Wave Phenomena

1. qi = qr Wave Phenomena InterferenceDiffraction Reflexion Refraction Diffraction is the bending of a wave around an obstacle or through an opening. qi Wavelenght dependence n1 n2 Diffraction at Slits qt q n1 sin (qi) = n2 sin (qt) w p=w sin(q)= ml bright fringes The path difference must be a multiple of a wavelength to insure constructive interference. q d q Diffraction at a lattice ml p=w sin(q)= ml p=d sin(q)= bright fringes

2. A central concept of Quantics:wave–particle duality is the concept that all matter and energy exhibits both wave-like and particle-like properties. The Wave Nature of Matter All material particles are associated with Waves („Matter waves“ E = hn E = mc2 mc2 = hn = hc/l or: mc = h/l De Broglie A normal particle with nonzero rest mass m travelling at velocity v mv = p= h/l Then, every particle with nonzero rest mass m travelling at velocity v has an related wave l l = h/ mv • The particle property is caused by their mass. • The wave property is related with particles' electrical charges. • Particle-wave duality is the combination of classical mechanics and electromagnetic field theory.

3. Schrödinger's cat It is a „Gedanken“ (thought experiment) often described as a paradox I don‘t like it and I regret that I got involved in it…. Superposition of two States: Broadly stated, a quantum superposition is the combination of all the possible states of a system. yAlife+ yDead Schrödinger Miau! yAlife yDead

4. Schrödinger wrote: „One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The Y-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.“

5. Schrödinger's famous tought experiment poses the question: when does a quantum system stop existing as a mixture of states and become one or the other? (More technically, when does the actual quantum state stop being a linear combination of states, each of which resemble different classical states, and instead begin to have a unique classical description?) If the cat survives, it remembers only being alive. But explanations of experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions. The purpose of the thought experiment is to illustrate this apparent paradox: our intuition says that no observer can be in a mixture of states, yet it seems cats, for example, can be such a mixture. Are cats required to be observers, or does their existence in a single well-defined classical state require another external observer? An interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location, or state of motion. In effect, the act of measurement causes the calculated set of probabilities to "collapse" to the value defined by the measurement. This feature of the mathematical representations is known as wave function collapse.

6. The Waves and the Incertainty Principle of Heisenberger „The measurement of particle position leads to loss of knowledge about particle momentum and visceversa.“ p y q Dy 2p sin (q) = Dpy=2pl/Dy Dpy . Dy ≈ 2pl = 2h v m x sin (q) = ±l/Dy The momentum of the incoming beam is all in the x direction. But as a result of diffraction at the slit, the diffracted beam has momentum p with components on both x andy directions.

7. The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. Rydberg Continuous spectrum emission lines absorptiom lines Rydberg formula for hydrogen • Where • λvac is the wavelength of the light emitted in vacuum, • RH is the Rydberg constant for hydrogen, • n1 and n2 are integers such that n1 < n2

8. Bohr‘s Atom Model Bohr • He suggested that electrons could only have certain classical motions: • The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies. • The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency n determined by the energy difference ΔE = E2 − E1 = hn of the levels according to Bohr's formula where his Planck‘s Constant. • the frequency of the radiation emitted at an orbit with period T is as it would be in classical mechanics--- it is the reciprocal of the classical orbit period: • n = 1/T = c/l

9. The angular momentum A is restricted to be an integer multiple of a fixed unit: A= n h / 2p = mvr [1] where n = 1,2,3,… and is called the principal quantum number. The lowest value of n is 1. This gives a smallest possible orbital radius of 0.0529 nm. This is known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlyke atoms and ions. Bohr's condition, that the angular momentum is an integer multiple of h/2p was later reinterpreted by de Broglie as a standing wave condition: nl =2p r = L [2] the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit. X=L X=0

10. To calculate the orbits requires two assumptions: • 1. Classical mechanics • The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulom force: • mv2/r = ke2 /r2 [3] • where m is the mass and e is the charge of the electron. This determines the speed at any radius: • v=√(ke2 /mr) [4] • It also determines the total energy at any radius: • E=mv2/2 - ke2 /r = -ke2 /2r [5] • The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. For larger nuclei, the only change is that ke2 is everywhere replaced by Z ke2 where Z is the number of protons.

