1.1 Stuff we associate with quantum mechanics

Transcription

1 Contents 1 Introduction Stuff we associate with quantum mechanics Schrödinger s Equation Planck s constant Heisenberg s Uncertainty Principle Wave-Particle Dualism Matter waves Schrödingers cat Entanglement Probability Amplitudes The Stern Gerlach Experiment The VV experiment The VH experiment The VHV experiment The situation around Black body radiation Photo Electric Effect Bohr-Sommerfeld Atom Double Slit experiments Matter waves Mathemical Excursion 19.1 Complex Numbers Vectorspaces Orthonormal basis Operators Eigenvectors and Eigenvales Multiplication of operators Linear Operators and Matrix Representations Spectral Decomposition The space C Eigenvectors Hermitian Operators Projectors on C The structure of quantum mechanical systems The Spin

2 .4. Column vector representation Observables and Hermitian Operators General apparatus orientation Expectation values Spin polarization principle The commutator Spinor Time passes Summary Particles Infinitely dimensional Hilbert-spaces Position Momentum Heisenberg s uncertainty principle The variance in measurements Time evolution and the Schrödinger equation Time independent Schördinger Equation Let s look at the Harmonic Oscillator More stuff on the Schrödinger equation Probability flux

3 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processes on small scales and it turns out that our intution, which is mainly derived from everyday experience, is no longer valid on small scales which makes it even more amazing that people came up with a theory of this quality.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 ψ(r, t = i h t ψ(r, t = Ĥψ(r, t [ ] h m + U(r i h ψ = H ψ. ψ(r, t The Schrödinger Equation describes the evolution of a quantum mechanical state ψ 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

4 1.1. 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 π 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 = hω where ω = πν is the radial frequency of the electromagnetic wave (ν 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 x p x h/. In this representation the product of the uncertainty in momentum, p x and the uncertainty in position x 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

5 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 iθ where i = 1. Don t worry, we will review complex numbers. 1. The Stern Gerlach Experiment One of the most important experiments in physics of all times is the Stern Gerlach experiment which was performed in 19 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

6 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 σz that is either up or down, in 6

7 Figure 1.: 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. 1.. 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 σ z = 1 state. The second experiment confirms that all atoms in the upper beam have the property σ z = 1. So it seems that the magnetic moments after the first experiment are all pointing upwards, confirmed by the second experiment. 7

8 Figure 1.3: The VH experiment, the second apparatus is rotated 90 degrees and splits the beam in two. 1.. 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 σ z = 1 atoms in it. In the VH experiment Stern Gerlach found that the σ 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 σ x. The first apparatus selects according to σ z and the second one according to σ x. In principle we still can think of the spin of the atoms, maybe as a dipol moment with three components σ = (σ x, σ y, σ 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 σ z = +1 were somehow altered with 50/50 chance into a state in which they now have σ 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 σ z and σ 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 σ z = 1 sub beam. We continue with this 8

9 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 σ x = +1 beam. At this stage we may think that we now have atoms that all have the properties σ z = 1 and σ 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 σ 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 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 σ z and σ x cannot be measured independently In fact is does not make sense to speak of the atoms having σ x AND σ z. The measurement of one quantity, e.g. σ x changes the propensity or likelihood of the system exhibiting a value for σ 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

10 Figure 1.5: The energy distribution u(ω of angular frequencies ω = πν of objects at different temperatures T 1 > T. 10

11 1.3 The situation around Black body radiation On of the key problems at the turn of the last century was black body radiation, explained in more detail below. Two of the most successful physical theories of the time were statistical mechanics as well as Maxwell s theory of Electromagnetism. Combinding both, people tried to predict the energy distribution of eletromagnetic waves emitted by a body in thermal equilibrium with its environment at temperature T. When we have an object that perfectly absorbs EM-radiation and that is in thermal equilibrium with its environment it also emits EM-radiation at different frequencies ν. For instance, when a body is very hot, it mostly emits bright light. When it s a bit cooler it the peak radiation may be in the infrared. We can measure the distribution of EM-radiation frequencies and will find something like depicted in Fig The distribution has a peak frequency at some value that depends on the temperature. Very low and very high frequencies are not present. One of the goals of physics around 1900 was to explain this type of distribution using classical theories in combination, like Boltzmann s statistical physics approach in combination with Maxwell s theory as mentioned Electromagnetic waves Maxwells equation for the electric and magnetic field in vacuum can be written has t E(r, t = c E(r, t t B(r, t = c B(r, t where r = (x, y, z is the position in 3d space, c is a constant and E and B are electric and magnetic fields, respectively and = ( x, z, z is the spatial vector derivative. These equations are wave equations like t ψ(r, t = c ψ(r, t that have travelling plane waves as solutions ψ(r, t = A cos(kr ωt The wave vector k points into the direction in which the wave travels, its magnitude is related to the wave length λ = π k and ω = πν is the angular frequency (ν is the regular frequency measure in Hertz. Frequency and wave length are related to c according to c = νλ = ω/k 11

