sure you know what it means when we say something is linear. This is the beginning of linear algebra. It is a very beautifully constructed algebra which physicists use in their daily life (like in Quantum Mechanics, Special theory of Relativity and Classical Mechanics). I don’t intend to start off with the abstract Hilbert Spaces (basically a set of linear vectors with some axioms) and define linear maps, inner product structure and so on. Instead I will try to demonstrate the linear algebra “at work (using Quantum Mechanics)” without worrying too much about mathematical vigour. Now I must warn you  that this mathematical vigour is absolutely necessary to not only appreciate the beauty of mathematics (I am a big fan of the water-tight derivations done by the mathematicians!) but to get a better understanding of Quantum Mechanics (least you land up in the situation where it might seem that Quantum Mechanics is inconsistent with itself). Basically in the following demonstration, I will define certain mathematical quantities in a very loose way (loose enough to drive any mathematician, with slightest respect for mathematics, crazy) and mix them with the postulates of Quantum Mechanics without explicitly stating them. For instance, the example I gave in last post, deals with the linear vectors belonging to a linear vector space equipped with inner product structure (or Hilbert Space). I believe that this way I wont make the subject abstract enough for the people new to Linear Algebra and Quantum Mechanics.

So Quantum Mechanics says that the physical states are linear. This might not surprise you and you might think that it is obvious (maybe after going through the example I gave in last post). But think again! This statement is powerful enough to lead to certain results which you might not consider obvious.

Before going further I want you to see this video and try to understand the problem it states in the end.

At first it might seem too confusing. But we will go, step by step, and try to reproduce the experimental results using Quantum Mechanics. I am considering the case when only one electron is shot out at a time from source S. I associate a linear state vector \left|S\right> to represent the electron at source. Now the states representing slit 1 and 2 be \left|1\right> and \left|2\right> respectively. The screen is basically a real number-line where each point represents a state written as \left|y\right>. Note that y is a continuous parameter here.

Now recall last post in which I gave an example of “creating” a set of persons “using” two persons (basis). In general a person can be written as \left|C\right> = \cos(\frac{\theta}{2})\left|A\right> + e^{i*\varphi}\sin(\frac{\theta}{2})\left|B\right>where the (|coefficient|)^2 will tell me the resemblance of person C with person A or B. Now I define a term inner product as a function (a linear map) which maps the ordered pair of linear vectors into a scalar complex number. This complex number gives the component of the linear vector on other linear vector ((|component|)^2 will give resemblance).

Consider the inner product of person A and B written as \left<A|B\right>. Now we know that they don’t resemble at all. Hence \left<A|B\right> should be equal to 0 (state B has no component on state A whatsoever). The states \left|A\right> and \left|B\right> are said to be orthogonal. Also we know that state \left|A\right> completely resembles with itself and so does state \left|B\right>. Thus \left<A|A\right> = \left<B|B\right> = 1 (I can assign any number to this inner product). I have normalised the state vectors \left|A\right> and \left|B\right> (because the number I assigned is 1). They are now called orthonormal linear vectors.

To find the component of \left|C\right> along \left|A\right>, I just need to evaluate the inner product \left<A|C\right>. So

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

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

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

The resemblance of state C with state A is (|\left<A|C\right>|)^2 and with state B is (|\left<B|C\right>|)^2. This is the idea of inner product.

In Quantum Mechanics it is used in a different way. We have state C (now representing a physical state and not a person) given in terms of state A and state B (again physical states). Now if a system is described by state \left|C\right> then \left<A|C\right> is the amplitude of state \left|C\right> to exist in state \left|A\right> (read it this way and understand it this way!). (|\left<A|C\right>|)^2 is the probability with which we will find a system described by state \left|C\right> in state \left|A\right>. The probability of state C to exist in state B is (|\left<B|C\right>|)^2. So instead of resemblance, the component of vector gives an idea about the probability. This is the basic rule which the nature seems to follow. Max Born was the guy who realised this.

The question which might arise here is that before measuring the system it was in state \left|C\right> = \cos(\frac{\theta}{2})\left|A\right> + e^{i*\varphi}\sin(\frac{\theta}{2})\left|B\right>. Mathematically it is called superposition of states \left|A\right> and \left|B\right>. But what is it physically? Does it mean that if a system is represented by state \left|C\right> then it is existing in both the states \left|A\right> and \left|B\right>, simultaneously? Quantum Mechanics gives no answer to this. It is something that you have to interpret yourself.

Although you might try to relate it to the example I gave in previous blog post, but you must understand that these physical states are very different from the states representing the persons in the sense that there are no “extra variables” (like lefty, hindi and lazy) which make up the states. These states are fundamental and nothing is implicit. For instance state \left| 1\right> represents slit 1 in double slit experiment and that is it. So from now on, the physical states are to be considered without any extra variables.

Now I define another linear entity “operator”. It is again a linear map (or function) which maps a linear vector into another linear vector. Operator is denoted by a letter with a hat \hat{A}. So the operation of a linear operator on a linear vector is defined as \hat{A}\left|\alpha\right> = \gamma\left|\beta\right>, where \gamma is a complex number. Now I can represent same operator \hat{A} by \left|\beta\right>\left<\alpha\right|. This is known as “outer product notation”. Let’s check weather this notation is compatible with the definition of operator.

\hat{A}\left|\alpha\right> = \left|\beta\right>\left<\alpha\right|\left|\alpha\right> = \left|\beta\right> \left<\alpha|\alpha\right> = \gamma\left|\beta\right>

Here \left<\alpha|\alpha\right> = \gamma. State \left|\alpha\right> might not be normalised.

In Quantum Mechanics all kind of actions on a state vector are represented by linear operator. There are operators corresponding to time translation, space translation, space rotations and observables. By definition theses operators map a linear vector (physical state) to another linear vector (another physical state).

