3. Basic Principles of Quantum Mechanics


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1 3. Basic Pinciples of Quantum Mechanics As mentioned in the pevious Chapte, all chemical eactions consist of changes of the electon cloud that suounds the nuclei. It is theefoe, of cental impotance that we ae able to descibe the popeties and behaviou of the electons in chemical systems. It tuns out that classical mechanics (as developed by e.g. Keple, Galilei, and Newton) succeeds in descibing the motion of macoscopic bodies vey accuately but fails to captue the behaviou of the electons that can only be epesented via a quantum mechanical desciption. It is fo this eason that we will discuss the basic pinciples that chaacteize quantum mechanics in this chapte. Quantum mechanics is a elatively new field of physics that was developed at the beginning of the last centuy as the common effot of diffeent scientists (Heisenbeg, Planck, de Boglie, Boh, Schödinge, Bon, Diac and othes). Up to that time, it was possible to descibe all known physical phenomena with the laws of classical physics as eithe paticles (classical mechanics, kinematics) o waves (optics, classical electomagnetism). The discovey of new phenomena such as e.g. the photoelectic effect has lead to the development of a new physics that has evolutionized the way we look at the wold aound us. Many insights fom quantum mechanics ae at fist sight counteintuitive (i.e. they diffe substantially fom what we would expect in a classical pictue!) which makes this field one of the most fascinating aeas of moden physics. The main goal of this chapte is an intoduction to the basic concepts of quantum mechanics so that we can undestand in what way a quantum mechanical pictue diffes fom its classical countepat. This will give us the basis fo an undestanding of the electonic stuctue of atoms that is intoduced in Chapte Useful Mathematical Notions As a fist intoduction, you can find hee a vey shot optional summay of some of the mathematical notions that ae useful fo an undestanding of Chapte 3 (and many othe fields of chemisty!). This is not diectly pat of the mateial of the exam but is meant fo you as a possibility to check and efesh you knowledge in mathematics. With the exception of opeatos, all of the basic mathematical concepts mentioned hee ae pat of the contents defined fo the Swiss matuity exams. Click hee if you want to access the summay of useful mathematical notions. 3. Classical Mechanics: Paticles and Waves All phenomena in classical physics can be descibed eithe as a paticle o a wave. The two following chaptes give you a shot summay as a eminde of the chaacteistic featues of classical paticle motion and wave phenomena. 0
2 3..1 The motion of point paticles In classical mechanics, the motion of a point paticle with mass m, position 0, and velocity v 0 is completely detemined at any instant in time t if we know the total foce f that acts on the paticle. f m 0,v 0 (t),v(t) Fig. 3.1 Motion of a classical point paticle with mass m unde the action of the foce f. The motion of the paticle can be descibed using Newton s law that elates the foce f acting on the paticle to the acceleation a it expeiences: f ma a = dv = dt = d dt v = d dt Fom this we get the equations of motion, i.e. the time evolution of the paticle position and its velocity as 1 ( t) v t at = o v( t) = v a t o + The paticle possess the kinetic enegy given by Ekin = 1 mv E kin can take any positive value E kin 0. We say, the paticle s kinetic enegy is continuous. This leads us to the impotant finding that: The position and the velocity of any classical paticle ae exactly detemined at any given time. Its enegy is a continuous function of the paticle velocity. 1
3 3.. Wave Phenomena Not all phenomena of classical physics ae descibable in a paticle pictue. A second lage class of physical events can be descibed as waves. A wave is a distubance that popagates though space o space and time, often tansfeing enegy. Fig. 3. Suface waves in wate (fom wikipedia) with highs and lows. Fom evey day live, you ae aleady familia with many wave phenomena, such as fo instance wate waves shown in Fig.3. (peiodical vaiations of the quantity of wate molecules) o sound waves (peiodical compessions of ai that tavel though space and can be detected by a mechanosensitive device in you inne ea). Wate waves and sound waves ae examples fo mechanical waves that exist in a medium (which on defomation is capable of poducing elastic estoing foces). On the othe hand, electomagnetic adiations such as visible light ae waves that ae chaacteized by peiodic oscillation of the electomagnetic field (see. Fig.3.3) that can tavel tough vacuum without the need of a tansmitting medium. Fig. 3.3 Electomagnetic wave. The electic field and the magnetic field oscillate at ight angles to each othe and to the diection of popagation (fom wikipedia). Electomagnetic waves ae classified accoding to how many peiodic oscillations they have pe second (i.e. thei fequency). Fig. 3.4 shows the entie span of electomagnetic waves that ange fom highfequency γays that we have aleady encounteed in chapte.5 on adioactivity, to xays, ultaviolet (UV) adiation, the visible spectum, infaed (IR), micowave, to adiowaves (FM and QM) and long adio waves.