12 and in the case of Maxwell equations we have c m/s. The success of Maxwell s theory clarified that light, e.g. is an electro-magnetic wave in vacuum. The cos wave is not the only solution that is a plane wave. Sine waves are also solutions and any combination of sine and cosine wave. For the sake of argument, it s ok to just use the cosine here Statistical mechanics According to Boltzmann theory, a body in thermal equlibrium with N 1 degrees of freedom and at temperature T will distribute its thermal energy equally among all its degrees of freedom, each degree of freedom getting an energy of E = k B T where k B = m kg/ks is the Boltzmann constant. A little bit of stastistical physics argument will tell us that every wave emitted by the black body will aquire an equal amount of energy proportional to E = k B T. If we want to know what the frequency distribution is, i.e. what is the concentration of EM waves in an interval [ω, ω + ω] this is equivalent to asking how many k vector fullfill [ k, k + dk]. Because k is a three-dimensional vector the number of waves in the inverval above scales as u( k k dk and because k ω we would expect an energy distribution u(ω E ω dω k B Tω dω This is called the Rayleigh-Jeans law. It predicts that more and more energy is emitted at higher and higher frequencies. This law, however, is inconsistent with observations and is only true for low frequencies. Sometimes this is called the ultraviolett catastrophy because it would imply an infinite amount of energy emitted by such a body Planck Planck approached the problem heuristically. This is often a good start. He proposed that energy bits k B T are NOT distributed equally among the possible degrees of freedom. Instead he assumed that the energy is given by E(ν, T = hν e hν/k BT 1 1

13 Figure 1.6: Photons as wave pakets with a frequency ω and particle like properties, e.g. localized in space. This function has the following interesting behavior. For small frequencies hν k B T we have E(ν, T = k B T consistern with the classical law. However when hν k B T then E(ν, T 0. So, physically, it means that frequencies that are larger than k B T/h are not excited and do not aquired energy and are thus not emitted. As such the law was a phenomenological law with a fitting parameter h. Planck used the letter h to symbolize Hilfskonstante. When fitting the new law to the observed data one finds an excellent fit and a numerical value for h: h Js This fitting parameter had the unit of an action. Later Planck showed that one could explain this law if one makes the physical assumption that EM radiation is quantized and comes in energy quanta that depend on the frequency E 0 (ν = hν = hω In other words light with a certain frequency comes in packets of quantized energy given by the above equation. This revolutionized physics. It seemed that classical, Maxwell EM theory predicted radiation to be waves and wave properties like frequency and have length were correct concepts. However, according to classical theory the enery in a wave is related to the amplitude of the wave and quantum theory shows that the energy of a quantum of EM radiation is related to the frequency. This was something quite different. The image people probably had in mind after this discovery is that light was a phenomenon that looked a bit like that in Fig Photo Electric Effect A few years later, in 1905, Einstein published 4 papers each one of them revolutionized physics. One paper was about Brownian Motion and a theory that proved the atomistic nature of matter and molecular collission causing the observed random movement patterns in Brownian motion. A second paper called zur Elektrodynamik bewegter 13

14 Körper Einstein introduced the special theory of relativity. In a sequel to this paper he discussed the consequences of this theory, and his famous formula E = mc. This paper is called Ist die Trägheit eines Körpers von seinem Energiegehalt anhängig?. In a forth paper Einstein explained the photo-electric effect, which is aside from Planck s paper the second paper that can be considered the foundation of quantum mechanics. Ironically Einstein didn t like the implications of QM that evolved in the following two decades. In particular the probabilistic aspect of quantum mechanics. The photo-electric effect is the following. If one shines EM radiation onto a thin metal plate, electrons will be emitted from the plate. IF one measures this carefully one finds that the kinetic energy of the emitted electron is related to the frequency of the EM radition that hits the metal plate 1 m ev e = hω W 0 hω > W 0 If the frequency exceeds a minimum value, such that hω > W 0 the kinetic energy increases with the frequency of the EM radiation. Einstein was able to explain this based on the assumption that the incoming light came in as energy quanta, photons, whose energy was hω. Then Einstein remembered some of his earlier work. According to the theory of relativity the energy of a particle is given by E = (p c + m c 4 1/ In this equation p is the momentum, c is the speed of light and m is the restmass of a particle. The speed of a moving body can be expressed in terms of its energy and restmass. pc v = (p m + m c 4 1/ This equations imply that for bodies at rest we have p = 0 and thus the famous E = mc. But what happens if we let v = c for instance what are these relations for photons that move at the speed of light? Then it follows that which in turn implies Recall that for EM radition we also have m = 0 E = pc E = ω h which also implies that p = ω h c = k h 14