Suppose you are standing on the other side of slit (opposite to electron source). Further suppose that slit 2 is closed. Now what state would you assign to the electrons coming out towards you? Now the slit 1 will act like a source. You will say that the state of electron is \left|1\right>. Thus the state \left|S\right> has been mapped to state \left|1\right>. This is the handiwork of slit 1 and thus we should associate an operator with it. How should the operator look like? Operator should be able to tell us the amplitude of state \left|S\right> to exist in state \left|1\right>. Consider the form \left|1\right>\left<1\right|. Now (\left|1\right>\left<1\right|)\left|S\right> = \left<1|S\right>\left|1\right>. This operator maps the state \left|S\right> to state \left|1\right> and also provides the required amplitude. So good! Similarly, on closing slit 1, we will find that the operator corresponding to slit 2 is \left|2\right>\left<2\right|. In hat notation, I assign operators \hat{M_1} and \hat{M_2} to slit 1 and 2 respectively.

Now we have necessary tools to work out the Quantum Mechanics of this experiment. The states for double slit experiment we have are \left|S\right>, \left|1\right>, \left|2\right> and \left|y\right> and the operators are \hat{M_1} and \hat{M_2}. Now if I shoot one electron after other, then (assuming no absorption by the slit wall) I should be able to get corresponding detection at certain state \left|y\right> (on the screen). So one electron is emitted out from source and we detect one electron on screen. I should evaluate the amplitude of state \left|S\right> to exist in state \left|y\right> or the number \left<y|S\right>. Then I can get the probability with which an electron will be detected at y by evaluating |\left<y|S\right>|^2.

We saw that the slits have the ability to map state \left|S\right> to state \left|1\right> or state \left|2\right> and these are the only two alternatives. This should make us conclude that state \left|S\right> is made up of state \left|1\right> and \left|2\right> with certain amplitude (which is nothing but probability for state S to exist in slit 1 or 2) for each of them in a linear way. Thus

\left|S\right> = c_1\left|1\right> + c_2\left|2\right>

\left|S\right> = \left<1|S\right>\left|1\right> + \left<2|S\right>\left|2\right>

and

\left<y\right|\left|S\right> = \left<1|S\right>\left<y\right|\left|1\right> + \left<2|S\right>\left<y\right|\left|2\right>

with little re-arrangement

\left<y|S\right> = \left<y|1\right>\left<1|S\right> +\left<y|2\right>\left<2|S\right>

Feynman’s way

In Feynman’s lectures volume III, he obtains the last equation in a very beautiful and intuitive way. He first states two rules

when an event can occur in several alternative ways, the probability amplitude of the event is the sum of  amplitudes of each alternative way considered separately (provided experiment is not capable of determining which alternative is actually taken). If the experiment is capable of determining, then the probabilities should be added.

and

if  a particle goes through a route then the amplitude for that route can be written as a product of amplitude to go part way and amplitude to go rest of the way.

In our double slit, the particle can reach state \left|S\right> by two ways (which can not be distinguish just by detecting the electron at the screen) thus the amplitudes must add

\left<y|S\right> = \left<y|S\right>_{slit 1} + \left<y|S\right>_{slit 2}

The experiment is going via superposition of two possible alternatives. Now using second rule we can say that \left<y|S\right>_{slit 1} = \left<y|1\right>\left<1|S\right> (I have changed the order since it is better to read from right to left). Similarly \left<y|S\right>_{slit 2} = \left<y|2\right>\left<2|S\right> and we end up with the same equation.

Now \left<1|S\right> is a complex number and so is \left<y|1\right>, but y is a continuous variable which can take any value like 3.4, 6, 0.12 and so on and for every y we should have corresponding amplitude. It is better to represent \left<y|1\right> by a function say \phi_1(y). Same goes for \left<y|2\right>. Thus

\left<y|S\right> = \phi_1(y)\left<1|S\right> +\phi_2(y)\left<2|S\right>

\left<y|S\right> = a\phi_1(y) +b\phi_2(y)

\left<y|S\right> = \phi_1(y) +\phi_2(y).

You can put a and b inside the functions. It won’t be a problem becase they are constant.

The probability to detect a particle on screen (at certain y) is |\left<y|S\right>|^2 = |\phi_1(y) +\phi_2(y)|^2. So now you see why we get interference pattern? The probability contains a cross term which precisely shows interference.

You see, how by associating linear vectors to physical systems (and using some additional assumptions) we found an astonishing result of interference. If you want you can watch the video again (keeping this treatment in mind). Now I am going to discuss the physical implications of this treatment.

First you might be wondering about the nature of electron. Is it particle? Or is it a wave? Well it is neither. Wave and particle are classical concepts which are often mixed with the name “wave particle duality” to describe quantum entities. I consider it as misleading specially for new people learning Quantum mechanics. You see in Quantum Mechanics we just say that it is a linear vector by which we mean that it is an entity that is linear in nature. There is absolutely no need to draw a picture of the electron in your mind (like a rigid ball or wavy curve or mixture of both).

Now since it is a linear vector it can exist in superposition of various states. So it is not necessary to imagine an electron going through a particular slit (when you are not observing the electrons at the slit). It is going through both the slits (if this is what you want to hear). And in Quantum Mechanics you will come across such examples many times.

In the video you have seen that interference pattern disappears when one tries to observe the slits through which electron goes. So the act of measurement destroys interference pattern. Of course we can show that using Quantum Mechanics, but it involves concept of Tensor Product or Direct Product of linear vectors, which is out of the scope of this article. Feynman, on the other hand shows (by following those two rules) how the measurement destroys interference without “explicitly” talking in terms of direct product. I dont want to repeat it, so I recommend Feynman Lectures Vol III chapter 3.

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.