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Physical Chemistry Why Study Quantum Mechanics? C HAPTER 8 - Q UANTUM T HEORY: I NTRODUCTION AND P RINCIPLES Classical physics is unable to explain why • an atom with a positively charged nucleus surrounded by electrons is stable. • graphite conducts electricity but diamond does not, why the light emitted by a hydrogen discharge lamp appears only at a small number of wavelengths, and why the bond angle in H2 O is different from that in H2 S. Professor Angelo R. Rossi Department of Chemistry Chemistry is a molecular science, and the goal of chemists is to understand macromolecular behavior in terms of the properties of individual molecules. For example, H2 is a good fuel Spring Semester H2 (g) + O2 (g) −→ H2 O (g) because the energy required to break the H-H and O-O bonds is much less than the energy that is released in forming two O-H bonds. Quantum Chemistry will provide the underlying explanation for the differences in bond strengths of these molecules. Spring Semester Last Updated: June 15, 2010 at 8:40am Quantum Mechanics: an Interplay of Experiment and Theory Distinguishing Between Classical Physics and Quantum Mechanics At the end of the 19th century, Experimental evidence points to two key properties that distinguish classical physics from quantum physics: Maxwell’s electromagnetic theory unified existing knowledge in the areas of electricity, magnetism and waves; and the well-established field of classical mechanics described the motion of particles. 1. quantization: energy at the atomic level is not a continuous variable but comes in discrete packets called quanta. But, a number of key experiments showed that the predictions of classical physics were inconsistent with experimental outcomes, and this stimulated scientists to formulate quantum mechanics. Spring Semester Last Updated: June 15, 2010 at 8:40am 2 2. wave-particle duality: at the atomic level, light waves have particle like properties, and atoms as well as electrons have wave-like properties. 3 Spring Semester Last Updated: June 15, 2010 at 8:40am 4 Blackbody Radiation Blackbody Radiation: Spring Model of a Metal A red-hot block of metal, with a spherical cavity in its interior emanates radiation through a hole small enough that the conditions inside the block are not perturbed. An ideal blackbody is shown below Spring Semester Last Updated: June 15, 2010 at 8:40am 5 Spring Semester Last Updated: June 15, 2010 at 8:40am Blackbody Radiation: Predictions from Classical Theory Blackbody Radiation: Predictions from Classical Theory Under the conditions of equilibrium between the radiation field inside the cavity and the glowing piece of matter, classical electromagnetic theory can predict the what frequencies (ν) of light are radiated and their relative magnitudes. Classical theory predicts that the average energy of an oscillator is simply related to the temperature: The spectral density is the energy at frequency ν per unit volume and unit frequency stored in the electromagnetic field of the blackbody radiator and is given by the equation: 8πν 2 ρ(ν, T )dν = 3 Ēosc dν c 6 Ēosc = kB T where kB is the Boltzmann constant. Combining the two equations results in an expression for ρ(ν, T )dν, • ρ, the spectral density in units of E × V −1 × ν −1 , is a function of the temperature (T ) and the frequency (ν). 8πkB T ν 2 dν c3 the amount of energy per unit volume in the frequency range between ν and ν + dν within a blackbody at temperature T ρ(ν, T )dν = • The speed of light is c, and Ēosc is the average energy of an oscillating dipole in the solid. • The factor dν provides the energy density observed in the frequency interval of width dν centered at frequency ν. The basis of the model is that the atomic nuclei and corresponding electrons act as oscillating dipoles. Spring Semester Last Updated: June 15, 2010 at 8:40am 7 Spring Semester Last Updated: June 15, 2010 at 8:40am 8 Blackbody Radiation: Experimental Results Blackbody Radiation: Quantum Hypothesis It is possible to measure the spectral density of the radiation emitted by a blackbody as shown in the adjacent figure. The first person to offer a successful explanation for blackbody radiation was the German physicist Max Planck in 1900. The experimental curves have a common behavior: Planck used a quantum hypothesis to derive the blackbody radiation law. • Implicit in the classical theory of radiation is the assumption that the energies of the electronic oscillators responsible for the emission of radiation could have any value whatsoever. 