11. . Quantum rule • Substituting the expression for the velocity [4] in the expression for the angular momentum [1] gives an equation for r in terms of n: • √(ke2mr) = nh/2p [6] • so that the allowed orbit radius at any n is: • r = n2 h2 / (4p2ke2 m) [7] • The smallest possible value of r is 0.51 x 10-10m (n=1) is called the Bohr radius The energy of the n-th level is determined by the radius (replacing [7] in [5] : • E= (2pke2) 2 m/(2n2 h2 ) • The combination of natural constants in the energy formula is called the Rydberg energy RH (n=1): • RH = (2pke2) 2 m/(2h2 )

12. Einstein Bohr Heisenberg Planck Photons de Broglie electrons Schrödinger Energy Waves Uncertainty diffraction

13. Quantum Mechanics

15. The book of nature you can only understand, if you have previously learnt its language and the letters. It is written in mathematical language and the letters are geometrical figures, and without these means it is impossible for human beings to understand even a word of it. Galileo Galilei, 16th century

16. Dirac Notation, Hilbert Space The mathematical concept of a Hilbert space, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. Dirac Hilbert Mathematically, a pure quantum state is typically represented by a vector in a Hilbert space. Bra-ket notation is a standard notation for describing quantum states composed of angle brackets and vertical bars. It can also be used to denote abstract vectors. It is so called because the inner product of two states is denoted by a bracket,     , consisting of a left part,   , called the bra, and a right part,   , called theket. The notation was invented by Paul Dirac and is also known as Dirac notation.

17. Operators An Operator transforms the original vector |u> in general by changing its amount (magnitude) and its direction into a new vector Eigenvalue: The Operator transforms the original vector |u> in particular by changing its amount (magnitude) (elongation or compression) but mantaining its direction into a new vector HermitianOperator: which fulfil the condition: (always realEigenvalues) If in addition an HermitianOperator also satisfies the condition: Adjugate

18. Conclusions, Summary, Some of more important Postulates • Duality Behaviour: Wavelike ↔ Particlelike: Associated with any particle is a wavefunction having wavelength related to a particle momentum by: l = h/p = h/√(2m(E-V))(de Broglie) • Wave Function:y: its absolute square is proportional to the probability density for finding the particle. It describes a state as completely as possible. First Postulate • Wave Function:y: it is an eigenfunction ofSchrödinger‘s Equation, which can be constructed from the classical wave equation requiring: l = h/p = h/√(2m(E-V)) • Wave Function:y: to be acceptable: it must be single-valued, continuous, nowhere infinite, with a piecewise continuous first derivative, square-integrable. • Wave Function:y - Normalization: • Operator:Ô:For any Observable there is an operator which is constructed from the classical expression according to a simple recipe. Second Postulate

19. Time-Dependent Schrödinger Equation:The State functions or Wave functions satisfy the equation: • Both, Hamiltonian and Wave functions are time-dependent. Third Postulate • If the Hamiltonian Operator for a system is time.independent, stationary eingenfunctions exist of the form: • The time-dependent exponential does not affect the measurable properties of a system in this state and is almost always completely ignored in any time-independent problem. • Operator:Ô: The Eigenvalues for such an Operator are the possible values we can measure for that quantity. Fourth Postulate • Operator:Ô: The act of measuring the quantity forces the system into a state described by an Eigenfunction of the Operator • Expectation Value m of an Operator M: Fifth Postulate • It is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence.

20. Completeness of Eigenfunctions of an Operator M: Sixth Postulate • The Eigenfunctions for any quantum mechanical operator corresponding to an observable variable constitute a complete set. • Briefly, a series of functions {f } having certain restrictions is said to be completeif an arbitrary function y having the same restrictions can be expressed in terms of the series: • Time-Dependent Schrödinger Equation: • Time-independent Schrödinger Equation: