1 Introduction. 1.1 Stuff we associate with quantum mechanics Schrödinger s Equation
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1 1 Introduction Quantum Theory and quantum mechanics belong to the greatest success stories of science. In a number of ways. First, quantum theories are among the most successful in terms of predicting the outcome of experiments. With very high accuracies. Also, quantum theory describes processes on small scales and it turns out that our intution, which is mainly derived from everyday experience, is no longer valid on small scales which makes it even more amazing that people came up with a theory of this quality. Well, these weren t ordinary people. Look up the following names: Einstein, Planck, Bohr, Schrödinger, Heisenberg, Born, Pauli, von Laue, Sommerfeld, Dirac, Oppenheimer. Please do. They are all very amazing. Not only because of the intellect but because of their courage. Quantum theory is weird in a number of ways because it shatters the way we are used to thinking about natural systems. No one can understand quantum mechanics and why the world behaves the way it does on a fine scale. Behavior of quantum systems is so strange this way. In this course you ll get a glimse of it. 1.1 Stuff we associate with quantum mechanics Schrödinger s Equation You may have heard about one of the most important equations that play a role in quantum mechanics, the Schrödinger equation. Here are a few variants of this equation, written in different representations or or i h t y(r, t) = i h t y(r, t) =Ĥy(r, t) " # h 2 2m r2 + U(r) y(r, t) i h hy = H hy. The Schrödinger Equation describes the evolution of a quantum mechanical state y and we will learn what that really is. It s something quite different from a state of a system in classical theories. One key difference is that the state of a system in QM quantifies the propensity of the system to behave in a certain way and the state is also something we prepare in a measurement. More about this later. 3
2 1.1.2 Planck s constant You may have heard that according to quantum mechanics a lot of things in nature come in quantized bits: Quantums. This is why quantum mechanics is called quantum mechanics. In fact, in this context you may have heard about Planck s constant (Planck sches Wirkungsquantum) often also provided as h-bar : h Js h = h 2p Js This constant is a universal natural constant and is an action (Wirkung) because it has units Energy times Time. Later we will learn that photons, little bits of electromagnetic radiation have an energy E = hw where w = 2pn is the radial frequency of the electromagnetic wave (n is the frequency in Hertz). h is a very important natural constants. In units that make sense on our scale, e.g. Joules and seconds, Planck s constant is very tiny Heisenberg s Uncertainty Principle You may have heard of this whole idea in QM that being able to measure one physical property of a system implies that another property cannot be measured with arbitrary precision. Typically this is stated for the properties momentum and position of say a particle DxDp x h/2. In this representation the product of the uncertainty in momentum, Dp x and the uncertainty in position Dx must be larger than a tiny number. This implies that knowning e.g. position with increasing precision implies increasing lack of knowledge of the momentum of the particle. This is because one of the fundamental principles in QM is that measurements are never gentle on small scales. If we measure a certain aspect of the system we disturb the system such that other things are no longer known about it Wave-Particle Dualism One of the weird things that you may have heard about that are connected with QM is that things may behave like particles and like waves at the same time which is strange from a classical point of view. You mave have heard that for example eletromagnetic radiation, e.g. light, ist described classically as a wave phenomenon, being a solution to Maxwell s equations in a vacuum, yet we speak of photons, i.e. light particles. Stranger yet, you may have heard about... 4
3 Matter waves which describe the phenomenon that massive particles, e.g. electrons, protons and even atoms like Helium may actually behave like waves and exhibit weird stuff, like interference Schrödingers cat Then, there are all these weird thought experiments, the most famous being Schrödingers cat which is all about physical systems being in two mutually exclusive states at the same time until some observer makes a measurement. And all this stuff is then connected to multiple universe interpretations of quantum mechanics and all sorts of sci-fi stuff. We will discuss this, too Entanglement Entanglement (in German Verschränkung) is one of the weirder aspects of QM. It touches one of the most fundamental aspects of physical theories which involves causality, locality and some other things. We will also talk about entanglement because it plays a role in quantum computing Probability Amplitudes In quantum mechanics we will deal with what is known as probability amplitudes. These amplitudes can change over time an govern with what probability a certain measurement will give a certain result. The strange thing about quantum mechanics is that these probability amplitudes are most conveniently described by complex numbers such as z = re iq where i 2 = 1. Don t worry, we will review complex numbers. 1.2 The Stern Gerlach Experiment One of the most important experiments in physics of all times is the Stern Gerlach experiment which was performed in 1922 and which with brute force killed all attempts to explain the world based on classical concepts alone. It clearly showed that Certain properties of systems are quantized That a measurement always consists of a system and an apparatus and that both cannot be understood as independent entities That measurements change the state of a system That an apparatus prepares a systems state or confirms it 5