1. The spectral density is peaked in a broad maximum and falls off to higher and lower frequencies. 2. The shift of the maxima to higher frequencies with increasing temperature is consistent with our experience that if a block of metal is heated to higher temperatures, the color will change from red to yellow to blue (i.e a trend toward increasing frequency and shorter wavelength). • Planck found that he could obtain agreement with theory and experiment only if he assumed that the energies of the oscillators were discrete and proportional to an integral multiple of the frequency E = nhν 3. The two curves show similar behavior at low frequencies (longer wavelengths), but the theoretical curve keeps on increasing with higher frequencies (shorter wavelengths). where h is Planck’s constant initially an unknown proportionality constant, and n is a positive integer (n = 0, 1, 2, . . . ). • For a given ν, energy is quantized. It is clear that the theoretical prediction is incorrect!!! Spring Semester Last Updated: June 15, 2010 at 8:40am 9 Spring Semester Last Updated: June 15, 2010 at 8:40am Blackbody Radiation: Quantum Hypothesis The Photoelectric Effect: Experimental Observations Planck obtained the following relationship for a quantum oscillator: The ejection of electrons from the surface of a metal by radiation is called the photoelectric effect. hν Ēosc = hν e kB T − 1 Incident light on a copper plate held in a vacuum can be absorbed, leading to the excitation of electrons into unoccupied energy levels. Using the equation, ρ(ν, T )dν = 8πν 2 Ēosc dν c3 Planck obtained the following general formula for the spectral radiation density from a blackbody: 8πhν 3 × ρ(ν, T ) dν = c3 1 hν Sufficient energy can be transferred to the electrons such that some of them leave the metal and are ejected into the vacuum and collected. dν e kB T − 1 The following trend is obtained: • At higher temperatures, hν ( kT << 1), the term e and Ēosc → kB T . The leads to ρ(ν, T )dν = predictions. hν kB T According to conservation of energy, the absorbed light energy must be balanced by the kinetic energy of the emitted electrons and the energy required to eject an electron at equilibrium. can be expanded in a Taylor-Mclaurin series, 8πkB T ν 2 dν c3 10 and approximately follows classical hν • At lower temperatures, ( kT >> 1), the denominator in the above equation becomes very large, and Ēosc → 0 which produces a drop in the spectral density. Last Updated: June 15, 2010 atthe 8:40am Spring Semester Thus, Planck’s Quantum Hypothesis seems to accurately predict behavior of blackbody radiation. 11 Spring Semester Last Updated: June 15, 2010 at 8:40am 12 The Photoelectric Effect: Experimental Observations The Photoelectric Effect: Predictions of Classical Theory 1. The kinetic energy per electron increases with the light intensity. 1. The number of emitted electrons is proportional to the light intensity, but their kinetic energy is independent of the light intensity. 2. Any one electron can absorb only a small fraction of the incident light. 3. Light is incident as a plane wave over the entire metal surface and is absorbed by many electrons in the solid. 2. No electrons are emitted unless the frequency ν is above a threshold frequency ν0 even for high light intensities. 4. Electrons are emitted for all light frequencies, provided that the light is sufficiently intense. 3. Electrons are emitted even at such low intensities such that all the light absorbed by the entire metal surface is barely enough to eject a single electron based on energy conservation considerations.. 4. The kinetic energy of the emitted electrons depends linearly on the frequency after the threshold frequency has been attained. Spring Semester Last Updated: June 15, 2010 at 8:40am 13 Spring Semester Last Updated: June 15, 2010 at 8:40am The Photoelectric Effect: Quantum Hypothesis Louis de Broglie: Matter Has Wavelike Properties Planck applied the concept of energy quantization (E = nhν) to the emission and absorption mechanism of the atomic electronic oscillators but believed that once the light energy was emitted, it behaved like a classical wave. de Broglie reasoned that, if light can display wave-particle duality, then matter which appears particle-like, might