15 Figure 1.7: Photons as wave pakets with a frequency ω and particle like properties, e.g. localized in space. So what Einstein derived was a relation of classical concepts like energy and momentum and related them to wave properties of classical EM waves ( E = hω p = hk Energy is proportional to the frequency and momentum proportional to the wavevector Bohr-Sommerfeld Atom Here s on last example of the quantization of action or energy introduced in the Bohr- Sommerfeld theory of the hydrogen atom. It was known at the time that atoms had internal structure, e.g. that the hydrogen atom consisted of a positively charged heavy proton and a negatively charged electron that circulated the central proton. However, the classical idea that the electron moved on circular paths around the atom s core implied that the electron, performing an accelerated motion, would have to emit electromagnetic radiation and energy which would slow the electron down which would imply that the electron would spiral towards the positive proton and collapse. Bohr and Sommerfeld proposed that the electron s orbits must be quantized at certain energy levels. When the electron is energetically excited it will move to a higher energy level and when it collapses to a lower level it would emit a quantized bit of energy in form of a photon. Once in the lowest energy level it cannot move any lower and is stuck to the minimal energy Double Slit experiments A bunch of experiments were designed to test the idea whether for example light is a wave or a particle. A class of problems are the so called double slit experiments. Where 15

16 Figure 1.8: The classical light experiment that showed that light behaves like waves because of the observed interference patterns when two circular waves that emerge from closely spaced slits emerge. When one slit is blocked, the interference pattern disappears. a source emits light, the light hits a wall with two slits, and at each slit a circular wave emerges that propages forward. The two waves interact and generate an interference pattern on a screen that measures the intensity. The interference can be calculated, and we will do that later, but essentially what happens ist that whem the wave maxima and minima of each wave meet the generate a larger maximum or minimum. On the other hand in locations where the maximum of one wave meets the minimum of another, they annihilate and cancel. On the screen this type of interference generates a sequenct of peaks and minima that are spaced at a certain distance, see Fig The interference pattern disappears, when one slit is closed and only one circular wave produces an intensity distribution on the screen. If we call this intensity I 1 and the intensity that we would get if the other slit is closed by I and the interference pattern intensity by I 1 we see that I 1 = I 1 + I. This is what we would get if we were to shoot particles through the holes (in a thought experiment. A particle would either go through one hole, maybe bounce off it, and then hit the screen. Many particles would generate a particle intensity which would be the sum of I 1 and I, see. Fig.1.9. Because this the original light experiments clearly proved the wave nature of light and electro-magnetic radiation The problem... But what happens if we turn down the intensity of the lamp that produces the light. The interference pattern will also go down in intensity but the pattern will not change. Now let s assume we can have a screen with photo-detectors that can measure very 16

17 Figure 1.9: If particles are shot at the board with two slits and either got through one or the other whole, no interference is observed, as expected. 17

18 tiny energy bits. What we see in such an experiment is that light actually hits the screen in particle fashion for instance every photon leaves a little mark, clear evidence that photons are particles. If we now wait long enough such that very many photons leave marks on the screen eventually an intensity pattern will emerge and this pattern is an interference pattern!!! This is cleary very strange. If photons are particles, they should go thtrough only one slit. But if that s the case, there should not be an interference pattern! Something is very strange. It seems like the photon is a particle that behaves as a wave as it goes though BOTH slits and the suddenly behaves like a particle when it hits the screen. This is actually the case. Noone knows why. If you actually try to figure out which slit the photon went through you ll find that as soon as you try to measure this, the interference patterns disappears. You cannot measure this without disturbing the system such that the effect is gone. Therefore the question: Which slit did the photon go through is meaningless in this experiment Matter waves Ok, you might think, photons, light, electromagnetic waves, all very strange stuff anyway. Let s look at stuff we can connect to more easily. Say electrons. Those are nice and massive point particles that are always somewhere, e.g. they have a position r in space. Let s shoot these at a wall with two slits. So, if you prepare the slits right, you will actually also see an interference pattern. Which means that even massive particles behave like wave in the sense that one observes interference of the electron with itself as it goes through both slits. You can, in principle do this with any body. However, the more massive the body, the more closely spaced the intensity peaks in the interference pattern to such a degree that you cannot resolve them. 18

