Failures of Classical Physics Some experimental situations where "classical" physics fails:

1. Failures of Classical Physics Some experimental situations where "classical" physics fails: Photoelectric effect Blackbody radiation Wave properties of electron Line spectra The remedies come from some inherently quantum ideas:

2. Blackbody Radiation Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating. The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency. But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Nature knew better.

3. The birth of Quantum Mechanics According to the Planck hypothesis, all electromagnetic radiation is quantized and occurs in finite “quanta" of energy which we call photons. The quantum of energy for a photon is not Planck's constant h itself, but the product of h and the frequency. The quantization implies that a photon of blue light of given frequency or wavelength will always have the same size quantum of energy. For example, a photon of blue light of wavelength 450 nm will always have 2.76 eV of energy. It occurs in quantized chunks of 2.76 eV, and you can't have half a photon of blue light - it always occurs in precisely the same sized energy chunks. But the frequency available is continuous and has no upper or lower bound, so there is no finite lower limit or upper limit on the possible energy of a photon. On the upper side, there are practical limits because you have limited mechanisms for creating really high energy photons. Low energy photons abound, but when you get below radio frequencies, the photon energies are so tiny compared to room temperature thermal energy that you really never see them as distinct quantized entities - they are swamped in the background. Another way to say it is that in the low frequency limits, things just blend in with the classical treatment of things and a quantum treatment is not necessary.

4. Blackbody Intensity as a Function of Frequency Energy per unit volume per unit frequency k = Boltzman constant

5. Rayleigh-Jeans vs Planck Comparison of the classical Rayleigh-Jeans Law and the quantum Planck radiation formula. Experiment confirms the Planck relationship

6. Radiation Curves

7. The Photoelectric Effect The remarkable aspects of the photoelectric effect when it was first observed were: 1. The electrons were emitted immediately - no time lag! 2. Increasing the intensity of the light increased the number of photoelectrons, but not their maximum kinetic energy! 3. Red light will not cause the ejection of electrons, no matter what the intensity! 4. A weak violet light will eject only a few electrons, but their maximum kinetic energies are greater than those for intense light of longer wavelengths! The details of the photoelectric effect were in direct contradiction to the expectations of very well developed classical physics. The explanation marked one of the major steps toward quantum theory.

8. The Photoelectric Effect Analysis of data from the photoelectric experiment showed that the energy of the ejected electrons was proportional to the frequency of the illuminating light. This showed that whatever was knocking the electrons out had an energy proportional to light frequency. The remarkable fact that the ejection energy was independent of the total energy of illumination showed that the interaction must be like that of a particle which gave all of its energy to the electron! This fit in well with Planck's hypothesis that light in the blackbody radiation experiment could exist only in discrete bundles with energy

9. Most commonly observed phenomena with light can be explained by waves. But the photoelectric effect suggested a particle nature for light.

10. The Line Spectrum Problem In the years before the beginning of the 20th century, the light emitted from luminous gases was found to consist not of a continuous range of wavelengths, but of discrete colours which were different for different gases. These spectral "lines" formed regular series and came to be interpreted as transitions between atomic energy levels. This presented a considerable problem for classical physics, because accelerated charges were known to radiate electromagnetic energy. It was expected that orbits of electrons about positive nuclei would be unstable because they would radiate energy and therefore spiral into the nucleus. No classical model could be found which would yield stable electron orbits. The Bohr model of the atom started the progress toward a modern theory of the atom with its postulate that angular momentum is quantized, giving only specific allowed energies. Then the development of the quantum theory and the Schrödinger equation refined the picture of the energy levels of atomic electrons. Helium spectrum Hydrogen spectrum integers RH = Rydberg constant = 1.097 10–7 m–1

11. Bohr atomic model - Classical Electron Orbit In the Bohr theory, this classical result was combined with the quantization of angular momentum to get an expression for quantized energy levels.

12. Angular Momentum Quantization In the Bohr model, the wavelength associated with the electron is given by the DeBroglie relationship and the standing wave condition that circumference = whole number of wavelengths. These can be combined to get an expression for the angular momentum of the electron in orbit. Thus L is not only conserved, but constrained to discrete values by the quantum number n. This quantization of angular momentum is a crucial result and can be used in determining the Bohr orbit radii and Bohr energies.