also display wave-like properties under certain conditions. In 1905, Einstein proposed that the radiation itself existed as small packets of energy (E = hν) now known as photons. Einstein had shown from relativity theory that the wavelength, λ, and the momentum, p, of a photon are related by Einstein showed that the kinetic energy (KE) of an ejected electron is equal to the energy of the incident photon (hν) minus the minimum energy required to remove an electron from the surface of a particular metal (Φ). KE = λ= 1 mv 2 = hν − Φ 2 Because the momentum of a particle is given by mv, the above equation predicts that for a particle of mass, m moving at velocity v will have a de Broglie wavelength given by The minimum frequency that will eject an electron is just the frequency to overcome the workfunction of the metal: hν0 = Φ. Combining the above two equations gives KE = hν − hν0 where ν > ν0 . λ= Last Updated: June 15, 2010 at 8:40am h p de Broglie argued that both light and matter obey this equation. where Φ is called the work function. Spring Semester 14 15 Spring Semester h mv Last Updated: June 15, 2010 at 8:40am 16 de Broglie Wavelength of the Electron de Broglie Wavelength of the Electron What is the de Broglie Wavelength λ of an electron that has been accelerated through a potential difference of 100 V? The energy of the electron is (1.602x10−19 C)(100V) = 1.602x10−17 J Thus, the momentum is p p = (2)(9.109 × 10−31 kg)(1.602 × 10−17 J) = 5.403 × 10−24 kg m s−1 The momentum after the acceleration process must be calculated first. This can be done by first calculating the energy of the electron. The energy of an electron of mass m moving with a velocity v well below the velocity of light is given by and the wavelength is 1 p2 E = mv 2 = 2 2m λ= λ = 1.226 × 10−10 m = 0.1226 m Thus, the momentum is given by p= Spring Semester 6.626 × 10−34 J s h = p 5.403 × 10−24 kg m s−1 √ Note that λ is of the same order of magnitude as the distance between the atoms in a crystal 2mE Last Updated: June 15, 2010 at 8:40am 17 Spring Semester Last Updated: June 15, 2010 at 8:40am The Atomic Spectrum of Hydrogen The Atomic Spectrum of Hydrogen For some time scientists had known that each atom has a characteristic line spectrum when subjected to high temperatures or electrical discharge. The atomic spectrum of hydrogen consists of several series of lines. 18 • The atomic spectra consist of only discrete frequencies and are called line spectra. • Hydrogen has the simplest spectrum because of its simplicity. The Swiss spectroscopist Johannes Rydberg accounted for all the lines in the hydrogen atomic spectrum by obtaining the equation 1 1 1 − = RH λ n21 n22 where RH is the Rydberg constant whose value is 109667.57 cm−1 . Hydrogen Discharge Tube Spring Semester Last Updated: June 15, 2010 at 8:40am 19 Spring Semester The Line spectrum of Hydrogen Last Updated: June 15, 2010 at 8:40am 20 What Determines if a System Needs to be Described Using Quantum Mechanics The Atomic Spectrum of Hydrogen It is important to realize that classical mechanics and quantum mechanics are not two competing ways to describe the world around us. Each theory has its usefulness in a different regime of physical properties that describe reality: quantum mechanics merges seamlessly into classical mechanics in the limit in which allowed energy values are continuous rather than discrete. It is not correct to state that whenever one talks about atoms, a quantum mechanical description must be used. The above image is shows • a continuous spectrum (top), • an absorption spectrum (middle), • and an emission spectrum (bottom), for some of the Balmer lines of hydrogen, i.e., those that lie within the visible part of the spectrum. Spring Semester Last Updated: June 15, 2010 at 8:40am 21 What Determines if a System Needs to be Described Using Quantum Mechanics Spring Semester Last Updated: June 15, 2010 at 8:40am 22 First Criterion for Classical or Quantum Mechanical Behavior: Magnitude of Particle Wavelength Relative to Dimensions of Problem. For example, the origin of pressure for a container filled with argon atoms at low pressure can be described by the collision of rapidly and randomly moving argon atoms with the container walls. This is the first criterion which is used to provide an understanding for when a particle or classical description of an atomic or molecular system is sufficient, or for