Quantum Mechanics I. Oscar Loaiza-Brito 1. Departamento de Física División de Ciencias e Ingeniería, Campus León, Universidad de Guanajuato

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1 Quantum Mechanics I Oscar Loaiza-Brito 1 Departamento de Física División de Ciencias e Ingeniería, Campus León, Universidad de Guanajuato January-July 2011 First course on Quantum Mechanics for undergraduates. Physics department, DCI, University of Guanajuato. 1 address: oloaiza@fisica.ugto.mx

2 Chapter 1 Introduction 1.1 Why do we need Quantum Mechanics? So far, we have learned that, by a theoretical model it is possible to describe most of the natural phenomena we observe in our local world. Even more, we can apply the same physic laws we have successfully used in our terrestrial laboratories, to predict the movement of celestial objects. From the mechanical state of a particle, to the dynamics of planets under the influence of gravity, or from the thermodynamical laws governing the dynamics of a state conformed by thousands of particles to the involved forces present on electric charged-objects, we could think that our description and comprehension of nature is almost complete. Indeed, we have learned that it is possible to predict the state of a particle (mechanical, thermodynamical or electrical) by fixing the initial conditions and by knowing the forces which act on. In order to know the final state of such a particle, it is enough to solve some differential equations and give a physical interpretation to the solutions. We all have some naive intuition about what a physical solution should be. For instance, if you compute the mass of a particle, and get an imaginary number as a result, you immediately should know that something is wrong in your analysis, whatever it is, or that the mathematical model you are using to describe the phenomena is non-physical.

3 Let us concentrate on another example. Consider for instance a box containing a hundred of smaller objects. These objects are divided by their color, in two groups. There are 50 red objects and 50 blue ones. There is also another way to divide the group of 100 objects. They are of two different shapes. 50 of them are small cubes, and the rest are spheres. So in general, there are red and blue spheres and red and blue cubes. What is then the probability to get, for instance, a blue sphere? Consider the easiest case in which there are 25 red cubes, 25 red spheres, 25 blue spheres and 25 blue cubes. Hence the probability to get a blue sphere is 25%. Suppose you can never open the box, and the only thing you can do is to introduce your hand in a small hole in the big box, and take an object. This is your experiment, and you would like to develop a mathematical model which would predict the probability to get a certain object (in our easiest case, this is always 25%). By sure, you would conclude, that, no matter the conditions the box is exposed to, you know some certain facts. There are a hundred objects, fifty red, fifty blue, fifty are spheres and fifty are cubes. So, if you take a blue sphere for instance, you perfectly know that now there are 49 blue objects and 49 spheres left in the big box. Right? Yes, right, you might answer, and by sure you are thinking how boring these lectures are...please keep concentrated and give me some more minutes. The first point to stress out is the following. It does not matter wether you are constructing a model which describes specific particle states or you are constructing a theory which predicts probabilities. Both of them are useful and are required to get a complete description of a certain experiment. In our big box, we can predict the probability to get certain object, after n number of measurements, and also we can say with a hundred percent of accuracy how many objects of such color and such shape are left in the box. Let us say that, after 40 measurements, i.e. after 40 small objects have been taken away from the box, we see that we took 25 blues ones out of those 40. How many blue objects are left in the box? ZERO! should be your angry answer. An-

4 other way to get this result is to develop a filter which takes away some specific objects. Let us say we have constructed a blue filter, i.e., a filter which removes all blue objects. After the filter application, let me ask you the same question. How many blue objects are left in the box? ZEEEEROOO should be again your answer (plus something I prefer not to listen). But your answer is based on our intuition, which in turn, can be written in terms of a mathematical model. In order to check that our intuitions are correct, we open (somehow) the box. And yes, your answer is correct. A second experiment consists in applying a cube filter. Hence, we are left with only red spheres at the end. What if we apply a blue filter at this stage? YOU GET NOTHING! is your last answer, and yes, you are right. All objects in the box are red and everyone passes through the filter. Finally, assume that our box, and the smaller red and blue objects are shrink into very small sizes. All this previous experiments, that are so intuitive and for that, so easy, are still valid in these scales? Let us perform the same experiment. In a box, put the same objects as before. Remove the blue ones. remove the spheres. And make and experiment to see if there are some blue ones left. Since you remove all blues from the beginning you would expect that there are not blue ones. However, nature is very tricky. It turns out that there is a possibility that there are blue ones left, even though you have removed all the blue objects from the very start. What does happen? Does the experiment fail, or is our notion of correctness in some sense, wrong? You can perform the experiment thousands of times, and you will see that there s nothing wrong about it. Even more, you can find many other experiments which point out that our, so far great intuition, has miserably failed down to describe these phenomena. We really need to construct another framework, which in some sense, let us to understand why our previous classical intuition has been worked so well in bigger scales. The development of such framework is the goal of these lectures. In the way, we shall see that many properties we associate to particles, are described, not in a continuously form as we liked, but in a discrete one, by compact little portions called quantum.

