Sunday, 24 February 2013

The Schrodinger's Equation and the Quantum Harmonic Oscillator…

   The mathematical construction behind the explanation of the behavior of a quantum mechanical system may seem rather complicated at first sight. This is simply because the former deals extensively with a rather abstract of mathematics known as calculus. And the worst is that it is not one, not even two but calculus in three or more dimensions thus making use of partial differential equations. Hopefully at this level we shall be encountering only linear differential equations in one dimension then extending these to three dimensions. Understanding what is happening through the mathematical procedures described in this short introduction is poised to give us a deeper insight of the behavior of some quantum entities.  And these quantum objects that I will explain here are the harmonic oscillators. Why harmonic oscillators? This is because the latter are extremely interesting from both a quantum mechanical and a classical point of view as well since they provide sets of general solutions which contribute to the modeling and understanding of many oscillatory systems. A simple example of such a system is the vibrational motion prevailing between two atoms in a diatomic molecule. Using the Lennard – Jones potential, one may have an idea of what is happening but this idea will be mostly perceived with a mechanistic point of view. This is actually a right start in order to understand oscillatory systems even at the microscopic level but the fact is that there will be observations which will be made which cannot be explained using the classical mathematical theory thus restricting our comprehension of the atom. However a complete picture is obtained by making use of quantum mechanics. Concerning the classical harmonic oscillator, I will not extend other the details as this is not topic of this discussion but if we have consider a damped (i.e: real) harmonic oscillatory system, e.g: a spring, then the general equation of motion is:

                                                              (1)
                                   (2)

The general solution of the above equation is rather complicated due to the large number of parameters involved, but the way of solving equation (1) is similar to the way a non - homogenous linear second order differential equation is solved, thus resulting in the general solution which is equation (2). The first part of the solution corresponds to the transient state (the envelope - due to damping) while the second part refers to the steady part of the system (the characteristic sinusoidal structure). This is the very basis of how a classical harmonic oscillator behaves, obviously these equation can be extended to produce the conditions needed for resonance to occur but this will not be discussed. 

  Considering now the quantum mechanical oscillator, since for the classical oscillator we have started with a general solution, its seems obvious to go through a similar procedure for a quantum oscillator. The general equation which does give a mathematical sense to this system is the Schrodinger's Equation. The latter was described by Schrodinger himself as an equation which would described the microscopic objects like the equations of Newton does in our everyday life. This is a good analogy but unlike the equations of Newton where the solutions can be predicted with much accuracy, the solutions of the Schrodinger's Equation gives us the PROBABILITY of finding a particle in a given quantum state in a region of space. This makes the solutions of the Schrodinger's Equations probabilistic ones while these are the equations of Newtons are deterministic ones. So it was probabilistic... and the one who inferred this idea at first was Max Born, an eminent physicist of the mid - twentieth century. This idea was then accepted by the physicist but not by Schrodinger who did refute it. Unfortunately for him it was the right interpretation.

   Schrodinger's equation actually is a postulate, that is something that exists simply because the observation made by scientists are coherent with the equation and nothing more.  That solving this equation provides us with an object which has no physical significance even though the mathematics may be fully consistent, this is the wave-function. Actually the wave-function is a useless object from a physical point view, it is the square of the modulus of the latter which is known as the Born Interpretation which gives the probability of finding a particle with a given amount of energy to exist in a given region of space, which is sensical. However what was making sense in the microscopic world was not at all coherent with our real macroscopic world and this really confused physicists, and the one which has difficulties in viewing this probabilistic version of our reality was Albert Einstein himself. This is actually seen in one of his famous phrases: "God does not play dice". However today physicists know that God does play dice! The weird behavior of discretionary that does prevail in the quantum theory of matter was not completely understood by many scientist because in the real world energy is continuous but it is discrete in the microscopic world, that is not all the solutions of the wave-functions and those of energy when solving the Schrodinger's equation are accepted. This bizarre picture of the behavior of quantum particles is today accepted and understood by all physicists on the planet.The Schrodinger's equation is something which is rather easy to treat with, its derivation is also not complicated and there are many ways of doing. I derived it starting with the quantum mechanical operators and obtained the final equation by doing some algebraic manipulations. Another way of deriving the Schrodinger equation is to start from the de Broglie wave-function. In a first case, the time derivative is taken and both sides of the equation are multiplied by . Then the wave-function is differentiated twice and both sides of the equation are multiplied by  . Equation (3) is the de Broglie wave-function and (4) is the the time dependent Schrodinger's equation in one dimension.