19 Mathemical Excursion.1 Complex Numbers In QM complex numbers play a key role. A complex number z looks like this z = x + iy where the imaginary unit i has the property i = 1 so in other words i = 1. So, because i = 1 does not have a real solution, we call i imaginary. A complex number has real and imaginary parts, x is the real part and y the imaginary part. We can visualize a complex number as a point in a two-dimensional space spanned by the x and y axis like this: The set of complex number is often denoted by C. We can add complex number, if z 1 = x 1 + iy 1 and z = x + iy then z 1 + z = (x 1 + x + i (y 1 + y. So that s very similar to adding d vectors. However, unlike vectors, we can multiply complex numbers and get another complex number z 1 z = (x 1 + iy 1 (x + iy = x 1 x + i y 1 y + i (x y 1 + y x 1 = x 1 x y 1 y + i (x y 1 + y x 1 An important notion is the complex conjugate of a complex number z = x iy so that s like z with the opposite imaginary part. Geometrically complex conjugation is reflection with respect to the real axis. Using the complex conjugate of a complex number we can compute z z = zz = z = x + y So this is geometrically the length of the complex number, or the absolute value. Sometimes instead of the bar in z we also use the star symbol for complex conjugation, so z. 19

20 Figure.1: The complex plane. 0

21 According to the above we use the pair of numbers (x, y to specify a complex number. Geometrically we could also use the length and angle variables in polar coordinates (θ, r. Theta is calle the phase. We know that r = x + y, θ = tan 1 y/x We also know that x = r cos θ and y = r sin θ so we can write z = r(cos θ + i sin θ Now it turns out that the series expansion of the cosine is given by and and because we can easily check that cos θ = 1 x! + x4 4! x6 6! +... sin θ = x 1! x3 3! + x5 5! +... e iθ = 1 + iθ 1! + (iθ + (iθ3! 3! so that any complex number can be written as You can check for yourself that + (iθ4... 4! e iθ = cos θ + i sin θ (.1 z = re iθ z = re iθ This makes live easy, when you want to operate with complex numbers. For instance multiplying two of them z 1 = r 1 e iθ 1 z = r e iθ then z 1 z = r 1 r e i(θ 1+θ. You can also use complex numbers and Eq. (.1 to find an amazing identity e iπ + 1 = 0 which relates the fundamental mathematical numbers, 0,1,e,π and i. 1

22 . Vectorspaces Let us repeat a bit of mathematics on vector spaces. We have learned that a vector r is a quantity that has a magnitude and a direction and we visualize it typically as an arrow in the vector space to which the vector belongs. Mostly we have seen the -d and 3-d vector spaces R and R 3 which we most readily identify with physical space, so a vector has components that correspond to the coordinates of the vector space for example in d we have ( x r = y and in 3d we have r = Of course we can generalize the concept to more than three dimensions. It s just no longer impossible to draw well. In what follows we will discuss what we know using d vector spaces. We know that we can do stuff with vectors. For example we can add them. Let s say u and v are d vectors then the sum x y z w = u + v is also a vector in the same space which we get by just adding the components of u and v. Another thing we can do with vectors is multiply them by a numer α for instance w = αu is also a vector. We just multiply all components of u with α. Geometrically this means the new vector w points into the same direction as u it is just shorter or longer depending on the value of α. If α < 0 then w points into the opposite direction. Formally, to have a nice proper vector space we also must have a 0 vector which, when we add it to a vector, doesn t change it u + 0 = u For every vector u in the vector space we also must have a vector u that if we add it to u we must get 0. u + ( u = 0. Some vector spaces are also equipped with a product which gives a number (not a vector. We call this product a scalar product and denote it by u v How do we compute this? We multiply the components and add the results, if u = (x 1, y 1 t and v = (x, y t then u v = x 1 x + y 1 y