when a wave or quantum mechanical description must be used. Classical mechanics provides a perfectly good description of the origin of pressure of a single argon atom colliding with the wall. For the diffraction of light of wavelength λ passing through a slit of width a, diffraction is observed only when the wavelength is comparable to the width of the slit: λ << a. However, if light of very short wavelength is passed through this same gas, a quantum mechanical description must be used to describe how much energy can be taken up by the argon atom. For the H2 molecule at room temperature, λ = The essence of quantum mechanics is that particles and waves are not really separate and distinct entities. Waves can show particle-like behavior. The localization of an electron to a small volume around the nuclei brings out the wave-like behavior of the electrons and protons making up a molecule. h p ≈ 1 × 10−10 m. Also, crystalline solids have atomic spacings that are appropriate for the diffraction of electrons as well as light atoms and molecules. Spring Semester Last Updated: June 15, 2010 at 8:40am 23 Spring Semester Last Updated: June 15, 2010 at 8:40am 24 Second Criterion for Classical or Quantum Mechanical Behavior: Degree to which Allowed Energies Form a Continuous Energy Spectrum The Boltzmann Distribution Because all values of the energy are allowed for a classical system, it has a continuous energy spectrum. In a quantum mechanical system, only certain values of the energy are allowed, and it has a discrete energy spectrum. Consider a one-liter container filled with an ideal gas of atoms at a pressure of 1 bar and temperature of 298.15 K. The broad distribution of kinetic energy in the gas is for the individual molecules is given by the Boltzmann Distribution: ni gi − (ǫi −ǫj ) kT = e nj gj which states that the number of atoms with energy ǫi relative to the number with energy ǫj depends on • The atoms have no rotational or vibrational degrees of freedom, and all of the energy is in the form of translational kinetic energy. 1. the difference in energies with an exponential dependence. The ratio of differences in energies relative to kT , the average energy that an atom at temperature T , is important. Fraction of Molecules • At equilibrium, there is a broad distribution of energies of atoms at p = 1 bar and T = 298.15 K. 3. the ratio of degeneracies of the two states, Lower Temperature . If kT is small compared to the spacing between allowed energies, the distribution of states of energy will be very different from those of classical mechanics where a continuous energy spectrum is present. It kT is much larger than the energy spacing, classical and quantum mechanics will give the same result for the relative numbers of atoms or molecules of different energy. 3 kT 2 Last Updated: June 15, 2010 at 8:40am gi gj The degeneracy of an energy value is the number of ways that an atom can have an energy ǫi within the interval ǫ − ∆ǫ < ǫi < ǫ + ∆ǫ Kinetic Energy • The root mean square energy is related to the absolute temperature by Ē = Spring Semester 2. the temperature with an exponential dependence. Higher Temperature 25 The Boltzmann Distribution: Quantum vs. Classical Behavior Spring Semester Last Updated: June 15, 2010 at 8:40am 26 The Boltzmann Distribution: Quantum vs. Classical Behavior A quantum mechanical system has a discrete energy spectrum with allowed values of energy called energy levels. For ∆E ≈ kT , each discrete energy level is sufficiently broadened by energy fluctuations that adjacent energy levels can no longer be distinguished. For a molecule containing vibrational energy levels, the allowed energy levels are equally space with interval, ∆E, and these energy levels are numbered with integers. In this limit, any energy that is chosen lies within the yellow area: the energy difference corresponds to an allowed value, and the discrete energy spectrum appears continuous. In a gas at equilibrium, the total energy of an individual molecule will fluctuate within a range of ∆E = kT through collisions of molecules with one another. Classical behavior will be observed under these conditions. Therefore, the energy of a molecule with a particular vibrational quantum number will fluctuate within a range of width kT . A plot of the relative number of molecules having a vibrational energy E as a function of E is shown in the adjacent figure for sharp