                                                                                                      (3)
                              (4)
   The quantity in the square bracket is known as the Hamiltonian operator, . The latter is the sum of the kinetic energy operator and the potential of a particle. And there are many other ways of doing so, for example one may start with the wave equation, all these methods are interlinked. OK now that we have encountered Schrodinger's equation, things will get a bit more rigorous, the latter shall be used as a template to model the system of the harmonic oscillator. (note: The above equation can easily be transformed for a 3 dimensional system but I will not deal with now, the calculations would become too long.) The detailed mathematical proofs are long and tedious to write using computer programs and they would rapidly make the discussion uninteresting, for this reason I will concentrate on the final results and equations.

  The harmonic oscillator is defined by the potential V(x), where V(x) is the elastic potential just like it is done with a spring. The latter is then inserted in the time independent Schrodinger's equation. This is given below:

                                       (5)

Thus the Schrodinger's equation become:

                                (6)

As one can notice, if we want the wave-function and energies for a quantum harmonic oscillator, using the above equation as it is would be rather inappropriate and complicated. Thus to make things simpler, some new operators are introduced along with coordinate transformation shown in (8):

                              (7)
      
                                                          
                                  (8)

Equation (7) and (8) are substituted in (6):

                        (9)

Now the wave-function is also changed:

                      

After expanding and factorizing the terms with the operators, one is able to obtained the final Schrodinger equation for a harmonic oscillator and the latter reads:

                     (10)

The operators used known as the annihilation and creation operator (the one with the +). This is because when applied to the wave-function with quantum number n, the effect is that b - cap decreases the quantum number to n - 1, thus destroying a quantum hence its name annihilation. The other operator b - cap - plus does the opposite by actually increasing the quantum number to n + 1, creating one quantum. We shall use these to obtained the expression for the wave-function of the oscillator. Lets consider the action of b - cap on wave-function with quantum number 0. Obviously:

                     (11)

          (12)

From equation (12) one can easily formed a first order linear separable differential equation, very easy to solve for the wave-function.

                          (13)

Thus the wave-function is given as:

               (14)

The value of the constant of integration is determined using normalization of the wave-function shown below:

                   (15)

The limits a and b corresponds to negative and positive infinity respectively. Now one can found the value of the integration constant. However I want to make one point, the integral here is non intuitive. If one tries to do this integral using the usual analytical method it would become too long and complex. This is a Gaussian integral and the best methodology is to make use of the multiple integrals methods to solve it. It is then easier after a transformation of coordinates form Cartesian to polar.

Finally, the full equation for the wave-function with quantum number 0 is given as:

                     (16)

In a similar way using the Schrodinger's equation, the state energy eigenvalue can be evaluated and is given as:

                                  (17)

Now we have to generalize the above results, this is the easiest part as it is almost purely mathematical. So, we want to find the wave-function of the first excited state, thus we have to use to creation operator applied onto the wave-function of the ground state. In fact if we want to find the wave-function with a given quantum number n, we have to apply the creation operator raised to the power of n on the eigenfunction with quantum number 0.

                   (18)

             (19)

            (20)

These calculations may be not difficult to understand but are rather long, there are many parameters with many change of sign occurring, I have thus stopped at quantum number n = 3 but for those interested you may continue if you have access to some computer programs. This would be quite easy if you use FORTRAN, you will get the values for the different quantum numbers but it would be difficult to figure the general final solution since the solutions would all be numerical.