23 Clearly We can also compute u v = v u. u u = x 1 + y 1 So the scalar product can be used to compute the length of the vector u u = u u Geometrically we can also show that u v = u v cos θ where θ is the angle between both vectors. This also shows that u v = 0 if the two vectors are perpendicular to one another. There are a bunch of other things that we require to be fullfilled by such a scalar product and you can check for the specific choise above that all of these are fullfilled, for example: u + w v = w v + u v and αu v = α u v..1 Orthonormal basis A basis of an N dimensional a vectorspace is a set of N vectors that can be used to represent any vector in the space as a linear combination of vectors in the basis. In R 3 for example a set of basis vectors are e 1 = e = e 3 = These are the vectors that point along the x,y,z directions and any vector v = (x, y, z t can be trivially written as v = xe 1 + ye + ze 3 Let s say we have an N-dimensional vector space and a basis b 1, b,..., b N we can write any vector as v = α i b i i where the coefficient α i are components of v along b i. Let s now focus on bases that are orthonormal. This means that all basis vectors are orthogonal, such that bi b j = δij

24 Figure.: where δ ij is the Kronecker symbol. This thing is 1 when i = j and 0 otherwise. Now we can compute bj v = α i bj b i i = α j. So we can write v = b i v b i. i.. Operators An operator  is like a machine that transforms vectors into other vectors in the vectorspace, so say we have a vector v and and operator  then w = Âv is the vector that we get when we apply the operator Â. Think of the operator as a box where you throw in a vector in the top and another comes out at the bottom. For example this is an operator v  v v (v + a (. where a is some constant vector. Let s say we are in R then the operator R θ that rotates a vector by θ is also an operator w = R θ v Another example is are projection operator P x and P y that project vectors onto the x or y axis, see Fig... Or maybe a general projection operator P a that projects onto a fixed operator a. Another interesting operator is the operator S x for example that takes a vector v and maps it onto the vector that is the mirror image reflected from the x axis. Clearly there are a lot of operators one can think of in R or also R Linear Operators One special class that will be important for quantum theory are linear operators for which we must have. Â(v + w = Âv + Âw Âαv = αâv 4

25 Let s see if the operator defined in Eq. (. is linear Â(v + w = v + w v + w (v + w + a = [ v v + w w + v w ] (v + w + a = v v (v + a + v v w + w w (v + a + w w v + v w (v + w + a = Âv + Âw + v v w + w w v + v w (v + w + a = Âv + Âw so this is a non-linear operator. How about the rotation R θ. We an really see that R θ (w + v = R θ w + R θ v geometrically. Rotating the sum of two vectors is the same as rotating them individually and then adding them. Rotation is linear. The same is true for projection operators P a...3 Eigenvectors and Eigenvales An important notion in quantum theory are eigenvectors and eigenvalues of linear operators. Given a linear operator  those are defined by the equation Âv = λv with v = 0 A vector v that is a solution to such an equation is called an eigenvector and the number λ is called an eigenvalue. Note that v has to be non-zero but λ can be zero. Operators in R can have from zero to an infinite number of eigenvectors. Let s look at rotation first. The general rotation operator R θ has no eigenvector, unless θ has certain values. An eigenvector only gets shrunk or expanded when operated on but still points in the same direction. Note also that if v is an eigenvector then so is αv so it s really a eigen- ray. How about the reflection operator Ŝ x that reflects vectors across the x-axis. Clearly all vectors that lie along the x-axis are eigenvectors, for example if we have v = α(1, 0 t we have Ŝ x v = v so their eigenvalue λ = 1. But also all vectors along the y-axis are eigenvectors, if v = α(0, 1 t we have Ŝ x v = v so for those vectors λ = 1. How about the projection operator P x. This one also has the two eigenvectors v 1 = (1, 0 t and v = (0, 1 t with P x v 1 = v 1 P x v = 0, so, very similar to the reflection operator except that one eigenvalue is zero. This is in general the case. All projection operators have eigenvalues that are either 0 or 1. The 5

26 Figure.3: also have another interesting property. If I take a general projection operator P a and apply it to a vector v getting w = P a v then w is parallel to a, see Fig..3. Which means that if I compute P a w = w the new vector is an eigenvector of P a. That also means that P a w = P a P a v = P a v so that P a = P a which we write as a multiplication of operators but at this stage we mean the successive application of P a. This is a general feature of projectors...4 Multiplication of operators If we have two operators  and B, we can investigate what happens if we first apply one than the other and compare to the reverse situation. So let s compute  Bv = w 1 and BÂv = w In general we have w 1 = w For example if we rotate a vector and then project to the x-axis we get something that is different from when we first project and then rotate. For example if we have the rotation operator R 90 and the projection operator P x and start with the vector v = (1, 1 t we have P x v = (1, 0 t R 90 P x v = (1/, 1/ t R 90 v = ( 1, 1 t P x R 90 v = ( 1, 1 t So clearly the combined operations aren t equal R 90 P x = P x R 90 6