energy levels. Spring Semester Last Updated: June 15, 2010 at 8:40am 27 Spring Semester Last Updated: June 15, 2010 at 8:40am 28 The Boltzmann Distribution: Quantum vs. Classical Behavior The Heisenberg Uncertainty Principle Classical mechanics does not involve any limitations on the accuracy with which observables can be measured. If ∆E >> kT , an arbitrarily chosen energy withing this range will lie within the blue area with high probability, because the yellow bars are widely separated. For example, the position and momentum of a classical particle may be simultaneously measured to an any desired accuracy. The blue area corresponds to forbidden energies, and the discontinuous nature of the energy spectrum will be observed. In 1927 Heisenberg formulated the principle that values of particular pairs of observables cannot be determined simultaneously with arbitrarily high precision in quantum mechanics. Quantum mechanical behavior will be observed under these conditions. Examples of pairs of observables that are restricted in this way are • momentum and position • energy and time Such pairs are referred to as complementary. Spring Semester 29 Last Updated: June 15, 2010 at 8:40am Spring Semester Last Updated: June 15, 2010 at 8:40am The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle de Broglie Relation The de Broglie wave for a particle is made up of a superposition of an infinitely large number of waves of the form Quantitative expressions for the Heisenberg uncertainty principle can be derived by combining the de Broglie relation (p = h/λ) and the Einstein relation (E = hν). ψ(x, t) = A sin 2π x λ − νt Substituting the de Broglie relation into the previous expression for the uncertainty principle gives the following for motion in the x-direction: where A is the amplitude, and λ is the wavelength. The waves that are added together have infinitesimally different wavelengths. The superposition of waves produces a wave packet as shown in the adjacent figure. It is possible to show that for wave motion of any type ∆x∆ then p 1 x ≥ h 4π ~ h ∆x∆px ≥ ~= 2 2π To determine the position of an electron, at least one photon would have to strike the electron, and the momentum would inevitably be altered by this process. ∆x∆ 1 4π where ∆x is the extent of the wave packet in space, λ the range of wavelengths, ∆ν the range of frequencies, and ∆t is the measure of the time required for the packet to pass a given point. ∆ is the above equations actually represents standard deviations. Spring Semester 1 px = λ h 1 1 ≥ λ 4π ∆t∆ν ≥ 30 Superposition of waves to give But, this process would limit our ability to measure the momentum of the electron. (a) a weakly localized and (b) a strongly packet. Last Updated: June 15, 2010 at 8:40am Photons of shorter wavelength would disturb the momentum to an even greater extent localized 31 Spring Semester Last Updated: June 15, 2010 at 8:40am 32 The Heisenberg Uncertainty Principle Experimental Error The Heisenberg Uncertainty Principle Lifetimes of Excited states Another form of the Heisenberg uncertainty principle may be derived by 1 substituting E = hν into the equation ∆t∆ν ≥ 4π 1 E ≥ ∆t∆ h 4π ~ 2 Suppose that excited atoms emit electromagnetic radiation in going to a lower state: ∆t∆E ≥ It is important to realize that the uncertainties in the equations for the Heisenberg uncertainty principle are NOT experimental errors that are dependent on the quality of the measuring apparatus but are inherent in quantum mechanics. 1. If these excited atoms live a long time, the radiation will be nearly monochromatic, and the spectral line will be sharp. 2. If the excited atoms have a very short life, the electromagnetic radiation will have a broader range of frequencies. Spring Semester Last Updated: June 15, 2010 at 8:40am 33 Spring Semester Last Updated: June 15, 2010 at 8:40am Quantum Mechanical Waves and the Schrödinger Equation Quantum Mechanical Waves and the Schrödinger Equation The time-independent Schrödinger equation can be justified by combining classical standing waves with the de Broglie relation. Quantum mechanics is introduced by using the de Broglie relation, λ = h/p for the wavelength: For classical standing waves, the wave function is a product of two functions: one which depends on spatial coordinates; the other depends only on time: The momentum is related to the total energy, E and the potential energy, V (x) by