Finally, yes finally...the general solution for the wave-functions and the energies of the quantum harmonic oscillator at different quantum numbers is given as:

                              (21)

The above equation may look scary but believe me it is not. The equation can be used to describe molecular vibration thus giving a more accurate perception on the behavior of molecular models. This equation is accurate for a diatomic molecule. A simpler way of expressing it given below:

                                   (22)

The right hand side of the equation is known as the Hermite polynomial. The final general equation for the energies is given:

                                                     (23)

After we have derived these equations which are rather impressive at first sight, lets go though their implications. Below is a plot of the various wave-functions with different quantum numbers at different energies based on equation (22) and (23).


This is what we get after all this mathematical analysis, the results are rather interesting even surprising I should say! I said above the wave-function itself is not physically interesting but what does the plot on the left hand side tell us is that the energy of a quantum system is not zero like it can be in classical mechanics. The energies are also distributed in a discrete manner and are indeed not continuous as indicated in our calculations. But it is the second graphical plot which is interesting. If we observe carefully the plot we can see that it is the square of the modulus of the wave-function, and the latter as said above gives us the probability of finding a particle with a given quantum of energy in a region of space. Thus one may argue that since we have started with the potential of a spring, then why are then wave-function not contained within that potential (i.e: the parabola in the plots) in the graph. Actually this is the beauty of quantum physics, our equations are telling us that a quantum particle has the potentiality to exist in classically forbidden regions! Particles with lower energy states has the possibility to traverse a barrier with higher energy. This is an absurd idea and we all this is not possible but the fact is that the equations do indicate a non - vanishing probability possessed by quantum entities in a quantum system. Particles have the possibility to Quantum Tunnel through a barrier with higher energy than they possess, the wave-function of the particle then propagates normally after it passed through the barrier! This part of quantum physics has opened new doors and it is only in the 21st century that physicists are trying to fully understand and exploit this phenomenon. The thing is that quantum tunneling has been existing well before us, it is actually because of this phenomenon that we are able to live on this planet. How? This is because it is occurring on the sun, the latter does not possess enough energy to make a large number of protons to undergo fusion, thus when the latter came close to each other they can quantum tunnel thus overcoming the highly repulsive forces due to like charges and fuse. Hence for sure quantum mechanics have enhanced our comprehension of the atom allowing us to understand how the microscopic world behaves and why does it differ so much with our known version of the macroscopic reality.





                                        

         



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Thursday, 14 June 2012

Reality, Philosophy and Spacetime


   Ever since the dawn of civilization ,man's scientific insight has been triggered by the
strange way nature has been designed, and trying to reveal some nature's secret is the goal of science in order to be able to understand the world in which we live. However there are many observations which we know form part of our reality but remain inexplicable, hence such observations raise questions such as : is the picture that we have of reality true ? A great example is the quest for finding the elusive Higgs boson , the particle which confer its mass to all particle, if the latter is not found this would imply that the well known standard model is untrue. There are many examples ( black holes, big bang singularity, geometry of the universe, dark matter and dark energy, and behavior of space and time at the limits of the expanding universe ) where the known laws of physics and mathematical theorems no more hold. However the intrinsically curious nature of the human mind has always forced mankind to be forward looking , this how today physicist have been able to reach the conclusion that only unified theory  will enable us progress in our understanding of deeper principles that contributed in modelling a correct picture about the creation , evolution and fate of the cosmos and probably the world beyond it.

  Throughout the history of humanity and since the breakthrough discoveries of the
nineteenth and twentieth century, that is the unification of electricity and magnetism by the Maxwell equations and the development of quantum physics along with the discovery of the theory of relativity, ideas in modern physics have completely changed our perception of the universe initially described by Newton in his  famous book (mathematical principles of natural philosophy). Since then these theories have enormously contributed to the development of what I will call "postmodern" physics, that is the quest for unification of quantum physics with the theory of gravity.  One of the candidate theories is the very popular string theory, which imply that matter consists of vibrating strings and this theory would be able to explain elegantly why the constants that prevails in the particle theory have the values they have. In this theory all the constants would be explained by two numbers only; the string tension (amount of energy per unit length of string) and the string coupling constant ( the probability of a string breaking into two strings giving rise to a force) , this theory also imply a multidimensional universe. However for the time being string theory remain a theory as for now there has been no experimental proofs of the above but there are both direct and indirect evidences that the theory may solve most of the problems in physics. Why this cannot be easily proved is because with unification comes instability and the amount of energy required to enable collisions which occurred  at time less than the Planck's time ( 10^-43 s ) is tremendous, just to have an idea the collider itself would have a diameter larger than that of our solar system! This is impossible for the time being! Other way to see things is by using symmetry, well this is more complex but just to have a notion of it , symmetry in physics has been initially described by Emmy Noether, this is known as the Noether's theorem - conservation laws are seen as symmetries. And there are more discussions and hypotheses about the topic. Eventually the more physicists study the theory of gravity the more they realize that modifications must be brought to the equations set by Einstein. However the laws of nature are completely specified without any prior assumption about the geometry of our space-time, this is the background independence principle . And the evolutionary geometry of the universe is the prime idea of Einstein' s theory. This is why a deeper theory is required. Now the problem with quantum physics is that observations made, even after having studied the subject for years, remains utterly baffling and go against our intuitions . This  is because our brains are unfamiliar with quantum phenomena as such things do not occur in everyday real  life. A nice example is the story of Alice in the wonderland , this little girl who falls in the rabbit hole before penetrating in world of incredible and unthinkable possibilities . In fact the correct title should have been Alice in the quantum wonderland !! This world would be totally unrealistic, but  what is reality ? As said initially comprehension of our reality remains still something complex and for now it is beyond us. This hence raises a lot of philosophical questions and there are numerous suggestions and discussions about the real picture of space and how this picture will evolve other time .

  One of the greatest philosophical questions ever asked by the human kind is undeniably the question about the creation of our universe, there are many other such questions and hypotheses which have emerge from imagination of physicists. Such questions are for now unanswered questions but they have enable us to understand how our reality is complex and subtle. Philosophy, throughout centuries, has allow us to see far beyond the equations. It has allow us to think differently and to be more open minded in order to explore the various possibilities that the theories of "postmodern" physics may offer. Philosophy tries to give explanations to events which for now the laws of science cannot explain. It hence indirectly contribute to the continuous progress of science. However this does not mean that the explanations suggested by philosophy must be all true. This is because when discussing about platonism in physics and mathematics everyone thinks that his answer is the right answer according to their intuition, as we live in a world where a breakdown of intuition rhymes with nonsense. However the most intuitive answer may not necessarily be the right answer! I will discuss some philosophical questions and some questions have high chances to be answered in a near future. As said in the beginning of this paragraph the understanding of the creation of our universe is not established
at all, therefore how our universe could have been created ? Most of us thinks it is the results of an intelligent creator and the Church agrees with this idea. However religion and science see things differently and according to the recent researches that are being made in theoretical and particle physics, there could be theories which may explain how our universe was born. One of these theories is that the creation of our universe may result from the big crunch of a previous universe which initiated our well known big bang ? I have discuss this idea with the professor of theoretical physics, Lee Smolin who is actually a researcher at the Perimeter Institute. An immediate question would be;  were the laws of physics in this previous universe similar to ours? . If yes then our universe has great chances to end up like the previous universe ,that is in a big crunch. Does it imply that our universe forms part of an ever ongoing cycle of creation of universes, where the behavior of space-time would be slighty different in the universes? Could such differences may have provided an adequate environment for formation of the elements of the periodic table and later to the formation of complex molecules like DNA and this leads to the evolution of intelligent life forms? This is what physicists called the Anthropic principle.