27 ..5 Linear Operators and Matrix Representations Here s the cool thing. For vectors in R we have representations like ( x v = y Now, one of the interesting things about linear operators is that they can be represented as matrices. In R matrices, so we can write them as ( a b  =. c d For example we can write the operators we discussed earlier as ( ( ( R cos θ sin θ θ =, S sin θ cos θ x = P 0 1 x = 0 0 So that means we can compute the operation, for example ( ( ( cos θ sin θ x x cos θ y sin θ R θ v = = sin θ cos θ y x sin θ + y cos θ It also means that finding eigenvales and eigenvectors is the same as finding eigenvectors and eigenvalues of the corresponding matrices Symmetric Operators A particular group of operators are represented by symmetric matrices, e.g. ( a b  =. b a One can show that these operators only have real eigenvalues, λ 1 and λ. Plus one has a basis of orthonormal eigenvectors, v 1 and v. Now there are two situations: 1. λ 1 = λ. That implies that for each eigenvalue we have a distinct 1d eigenvector and v 1 and v are orthogonal and form a basis for R. For example take ( 1 0 P x = 0 0 Eigenvalues are λ 1 = 1 with v 1 = (1, 0 t and and λ = 0 with v = (0, 1 t. if λ 1 = λ, for example there s only one eigenvalue, then all vectors are eigenvectors. For example the operator ( ( I = R = 0 1 are such.. 7

28 ..6 Spectral Decomposition One important notion is that of a spectral decomposition. Let s say we have a d symmetric operator ( a b  = b a and let s assume it has two eigenvalues λ 1 and λ with eigenvectors v 1 and v that are orthonormal, we can write the operator  in the following form  = λ 1 P 1 + λ P where P i are projectors onto the eigenvectors v 1. This is called the spectral decomposition of a symmetric d linear operator. It makes sense, because Âv 1 = ( λ 1 P 1 + λ P v1 = λ 1 P 1 v 1 = λ 1 v 1 likewise for v. For example, if we have ( 3 A = 0 this guy has eigenvalues with eigenvectors λ 1 = 4 v 1 = (, 1 t λ = 1 v = (1, t These vectors are orthogonal clearly. Let s make them normal by requiring v i v i = 1 so v 1 = 1 5 (, 1 t v = 1 5 (1, t are an orthonormal basis. Here s a little trick how to compute a projector on the these vectors we compute P 1 = v 1 v1 t = 1 ( (, 1 = 1 ( Let s see if P1 = P 1: 1 ( 4 ( = 1 5 ( = 1 5 ( 4 1 = P 1 Likewise we get P = v v t = 1 5 ( 1 (1, = 1 (

29 and A = ( 3 0 = 4 5 This will become very important later. ( ( The space C We have now learned how things work in R and generalizations to R 3 or R n are straighforward. It s time for a generalization. We ve learned about complex numbers z that have the shape z = x + iy. We know how to multiply them and add them. So we can also from say two dimensional column vectors v = ( z1 where now the elements are no longer real but complex numbers. The vectors that we form this way are elements of the vector space C. We can add them according to the same rules v + u = w by adding components. we can multiply them by scalars z w = αv but this time α is a complex number. In C however we can also form the complex conjugate of a vector, so if then v = ( z1 z ( ( x1 + iy = 1 r1 e = iθ1 x + iy r e iθ ( ( ( v z = 1 x1 iy z = 1 r1 e = iθ1 x iy r e i θ Because of this, we are going to do something a little different in terms of notation. we write for a vector v = v and we call this a ket vector. we call the associated conjugate vector a bra vector v = v So to every ket vector we associate a bra vector v v 9

30 in fact we will sometimes call this vector a dual vector and think of it as being an element of a dual vector space. Let s now assume that we have a vector w = α v where α is a complex number. What s the bra vector associated with w? The answer is w = α v = v α because if and ( αz1 w = α v = αz w = ( α z 1 α z = α v Now the clue is: How do we define a scalar product between vectors v and w. The way this is done is by multiplying w v = w v Now we see the reason for the bra ket notation. With this we see that if v = (z 1, z t w = (q 1, q t then w v = q 1 z 1 + q z Note that this means w v is a complex number and also that w v = w v The reason why we have to do this, is because we want v v to be a real number, and it is v v = z 1 z 1 + z z = z 1 + z and if z 1 = x 1 + iy 1 and z = x + iy then We define the length of a vector in C as v v = x 1 + y 1 + x + y v = v v In R, if we have a vector the vector r 1 = ( x y r = r 1 = ( 1r 1 30