13. Combining the energy of the classical electron orbit with the quantization of angular momentum, the Bohr approach yields expressions for the electron orbit radii and energies: substitution for r gives the Bohr energies and radii:

14. Hydrogen Energy Levels The basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures are those of a shell structure with each main shell associated with a value of the principal quantum number n. This Bohr model picture of the orbits has some usefulness for visualization so long as it is realized that the "orbits" and the "orbit radius" just represent the most probable values of a considerable range of values.

15. The Bohr model for an electron transition in hydrogen between quantized energy levels with different quantum numbers n yields a photon by emission, with quantum energy This is often expressed in terms of the inverse wavelength or "wave number" as follows:

17. Failures of the Bohr Model While the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are: 1. It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities. 2. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle which dictates that position and momentum cannot be simultaneously determined. The Bohr model gives us a basic conceptual model of electrons orbits and energies. The precise details of spectra and charge distribution must be left to quantum mechanical calculations, as with the Schrödinger equation.

18. Particle nature of light - Compton Scattering Arthur H. Compton observed the scattering of x-rays from electrons in a carbon target and found scattered x-rays with a longer wavelength than those incident upon the target. The shift of the wavelength increased with scattering angle according to the Compton formula: Compton explained and modeled the data by assuming a particle (photon) nature for light and applying conservation of energy and conservation of momentum to the collision between the photon and the electron. The scattered photon has lower energy and therefore a longer wavelength according to the Planck relationship.

19. The above expression for Δλ can be obtained by exploting energy-momentum conservation

20. Wave Nature of Electron As a young student at the University of Paris, Louis DeBroglie had been impacted by relativity and the photoelectric effect, both of which had been introduced in his lifetime. The photoelectric effect pointed to the particle properties of light, which had been considered to be a wave phenomenon. He wondered if electons and other "particles" might exhibit wave properties. The application of these two new ideas to light pointed to an interesting possibility: Confirmation of the DeBroglie hypothesis came in the Davisson- Germer experiment which showed interference patterns – in agreement with DeBroglie wavelength – for the scattering of electrons on nickel crystals.

21. When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions: The angle of incidence = angle of scattering. The pathlength difference is equal to an integer number of wavelengths. The condition for maximum intensity contained in Bragg's law above allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the x-rays incident upon the crystal. The Davisson-Germer experiment showed that electrons exhibit the DeBroglie wavelength given by:

22. DeBroglie Wavelengths

23. Wave-Particle Duality: Light Does light consist of particles or waves? When one focuses upon the different types of phenomena observed with light, a strong case can be built for a wave picture: Reflection Refraction Interference Diffraction Polarization Photoelectric effect Compton scattering Most commonly observed phenomena with light can be explained by waves. But the photoelectric effect and the Compton scatering suggested a particle nature for light. Then electrons too were found to exhibit dual natures.

25. Wavefunction Properties

26. Schrödinger Equation The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrödinger equation will predict the distribution of results. The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrödinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

27. Time-independent Schrödinger Equation For a generic potential energy U the 1-dimensional time-independent Schrodinger equation is In three dimensions, it takes the form for cartesian coordinates. This can be written in a more compact form by making use of the Laplacian operator The Schrodinger equation can then be written: HΨ = EΨ

28. Time Dependent Schrödinger Equation The time dependent Schrödinger equation for one spatial dimension is of the form For a free particle where U(x) =0 the wavefunction solution can be put in the form of a plane wave For other problems, the potential U(x) serves to set boundary conditions on the spatial part of the wavefunction and it is helpful to separate the equation into the time-independent Schrödinger equation and the relationship for time evolution of the wavefunction

29. The Postulates of Quantum Mechanics 1. The Wavefunction Postulate: Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system. It is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wavefunction. This wavefunction determines everything that can be known about the system. The wavefunction may be a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state.

30. Probability in Quantum Mechanics The wavefunction represents the probability amplitude for finding a particle at a given point in space at a given time. The actual probability of finding the particle is given by the product of the wavefunction with it's complex conjugate (like the square of the amplitude for a complex function). Since the probability must be = 1 for finding the particle somewhere, the wavefunction must be normalized. That is, the sum of the probabilities for all of space must be equal to one. This is expressed by the integral infinitesimal volume Part of a working solution to the Schrodinger equation is the normalization of the solution to obtain the physically applicable probability amplitudes

31. 2. The Operator Postulate With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction. With every physical observable there is associated a mathematical operator which is used in conjunction with the wavefunction. Suppose the wavefunction associated with a definite quantized value (eigenvalue) of the observable is denoted by Ψn (an eigenfunction) and the operator is denoted by Q. The action of the operator is given by The mathematical operator Q extracts the observable value qn by operating upon the wavefunction which represents that particular state of the system. This process has implications about the nature of measurement in a quantum mechanical system. Any wavefunction for the system can be represented as a linear combination of the eigenfunctions Ψn (see basis set postulate), so the operator Q can be used to extract a linear combination of eigenvalues multiplied by coefficients related to the probability of their being observed (see expectation value postulate).

32. Operators in Quantum Mechanics Associated with each measurable parameter in a physical system is a quantum mechanical operator. Such operators arise because in quantum mechanics you are describing nature with waves (the wavefunction) rather than with discrete particles whose motion and dynamics can be described with the deterministic equations of Newtonian physics. Part of the development of quantum mechanics is the establishment of the operators associated with the parameters needed to describe the system. Some of those operators are listed below. It is part of the basic structure of quantum mechanics that functions of position are unchanged in the Schrödinger equation, while momenta take the form of spatial derivatives. The Hamiltonian operator contains both time and space derivative.

33. 3. Hermitian Property Postulate Any operator Q associated with a physically measurable property q will be Hermitian. The quantum mechanical operator Q associated with a measurable property q must be Hermitian. Mathematically this property is defined by where Ψa and Ψb are arbitrary normalizable functions and the integration is over all of space. Physically, the Hermitian property is necessary in order for the measured values (eigenvalues) to be constrained to real numbers. if Q is Hermitian, then all qi are real numbers

34. 4. Basis Set Postulate The set of eigenfunctions of Hermitian operators Q will form a complete set of linearly independent functions. The set of functions Ψj which are eigenfunctions of the eigenvalue equation form a complete set of linearly independent functions. They can be said to form a basis set in terms of which any wavefunction representing the system can be expressed: This implies that any wavefunction Ψrepresenting a physical system can be expressed as a linear combination of the eigenfunctions of any physical observable of the system.

35. 5. Expectation Value Postulate For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction. For a physical system described by a wavefunction Ψ, the expectation value of any physical observable q can be expressed in terms of the corresponding operator Q as follows: It is presumed here that the wavefunction is normalized and that the integration is over all of space. This postulate follows along the lines of the operator postulate and the basis set postulate. The function can be represented as a linear combination of eigenfunctions of Q, and the results of the operation gives the physical values times a probability coefficient. Since the wavefunction is normalized, the integral gives a weighted average of the possible observable values.

36. A physical system is described by the wave function Ψ, which can always be written as a linear combination of the eigenfunctions of a Hermitian operator Q: A measure of Q for the state Ψwill give as a result any of its eigenvalues qn,each with a probability |cn|2,so that The normalization condition of the wavefunction implies that A measurement of Q forces the system tobe in one of the eigenstates, Ψn, of Q: any subsequent measure of Q will give the result qn

37. 6. Time Evolution Postulate The time evolution of the wavefunction is given by the time dependent Schrödinger equation If Ψ(x,y,z; t) is the wavefunction for a physical system at an initial time and the system is free of external interactions, then the evolution in time of the wavefunction is given by where H is the Hamiltonian operator formed from the classical Hamiltonian by substituting for the classical observables their corresponding quantum mechanical operators. The role of the Hamiltonian in both space and time is contained in the Schrödinger equation.