   Last year the Nobel prize in physics was awarded to two teams of astronomy for their incredible discovery of the acceleration of the expansion of the universe. We know that objects which are found further from us recedes at higher and higher velocities than objects nearer to us, this implies that matter in space-time which found much beyond the observable universe, that is at probably billions billions light years from us, are receding at velocities which are beyond the speed of light ! Well we hence do not know what is happening in such regions of our space-time. Imagine yourself now found in a region of the cosmos where the fabric of space-time evolves at speeds higher than speed of light? Does matter and energy still behave as they do in our known regions of the universe? What would happen to a beam of light in such regions? How would we perceive time? Would the effects be opposite to that of light entering a black hole? If the universe is expanding at such a fast rate, what is fueling such an accelerating expansion? Many physicists think that it could be the effect of dark energy. Dark matter and dark energy are for now very mysterious and they are not fully understand and dark energy could be something which generates repulsive gravity as Einstein predicted in his theory of gravity. Could this account for the accelerating expansion of the cosmos? Mass being known as gravitational charge, does dark matter possess negative mass? This could be a plausible idea. For example an experiment could be carried out to produce microscopic amount of dark matter and trying to measure the force of interaction between this amount and an equivalent amount of matter. This would be a logical procedure but the problem is that the properties of dark matter are completely unknown hence making all experimentation impossible. The expansion of the cosmos raises another question : if the universe is expanding, then in what is it expanding? Could it be in dark matter itself? And the discussions goes a step further to what I have explained above, if the unified theory is found to be true then the results may predict the presence of other universes. This is an idea which is taken seriously by many cosmologists as it can be true. If there are other universes which exist, are they similar to ours in terms of topology? Is then the speed of light in these other worlds similar to ours? Are there interactions among the other universes and our universe? Could such interactions result in the creation of universes? These interactions are for sure beyond our level of comprehension but they are very interesting in the way that they would provide a totally new conception of what is called the cosmos, the universe will be far less mysterious !

  Many people may think that these philosophical questions derived from masterpieces of great physicists may seem to complicate physics and mathematics as we already know them! In fact to be able to see beyond the actual theories of science is dictated by an intellectual property which is of higher order and nobler than intelligence itself, this is our imagination! Imagination is the most important tool for a scientist as science generally implies seeing with the mind first then with the eyes. Some extraordinary examples where imagination has been part of breakthrough discoveries  are the theory of evolution by natural selection by Charles Darwin, the discovery of electromagnetism by Michael Faraday and finally the discovery of the theory of gravity by Albert Einstein! This is what that makes the intrinsic beauty of the greatest feats throughout the history of humanity. This is why philosophy is considered to be of higher order than any other branch of knowledge, because it involves thoughts derived from our imagination concerning deeper aspects of abstract theories. However there are simple things which do not attract our mind but forms an intrinsic part of our reality, this is time! Have you ever asked yourself why do you have a notion of time? This is because events do not occur instantaneously in our observable space-time fabric hence this allow us perceive the forth dimension, that is time. Since the big bang time has been evolving in a unique direction, can it be considered as a continuum quantity? It would be really interesting to understand how time evolves when the universe has entered in its big crunch  "phase" , well gravitational forces would become important thus making time to elapse faster. One great question about time is why do we remember our past but not our future? I think that it is linked to the fact that we feel time the way it is in this universe, if it was to be different then I think that we would not be here to observe it! These are also complex problems which physics and philosophy are trying to understand. I found it very remarkable how we are able to ask great questions about our own existence and the existence of such a complex reality in this particular world and to be able to have great thoughts about worlds outside our universe . It was as if our neural architecture has been designed in a way that it enables us to have an undescriptible link with the cosmos.