4 Figure 1.1: The Stern-Gerlach experiment. That the outcome of an experiment can only be predicted probabilitistically That these probabilistic outcomes follow regular laws That systems exist that can possess two quantities but not at the same time, that it is meaningless for example to speak of an electrons spin having a component in one direction AND in another direction. In the original Stern Gerlach experiment atoms were emitted from an oven and directed in a beam through an inhomogenous, strong magnetic field. When the beam went through the field it split into two subbeams. Each sub beam had the same intensity, i.e. half the atoms went up the other went down. These atoms were electrically neutral so another explanation was that maybe they had a magnetic dipol moment ~µ caused by an intrinsic angular momentum of one of the electrons in the atom. A magnetic dipol in an inhomogeneous magnetic field gets deflected. Now the first puzzling result was the split into two sub beams. If the magnetic dipol explanation was correct we would expect that the atoms leaving the oven would possess dipols in random directions. The deflection by the magnetic field is classically proportional to ~B ~µ so we would expect a continuous line of deflections. So the first interpretation we would deduce is that either All atoms exiting the oven have a property sz that is either up or down, in 6
5 Figure 1.2: Two Stern Gerlach machines in series. other words +1 or 1 with equal chance. Whatever it is that is different between these atoms exiting the oven it is quantized. or We have not idea what the state is of the atoms that exit the oven, but the apparatus makes the atoms go into two different states ±1 with equal probability. Maybe some atoms exit the oven and lean towards to +1 state and other lean towards the 1 state and the apparatus makes them collapse onto these quantized states The VV experiment The next version could shed some light on the matter by setting two Stern Gerlach machines in series, see Fig Blocking all the atoms that went down and filtering only the ones that went up we can pipe that up beam through the secon apparatus and - voila - we see confirmed that in the second apparatus 100% of the up atoms are again all deflected upwards. So we conclude The first experiment can be considered as preparing two beams, e.g. one beam in the s z = 1 state. The second experiment confirms that all atoms in the upper beam have the property s z = 1. So it seems that the magnetic moments after the first experiment are all pointing upwards, confirmed by the second experiment. 7
6 Figure 1.3: The VH experiment, the second apparatus is rotated 90 degrees and splits the beam in two The VH experiment But what happens if we rotated the second experiment by 90 degrees so it is aligned horizontally, see Fig Again, just like in the VV experiment we prepare the atoms and just use the upper beam that presumably only has s z = 1 atoms in it. In the VH experiment Stern Gerlach found that the s z = 1 beam is split 50/50 into a beam that turns left, and one that turns right. So from this experiment we could conclude In addition to the z Component of the spin, the atoms also have an x-component s x. The first apparatus selects according to s z and the second one according to s x. In principle we still can think of the spin of the atoms, maybe as a dipol moment with three components s =(s x, s y, s z ) but this vector behaves in an awkward way, i.e. being quantized in all directions. An alternative interpretation would be that the second apparatus impacts the property of the atoms in such a way, that all the atoms that went into the second system with s z =+1 were somehow altered with 50/50 chance into a state in which they now have s x = ± The VHV experiment If we want to keep an interpretation in which the atoms may behave strangely but are unaltered by the measurement, which would be the classical explanation, and which would imply that every atom has the two properties s z and s x simultaneously we could run a third experiment which we term VHV so we let the beam from the oven enter the first, vertically aligned apparatus, filter out the s z = 1 sub beam. We continue with this 8
7 Figure 1.4: The VHV experiment, the second apparatus is rotated 90 degrees and splits the beam in two and the third apparatus is again vertically aligned. Classically, the beam should not split. But it is. beam into the horizonally aligned apparatus and filter out the s x =+1 beam. At this stage we may think that we now have atoms that all have the properties s z = 1 and s x = 1. With this beam we now go into a third apparatus that is again vertically aligned. Classically we would now expect that all the atoms are deflected updwards because they all should have s z = 1. However: This is not what is observed, see Fig. 1.4 In fact, the beam that goes into the third apparatus is again split into 2 sub beams at 50/50 change. We will come back to this puzzling result again. What this implies is: The x and z components (and this is also true for y) of the spin s z and s x cannot be measured independently In fact is does not make sense to speak of the atoms having s x AND s z. The measurement of one quantity, e.g. s x changes the propensity or likelihood of the system exhibiting a value for s z in a following experiment. A measurement prepares a system to behave in a certain way in a subsequent experiment. The Stern-Gerlach experiment did not start the development of QM. The issues with physics at small scales arose much earlier and a number of revolutionary things happened at the turn of last century. Let s go over some of the developments in fast forward. 9
8 Figure 1.5: The energy distribution u(w) of angular frequencies w = 2pn of objects at different temperatures T 1 > T 2. 10
1.1 Stuff we associate with quantum mechanics
Contents 1 Introduction 3 1.1 Stuff we associate with quantum mechanics................. 3 1.1.1 Schrödinger s Equation........................ 3 1.1. Planck s constant............................ 4 1.1.3
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