31 has the same length. In C a similar relation exists. Let s say we have a vector ( z1 v = the vector w = e iα v that is multiplied with a phase factor has the same length because w = w w = v e iα e iα v = v v z Unit Vectors in C have the shape v = 1 e iα ( e iθ 1 e iθ because v = 1 e iα (e iθ 1, e iθ In analogy to R we can define linear operators on C where these are represented by matrices that have complex numbers as entries for example ( 1 i 1 Â = + i i is an operator on C. So let s say we have such an operator and form w = A v We know their are dual vectors v and w that correspond to v and w respectively. We can also find an operator that transforms v into w and call this operator the adjoint operator A w = v A The writing is a little confusing because we write A on the right of v but that s only a notational convention that will make sense later. For example let s say ( i 1 Â = 0 i and v = (i, 1 then w = A v = ( i 1 0 i ( i 1 = ( i and v = ( i, 1 w = (, i 31

32 The adjoint operator is an operator that is the transpose plus complex conjugation so ( A i 0 = 1 i So let s see.3.1 Eigenvectors ( v A i 0 = ( i, 1 1 i = (, 1 = w In C we can also do the whole deal of eigenvectors and eigenvalues, so the eigenvalue equation reads in the new notation A v = λ v and in general eigenvectors and eigenvalues are complex. Now let s look at.3. Hermitian Operators v A = λ v A hermitian operator is such that it is self-adjoint So they have to have the form ( α γ A = A ( β α γ = δ which means the structure has to look like ( a β β d where a, b are real. An interesting property that is super important for quantum theory is that a self hermitian operator only has real eigenvalues (despite the fact that it s a complex operator. Let s see why and thus Also and thus Now because A = A we have β A v = λ v δ v A v = λ v v v A = λ v v A v = λ v v λ = λ which means that λ is real. Awesome. Also if a Hermitian operator in C has two distinct eigenvectors, they form an orthnormal basis in C. 3

33 .3.3 Projectors on C We can also have projectors in the complex d vector space. Let s say we want to construct a projector on the ray spanned by the unit vector u = 1 ( e iθ 1 e iθ Here s how this works, we say So P u = u u P u w = u u w because u w is the component of w in the subspace spanned by u. We also see that P u u = u and if P u v = 0 u v = 0 i.e. when u and v are orthogonal. Let s say u = 1 ( e iθ 1 e iθ then u = 1 (e iθ 1, e iθ so P u = 1 For example if θ 1 = π/ and θ = 0 then ( 1 e i(θ 1 θ e i(θ 1 θ 1 u = 1 ( i 1 and P u = 1 ( 1 i i 1 We see that P u is hermitian and we also see that P u = P u Now what s the projector that projects orthogonally? We now that 1 = P u + P v 33

34 so P v = 1 u u ( 1 0 = = 1 ( 1 i i 1 ( 1 i i 1 Let s check this by construction so for instance θ = 0 and θ 1 = π/. u v = 1 (e iθ 1, e iθ 1 ( i 1 = 1 ( e iθ1+π/ + e iθ u = 1 ( i 1 and P v = v v = 1 ( 1 i i 1 34

35 .4 The structure of quantum mechanical systems Having discussed the vector space C we can now start talking about physics. It turns out that C is a Hilbert space. Hilbert spaces are vector spaces over the complex numbers with an inner product like the one we defined above for C. Hilbert spaces are used to describe quantum mechanical systems in the following way: 1. The state of a quantum system is a normalized vector v in a Hilbert space.. Physical observables, like energy, position, momentum or spin are linear, hermitian operators  in these spaces that act on quantum mechanical state. 3. When a measurement of a physical quantity is made the value of the measurement is on of the eigenvalues λ i of the operator A that represents that quantity. 4. The eigenvectors λ i are quantum mechanical states for which a measurement of A gives λ i with certainty. 5. The orthogonal eigentstates of an observable A represents the different certain possible outcomes of measuring A. 6. If a system is in a state v then the probability P(λ i of measurung λ i is given by.4.1 The Spin v P λi v = v λ i λ i v = v λ i Let s now do quantum mechanics and use the above abstract notions and apply time to spin that is measured in the Stern Gerlach experiment. We found that the measurements of spin always gave +1 or 1 regardless of the orientation of the apparatus. Now obviously, the values for the spin are phsyical quantities because the measure something like a magnetic moment. in fact, the magnetic moment of the atoms are s = ± h [Js] which has units of an action. These are the natural unis of an angular momentum which has units of Mass Length Time Length = Mass Length Time We discussed this initial for 3 orthogonal orientations = Mass Length Time Time = Energy Time. 1. oriented in the z-direction we measured σ z and found s ± h/ and afterwards the state up or down was prepared. up corresponding to having measured a spin with and down when having measure 1 35