5 1.2 Towards a consistent quantum framework We have based our motivation to study new phenomena on the possibility that our analogies come true at some energy scale. In this section we shall comment on some real experiments and phenomena, which were the starting point through the construction of a new mechanic Do we exist? At the beginning of the twentieth century, it was clear that every massive body was made of some fundamental objects called atoms. People also knew that atoms are neutral, i.e., they do not carry a net electric charge. However there are several processes by which one can put some electric charge on such objects. We usually refer to that, as charging. Well, if a non-charged object turns charged, that means that some of its atoms are also charged. This tells us that atoms are also made of smaller particles and that such particles carry electric charge. An atom is then, you might conclude, made of positive-charged particles (protons) and negative-charged particles (electrons). So, it seemed very easy to understand the physics of atoms, after all, it was just another application of electrodynamics summarized in Maxwell s equations. It was straightforward to compute the stability of an atom. There were two possibilities: the electrons are in rest with respect to (wrt) the positive-charged particles, or they are in relative motion. The first case was ruled out by a series of important experiments, as the ones performed by Thompson et. al. They showed that the atom was formed by a nucleus, where all positive charge was placed, and negative charges around the nucleus. By electrodynamics we can immediately say that such negative charged particles could not be at rest wrt to the nucleus, unless some neutral material exist between electrons and the nucleus such that the electric force was cancelled. This was proved false as well. So the only option was to have a punctual positive charged nucleus and various electrons moving around it. However, electrical charged objects emit electromagnetic radiation when ac-

6 celerate and therefore, they gradually loose their energy. The atom, must be unstable after all. This is not a problem if the associated lifetime is bigger than the life of the entire universe. The problem is that, after some simple electrodynamics calculations, it turns out that the atom life time should be around some minutes. We should not exist! There is another experimental fact which contradicts our intuition. All material when heated, radiate electromagnetic waves. According to our intuition, if we heated the material a little bit, they will emit a small amount of energy. The bigger the energy we give to the system, the bigger the intensity of the radiation it emits, which after all, it is the same initial energy. Do you agree? Well, it turns out that such premise is false. Nature does not behave in such way and once again, our intuition is meaningless. Different materials, chemical elements, emit electromagnetic radiation in specific colors (wave lengths). Let us say you take some hydrogen and heat it up till radiates. You will see and (emission) spectra in which some very specific length waves were turned on. It does not matter if you heat the hydrogen a little bit more, the same lines will appear. But if you provide significantly more heat than before, another line will appear in the emission spectra, but not a continuous. Why? There were some interesting proposals in order to solve this problem. Bohr was one of the first physicists working on such topic. At the beginning, the solutions appeared as dogmas. Under their assumption, people were able to reproduce in theory, what experiments showed, as the two above cases. But a complete construction of a consistent framework required the work of many brilliant physicists and a couple of decades. Quantum mechanics is the result of such enterprise. Since then, we have learn a lot. And one of those learnings, tells us that physics must consider quantum phenomena. Real physics is quantum physics. Mot of our technology is based on it. It is real and powerful. Physics which do not consider quantum aspects, is just an approximation of reality. It could be good enough to be applied in some range of energies, but it is not a fundamental theory. Quantum physics is the keystone of our research of fundamental natural laws. Welcome to the study of Quantum Mechanics. In this course, we shall concentrate on how to

7 describe the atom. For that, we shall develop some mathematical tools as well as some new quantum physical intuition. There are more famous experiments which contradict the results expected from classical physics. They are the black-body radiation and the Compton effect. Forming groups of no more than 4 people, work out one of them. describe the experiment, the expected theoretical results (give a logical deduction of such results) and the solution. Expose your results at the end of the course, and if the exposition is good, you will earn up to 2 points on your final note!

8 1.2.2 Bibliography There are hundreds of books and articles about Quantum Mechanics. This is a very short list of some of them. You are free to pick up your favorite one. 1. Introducción a la Mecánica Cuántica. Luis de la Peña, UNAM. 2. Quantum Mechanics, Cohen 3. Mecánica Cuántica, Landau y Lifshitz 4. Quantum Dynamics, Sakurai. También existen videos en la red que pueden ser de gran utilidad, como 1. Video: Los 5 secretos del oficinista. Guión de Alberto Guijosa y Alejandro Corichi, UNAM. (en Observen especialmente el experimento de la difracción de electrones Timetable There will be 3 exams during the course. They will be applied on February 24th, Abril 12th and May 19th. Exams cover the 70% of the final note, while exercises will cover the rest.