38. 1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system. 2. With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction. 3. Any operator Q associated with a physically measurable property q will be Hermitian 4. The set of eigenfunctions of each Hermitian operator Q will form a complete set of linearly independent functions. 5. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction 6. The time evolution of the wavefunction is given by the time dependent Schrödinger equation.

39. Free particle approach to the Schrödinger equation Though the Schrodinger equation cannot be derived, it can be shown to be consistent with experiment. The most valid test of a model is whether it faithfully describes the real world. The wave nature of the electron has been clearly shown in experiments like the Davisson-Germer experiment. This raises the question "What is the nature of the wave?". We reply, in retrospect, that the wave is the wavefunction for the electron. Starting with the expression for a traveling wave in one dimension, the connection can be made to the Schrödinger equation. This process makes use of the deBroglie relationship between wavelength and momentum and the Planck relationship between frequency and energy.

40. It is easier to show the relationship to the Schrödinger equation by generalizing this wavefunction to a complex exponential form using the Euler relationship. This is the standard form for the free particle wavefunction. One can check that Ψ is eigenunction of momentum and energy operators The connection to the Schrodinger equation can be made by examining wave and particle expressions for energy Asserting the equivalence of these two expressions for energy and putting in the quantum mechanical operators for both brings us to the Shrödinger equation

41. The Uncertainty Principle The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time This is not a statement about the inaccuracy of measurement instruments, nor a reflection on the quality of experimental methods; it arises from the wave properties inherent in the quantum mechanical description of nature. Even with perfect instruments and technique, the uncertainty is inherent in the nature of things.

42. Uncertainty Principle Important steps on the way to understanding the uncertainty principle are wave-particle duality and the DeBroglie hypothesis. As you proceed downward in size to atomic dimensions, it is no longer valid to consider a particle like a hard sphere, because the smaller the dimension, the more wave-like it becomes. It no longer makes sense to say that you have precisely determined both the position and momentum of such a particle.

43. The exact definition of Δx and Δp is:

44. Particle Confinement

45. Confinement Calculation

46. The Hydrogen Atom The solution of the Schrodinger equation for the hydrogen atom is better achieved by using spherical polar coordinates and by separating the variables so that the wavefunction is represented by the product: The separation leads to three equations for the three spatial variables, and their solutions give rise to three quantum numbers associated with the hydrogen energy levels.

47. Quantum Numbers from Hydrogen Equations The hydrogen atom solution requires finding solutions to the separated equations which obey the constraints on the wavefunction. The solution to these equations can exist only when a few constants which arise in the solution are restricted to integer values. This gives the hydrogen atom quantum numbers: n = principal quantum number l = orbital quantum number ml = magnetic quantum number

48. Vector Model for Orbital Angular Momentum The orbital angular momentum for an atomic electron can be visualized in terms of a vector model where the angular momentum vector is seen as precessing about a direction in space. While the angular momentum vector has the magnitude shown, only a maximum of l units of ħ can be measured along a given direction, where l is the orbital quantum number. While called a "vector", the orbital angular momentum in Quantum Mechanics is a special kind of vector because its projection along a direction in space is quantized to values one unit of angular momentum (ħ) apart. The diagram shows that the possible values for the "magnetic quantum number" ml (for l =2), can take the values

49. Electron Spin An electron spin s = 1/2 is an intrinsic property of electrons. In addition to orbital angular momentum electrons have intrinsic angular momentum characterized by quantum number 1/2. In the pattern of other quantized angular momenta ms= ½ “spin up” ms= – ½ “spin down” Spin "up" and "down" allows two electrons for each set of spatial quantum numbers

50. Pauli Exclusion Principle No two electrons in an atom can have identical quantum numbers. This is an example of a general principle which applies not only to electrons but also to other particles of half-integer spin (fermions). It does not apply to particles of integer spin (bosons). The nature of the Pauli exclusion principle can be illustrated by supposing that electrons 1 and 2 are in states a and b respectively. The wavefunction for the two electron system would be but this wavefunction is unacceptable because the electrons are identical and indistinguishable. To account for this we must use a linear combination of the two possibilities since the determination of which electron is in which state is not possible to determine.