36 . oritented in the x-direction. we measured σ x and found h/ and afterwards the state right or left was prepared, right corresponding to having measured + h/ and left when having measure 1 3. oritented in the y-direction. we measured σ y and found ± h/ and afterwards the state in or out was prepared. in corresponding to having measured + h/ and out when having measure h/ We also had the notion that for instance up and down are mutually exclusive for example that after a measure σ z that resulted in s = + h/ we can measure σ z again and find s = h/ with 100% chance and σ z = h/ never. when we prepare a state right however, that is measured σ x and then measure σ z we find s = h/ with 50% chance and likewise for s = h/. Bottom line Any spin measurement has only two outcomes, + h/ and h/ that occur with different probabilities We now make use of the fundamental idea of quantum mechanics The state of a quantum mechanical system quantifies the tendency of the system to behave in a certain way and we describe this state by a vector in a Hilbert space, specifically the spin state is a vector s C. Let s see how this works. The above obersvations on spin suggest that because we only have two different outcomes maybe the general spin state can be describe by a two-dimensional vector in C. We know that after measuring σ z the system is in a state u if the measurement gave + h/ and in a state d if it gave h/. Now we assume that two mutually exclusive states are represented by orthogonal vectors in the state space C u d = 0 It is also useful to say that both u and d are normalized u u = d d = 1 This is already hinting at a relation between the scalar product and a probability probability has something to do with w v 36

37 However quanties like w v are just complex numbers and thus no probabilities because probabilities are between 0 and 1. It turns out that we call quanties like u v probability amplitudes. Because C is two dimenional and because the vectors u and d are orthonormal and form a basis any spin state can be represented a linear combination of those two basis vectors s = α u u + α d d where α u = u s α d = d s in other work. s = u u s + d d s α u/d are complex numbers. Now the idea is that their absolute value quantities the probabiliy of measuring + h/ or h/ and preparing the states u and d in these measurements so that α uα u = the probability of measuring + h/and preparing the state u α d α d = the probability of measuring h/and preparing the state d That also means that α uα u + α d α d = 1 which we can write as Now let s look at this We can also write this using s u u s + s d d s = s ( u u + d d s = s (P u + P d s = s s = 1 p(σ z, + h/ = α uα u = s u u s = s u p(σ z, h/ = α d α d = s d d s = s d P u = u u P d = d d as p(σ z, + h/ = s P u s p(σ z, h/ = s P d s 37

38 so the projector is sanwiched between the general state. Let s see if this is consistent with what we know. Let s assume that we have prepared a system in state u so s = u which means that α u = 1 and α d = 0 and we measure σ z again. It follows that p(σ z, + h/ = s P u s = u P u u = u u u u = 1 p(σ z, h/ = s P d s = u P d u = u d d u = 0 Nice. That works. Now let s assume that we ve prepared the system in a state r by having measured + h/ in a measurement of σ x. If what we said is true then the state r must be a superposition of states u and d so r = α u u + α d d We also know that if we now make a σ z measurement we will find p(σ z, + h/ = r P u r = r P u r = r u u r = α uα u = 1/ p(σ z, h/ = r P d r = r P d r = r d d r = α d α d = 1/ This is what we observe and we can accomplish this by r = 1 u + 1 d Clearly we have r r = 1 This means that the r state can be considered a superposition of u and d states. Now if we have the system in a state r and measure σ x again we find the state l with probability zero so the state l must be a vector that looks like this l = 1 u 1 d which fullfills l r = 0. This also implies that the apparatus σ x can be associated with the projectors P l = l l and P r = r r which we can write as P l = ( 1 u 1 ( 1 d u 1 d = 1 P u + 1 P d 1 ( d u + u d 38