p2 = E − V (x) 2m Ψ(x, t) = ψ(x) cos ωt p 2m(E − V (x)) d2 ψ(x) 8π 2 m + [E − V (x)] ψ(x) = 0 dx2 h2 Using the abbreviation ~ = h/2π and rewriting the above equation, we obtain d2 ψ(x) 4π 2 + 2 ψ(x) = 0 dx2 λ − ~2 d2 ψ(x) + V (x)ψ(x) = Eψ(x) 2m dx2 which is the time-independent Schrödinger equation in one dimension. All of the above pertains to a classical wave. Last Updated: June 15, 2010 at 8:40am p= Introducing the above expression for momentum into the de Broglie relation and substituting the expression for λ into the wave equation yields If the above function is substituted into the classical wave equation, the following is obtained d2 ψ(x) ω 2 + 2 ψ(x) = 0 dx2 v Using the relations ω = 2πν and νλ = v, where v is the velocity, ν the frequency, and λ the wavelength of the wave, the above equation then becomes Spring Semester or 34 35 Spring Semester Last Updated: June 15, 2010 at 8:40am 36 The Born Interpretation of the Wave Function The Physical Meaning Associated with the Wave Function The Born interpretation of the wave function focuses on the square of the wave function (ψ 2 ) or the square of the modulus: The association of the wave function with probability has an important consequence called normalization. • The probability that a particle is found in an interval of width dx centered at the position x must lie between 0 and 1. | ψ |2 = ψ ∗ ψ if ψ is complex. • The sum of the probabilities over all intervals accessible to the particle is 1, because the particle is somewhere in this range. This states that the value of | ψ |2 at a point is proportional to finding the particle at that point. Consider a particle that is confined to a one-dimensional space of infinite extent which leads to the following normalization condition: Z ∞ ψ ∗ (x)ψ(x)dx = 1 | ψ |2 is a probability density and must be multiplied by an infinitesimal region dx to obtain the probability | ψ |2 dx −∞ Such a definition is meaningless if the integral did not exist. Therefore, ψ(x) must satisfy important mathematical conditions to ensure that it represents a possible physical state. The wave function ψ is called the probability amplitude and does not have any physical significance. Spring Semester Last Updated: June 15, 2010 at 8:40am 37 The Physical Meaning Associated with the Wave Function Spring Semester Last Updated: June 15, 2010 at 8:40am 38 Operators Another basic quantum mechanical concept is an operator which is a mathematical operation that is applied to a function. 1. The wave function must be a single valued function of the spacial coordinates. If this were not the case a particle would have more than one probability of being found in the same interval. ∂ • For example, ∂x is the operator that indicates that the function is to be differentiated with respect to x. • x̂ is the operator that indicates the function is to be multiplied by x. 2. The first derivative of the wave function must be continuous so that the second derivative exists and is well-behaved. If this were not the case, we could not set up the Schrödinger equation. • Operators are designated with a caret, as in  or Ĥ. The three-dimensional Schrödinger equation can be written in operator form: ~2 2 − ∇ + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z) 2m {z } | 3. The wave function cannot have an infinite amplitude over a finite interval. If this were the case, the wave function could not be normalized. Ĥ where For example, the function ψ(x) = tan 2πx , 0 ≤ x ≤ a cannot be a normalized. ∇2 = ∂2 ∂2 ∂2 + 2+ 2 ∂x2 ∂y ∂z The quantity in square brackets is called the Hamiltonian operator. Spring Semester Last Updated: June 15, 2010 at 8:40am 39 Spring Semester Ĥψ(x, y, z) =June Eψ(x, Last Updated: 15, 2010y, atz) 8:40am 40 Operators, Eigenfunctions, and Eigenvalues Classical Functions to Quantum Mechanical Operators Suppose an operator, e.g.  operating on a function, e.g. ψn , yields a constant, an multiplied by that function For a particle of mass m moving in one dimension subject to a potential energy V (x), the classical Hamiltonian function is Âψn = an ψn H= The appropriate statement is that ψn is an eigenfunction of operator  with an eigenvalue of an . There is no caret over H because this is a classical function. The above equation should be compared with the corresponding quantum mechanical equation. ~2 ∂ 2 − + V (x) ψ(x) = Eψ(x) 2m ∂x2 Thus, for the one-dimensional Schrödinger equation, ψ(x) is an eigenfunction of Ĥ with eigenvalue E. Ĥψn (x) = En ψn (x) Spring Semester Last Updated: June 15, 2010 at 8:40am 41 Spring Semester Last Updated: June 15, 2010 at 8:40am 42 Classical Mechanical Observables and Corresponding Quantum Mechanical Operators Classical Functions to Quantum Mechanical Operators The process of converting a function for a classical system to the corresponding operator for the quantum mechanical system is formalized by the following rules: One-Dimensional Systems Classical Observables QM Operators Name Symbol Symbol Operation Position x x̂ Multiply by x Position Squared x2 x̂2 Multiply by x2 ∂ Momentum px p̂x −i~ ∂x 2 2 2 ∂2 Momentum Squared px p̂x −~ ∂x2 p2x ~2 ∂ 2 Kinetic Energy Tx = 2m Tˆx − 2m ∂x2 Potential Energy V (x) V̂ (x) Multiply by V (x) ~2 ∂ 2 + V (x) Total Energy Tx + V (x) Ĥ − 2m ∂x2 1. Each Cartesian coordinate in the Hamiltonian function is replaced by that coordinate and a multiplication operator. q̂ → q× 2. Each Cartesian component of a linear momentum, pq , q = x, y, z, in the Hamiltonian function is replaced by the operator p̂q = −i~ p2 + V (x) 2m ∂ ∂q 3. The potential energy function is not changed because of the first statement given above: V̂ (q̂) =→ V (q)× Spring Semester Last Updated: June 15, 2010 at 8:40am 43 Spring Semester Last Updated: June 15, 2010 at 8:40am 44 Classical Mechanical Observables and Corresponding Quantum Mechanical Operators Operators and Observables Each classical observable is associated with a quantum mechanical operator. Three-Dimensional Systems Classical Observables QM Operators Name Symbol Symbol Operation(a) Position r r̂r Multiply by r ∂ ∂ ∂ + j ∂y + k ∂z Momentum p p̂p −i~ i ∂x Kinetic Energy Potential Energy Total Energy Angular Momentum T V (x, y, z) E =T +V lx = ypz − zpy ly = zpx − xpz T̂ V̂ (x, y, z) Ĥ ˆlx ˆlx ˆlx lz = zpy − ypx 2 ∂ (a) ∇2 = ∂x 2 + Spring Semester ∂2 ∂y 2 + ∂2 ∂z 2 Another postulate of quantum mechanics is that the only possible measured values of an observable are the eigenvalues of the operator representing that observable. ~ ∇2 − 2m Multiply by V (x, y, z) ~2 2 − 2m ∇ + V (x, y, z) 2 It is a postulate of quantum mechanics that the average value < a > of an observable corresponding to an operator  is given by Z < a >= ψ ∗ Âψdτ ∂ ∂ −i~ y ∂z − z ∂y ∂ ∂ −i~ z ∂x − x ∂z where ψ ∗ is the complex conjugate of ψ. ∂ ∂ − y ∂x −i~ x ∂y Last Updated: June 15, 2010 at 8:40am A complex complex conjugate of a complex function is obtained by replacing i with −i. 45 Spring Semester Last Updated: June 15, 2010 at 8:40am Operators and Observables Hermetian Operators, Eigenfunctions, and Eigenvalues Taking the energy of a one-dimensional system as an example of an observable, energy eigenvalues can be obtained from the Schrödinger equation: Ĥψn (x) = En ψn (x) Operators and wave functions may be complex, but eigenvalues of quantum mechanical operators must be real because they are the only positive measured values. Operators which yield real values are called Hermitian and have the following property: Z Z where n is an index that labels the different eigenfunctions and eigenvalues. Multiplying the above equation by the complex conjugate of the wave function and integrating over all values of x yields Z Z Z ∗ ∗ ψn (x)Ĥψn (x)dτ = ψn (x)En ψn (x)dτ = En ψn∗ (x)ψn (x)dτ = En ψi∗ Âψj dτ = Last Updated: June 15, 2010 at 8:40am ψj (Âψi )∗ dτ for any two well-behaved functions ψi and ψj , where (Âψi )∗ is the complex conjugate of Âψi If ψi = ψj = ψ and an eigenfunction of Â, then Âψ = aψ and Z Z ψ ∗ Âψdτ = a; ψ(Âψ)∗ dτ = a∗ since the wave functions are normalized. Spring Semester 46 But, the Hermetian property requires that a = a∗ , proving that the eigenvalues are real. 47 Spring Semester Last Updated: June 15, 2010 at 8:40am 48 Hermetian Operators, Eigenfunctions, and Eigenvalues Commuting and Non-commuting Operators Eigenfunctions of an Hermitian operator corresponding to different eigenvalues are orthogonal. If As discussed on a previous slide, classical physics predicts that there is no limit to the amount of information (or observables) that can be known about a system at a given instant of time. Âψn = an ψn then Z and Âψm = am ψm This is not the case in quantum mechanics. • Two observables can be known simultaneously only if the outcome of the measurements is independent of the order in which they were measured. ψn∗ ψm dτ = 0 m 6= n Since the ψm can always be normalized, they form an orthonormal set of functions: Z ψn∗ ψm dτ = δmn • An uncertainty relation limits the degree to which observables of non-commuting operators can be known simultaneously. where δmn is the Kronecker delta which is defined by ( 0 for m 6= n δmn = 1 for m = n Spring Semester Last Updated: June 15, 2010 at 8:40am 49 Spring Semester Last Updated: June 15, 2010 at 8:40am Commuting and Non-commuting Operators Commuting and Non-commuting Operators The values of two different observables α and β, which correspond to two operators  and B̂, can be simultaneously determined only if the measurement process does not change the system. The only case in which the measurement does not change the state of the system is if ψn (x) is simultaneously an eigenfunction of  and B̂. B̂[Âψn (x)] = αn B̂ψn (x) = αn βn ψn (x) Otherwise the system on which the two measurements are carried out is not the same. Â[B̂ψn (x)] = βn Âψn (x) = βn αn ψn (x) Let ψn (x) be the wave function that characterizes the system, but possibly not an eigenfunction of the operators  and B̂. Because the eigenvalues αn and βn are constants, βn αn ψn (x) = αn βn ψn (x) Now perform a measurement of the observables corresponding to the first operator  and subsequently to the operator B̂ which is equivalent to B̂[Âψn (x)] = αn B̂ψn (x) and [ÂB̂ − B̂ Â]ψn (x) = 0 We have just shown that the act of measurement changes the state of the system unless the system wave function is simultaneously an eigenfunction of two different operators. and Â[B̂ψn (x)] = βn Âψn (x) Spring Semester Last Updated: June 15, 2010 at 8:40am 50 51 Spring Semester Last Updated: June 15, 2010 at 8:40am 52 Commuting and Non-commuting Operators Symmetry: Commuting and Non-commuting Operators Two operators commute, if they have a common set of eigenfunctions The Hamiltonian, Ĥ, for a molecule is unchanged under other symmetry operation of a group, because the total energy remains the same. Â[B̂f (x)] − B̂[Âf (x)] Thus, Ĥ belongs to the totally symmetric representation. If a molecule that contains a plane of symmetry undergoes the reflection symmetry operation (σ̂), is indistinguishable from the original molecule. which is abbreviated as [Â, B̂]f (x) Ĥ and σ̂ commute because the order of operating on the wave function is unimportant. The expression in square brackets [ ] is called the commutator of operators  and B̂. [Ĥ, σ̂] = Ĥ σ̂ − σ̂ Ĥ Thus, eigenfunctions of Ĥ can be found that are simultaneously eigenfunctions of σ̂ and all other symmetry operarions of the point group. If the value of the commutator is not zero for an arbitrary function f (x), then the corresponding observables cannot be determined simultaneously and exactly. Spring Semester Symmetry-adapted MOs which exhibit the symmetry of the molecule can be obtained. 53 Last Updated: June 15, 2010 at 8:40am Expectation Values and Superposition Spring Semester 54 Last Updated: June 15, 2010 at 8:40am Expectation Values and Superposition The average value of  is given by The only possible measured values of an observable are the eigenvalues of the operator representing that observable. < a >= The average value of an observable, Â, when the system is in the state ψ is Z < a >= ψ ∗ Âψdτ Z X n cn φ n !∗  X cm φ m m Then < a >= X n ! dτ = XX n c∗n cm m Z φ∗n Âφm dτ |cn |2 an In most cases, ψ is not an eigenfunction of Â, but it can always be written as a linear superposition of the eigenfunctions of Â: X cn φn ψ= Thus, the average value < a > is the sum of possible measured values (an ) multiplied by |cn |2 , a non-negative number. |cn |2 is the probability of measuring the value an . where the cn are constants. The probability of measuring the eigenvalue am is given by cm = n Spring Semester m n Last Updated: June 15, 2010 at 8:40am φ∗m ψdτ = Z φ∗m X cn φn dτ n 2 Z | cm |2 = φ∗m ψdτ Since the φn are orthonormal, ψ has to be normalized ! !∗ Z Z Z X X X X |cn |2 = 1 c∗n cn φ∗n φn dτ = cm φm dτ = cn φn ψ ∗ ψdτ = n Z Note that the probability amplitude may be complex, but the probability | cm |2 is always real. When ψ is an eigenfunction of Â, e.g. φk , then the probability of measuring ak is 1, and the probability of measuring any other eigenvalue is 0. n 55 Spring Semester Last Updated: June 15, 2010 at 8:40am 56 What day is it? A test <?php if ($day == "monday") { $callInSick = true; } else { $callInSick = false; } ?> Spring Semester Last Updated: June 15, 2010 at 8:40am 57