4 Fig. 3.4 Spectum of electomagnetic waves (fom wikipedia). Mathematical desciption of the simplest wave: the hamonic oscillation If we attach a mass m to a hamonic sping with sping constant k, the sping is elongated to an equilibium length x 0 fo which the gavitational foce that acts on the mass is exactly compensated by the elastic foce of the sping mg = kx 0 If we pull on the mass and elongate the sping futhe, the estoing foce of the sping is popotional to the elongation Δx = xx 0 f = kδx If we elease the mass, the sping stats to oscillate aound the equilibium length x 0 by peiodic elongations and compessions. If we ecod the instantaneous displacement Δx(t) of the sping as a function of time we find the fom shown in Fig ν Δx x 0 ν Fig. 3.5 Hamonic oscillation of a mass on a sping. 3
5 We can use Newton s law f = ma to find the diffeential equation that detemines Δx(t) d Δx() t m dt = kδ x() t A possible function Δx(t) that fulfils this equation is fo instance a simple tigonometic function Δ xt () = Asin( ωt+ δ ) If we inset this function in the diffeential equation above, we obtain ( ) ( ) maω sin ωt = kasin ωt ω = ω is the angula velocity (i.e. the change of angle [adians] as a function of time) of the oscillation, ω k m δϕ δ t =. A is the amplitude (i.e. the maximal displacement fom the equilibium length), δ is called the phase (the phase detemines at which point the oscillation stats, e.g. in a minimum o at half a peiod etc ) and Δx(t) is called the wavefunction. Othe chaacteistic quantities of a hamonic wave ae its peiod T [s] (the time it takes between two equivalent points. e.g. maxima o minima, of the oscillation). The fequency ν of the oscillation is the numbe of oscillation pe second measued in Hetz (1 Hetz = 1 s 1 ). The fequency ν is elated to the peiod T by 1 υ = T The wavelength λ [m] of the oscillation is the length between two equivalent points. The wave tavels with the velocity (also called goup velocity ) c λ c = = λυ T 4
6 Fo mechanical waves, c (and λ and ν) ae a function of the medium in which the wave tavels, e.g. the velocity of sound in ai at 0 C is 344 m/s, at the same tempeatue, the speed of sound in helium is c = 97 m/s. Electomagnetic waves can tavel in vacuum (without the help of any medium) with the speed of light c = 3x10 8 m/s. Wave Phenomena: Intefeence and diffaction Waves can exhibit phenomena that cannot be explained in a classical paticle pictues. Typical such wave phenomena ae intefeence and diffaction. Two o moe waves can supeimpose and fom a new wave, this effect is called intefeence. When two sinusoidal waves supeimpose, the esulting wavefom depends on the fequency, the amplitude and in paticula on the elative diffeence in phase (the phase shift) of the two waves. If the two waves have the same amplitude A and wavelength the esultant wavefom will have an amplitude between 0 and A depending on whethe the two waves ae in phase (i.e. both waves have thei minima and maxima at exactly the same time) o out of phase (i.e. one wave has a maximum when the othe one has a minimum). Conside two waves that ae in phase, with amplitudes A 1 and A. Thei toughs and peaks line up and the esultant wave will have an amplitude A = A 1 + A. This is known as constuctive intefeence. Constuctive Intefeence Destuctive Intefeence Fig. 3.6 Constuctive and destuctive intefeence of two waves. If the two waves ae π adians, o 180, out of phase, then one wave's cests will coincide with anothe wave's toughs and so will tend to cancel out. The esultant amplitude is A = A 1 A. If A 1 = A, the esultant amplitude will be zeo. This is known as destuctive intefeence. Light o othe waves such as sound waves can bend when they pass aound an edge o though a slit. This bending is called diffaction. The chaacteistic patten of high and low intensity aeas shown in Fig. 3.7 is called diffaction patten. While diffaction always occus, its effects ae geneally only noticeable fo waves whee the wavelength is on the ode of the featue size of the diffacting objects o apetues. 5
7 a Fig. 3.7 Double slit diffaction and intefeence patten (fom wikipedia). If we have two apetues, a paallel wave font that is aiving at these two slides will ceate two cicula waves that as a function of the phase shift will supeimpose constuctively o destuctively. The angula positions of the intensity minima coespond to path length diffeences of an odd numbe of half wavelengths λ asinθ = ( m+ 1) and the coesponding maxima ae at path diffeences of an intege numbe of wavelengths asinθ = λm. 3.3 Quantum Mechanics One of the most impotant findings that lead to the development of quantum mechanics as a new banch of physics was the obsevation that paticles can behave as waves, a phenomena that is not explainable with the laws of classical mechanics that we just evisited. Such a paticlewave dualism is fo instance obseved when a beam of electons (i.e. paticles!) tavels though a small apetue o is eflected by a suface and a diffaction patten esults that is a typical chaacteistic of waves (Fig. 3.8). Consequently, electons can behave at the same time as paticles and waves! Fig. 3.8 Diffaction of electons PaticleWave Dualism One of the cental postulates of quantum mechanics is that evey paticle possesses also a wave chaacte and vice vesa. The wave chaacte is chaacteized by the de Boglie wavelength λ. de Boglie could demonstate that in the nonelativistic limit (i.e. fo velocities v << c (the velocity of light)) the elation h p = mv= λ 6
8 that was oiginally deived fo electomagnetic adiation only is in fact valid fo all matte. The de Boglie equation elates the linea momentum p of a paticle with mass m and velocity v to its wavelength λ. h = 6.63x1034 Js is the Planck constant. The de Boglie elation tells us that the associated wavelength of a moving paticle is indiectly popotional to its mass, i.e. vey light masses have vey long wave lengths. Question 3.1: What is the wavelength of an electon that moves with a velocity of 100 m/s? Answe 3.1: 7.8 μm. De Boglie could also demonstate that the enegy of one quantum of any wave (electomagnetic wave o matte wave) is given by the elation E = hv. A quantum is the smallest, indivisible potion of enegy associated with a wave. Fo instance, in a paticle pictue one can think of light as being composed of photons. A photon is a massless paticle that caies the enegy of one quantum of light given in the equation above. The highe the fequency of a wave the highe its enegy. Question 3.: Which adiation has the highe enegy: micowaves o visible light? Answe 3.: Visible light Enegy Quantization The notion of quanta that we just intoduced in the peceding chapte is a futhe impotant chaacteistic of quantum mechanics, in fact the one that gives it its name! The enegy of a quantum system cannot be divided into abitaily small pats, instead thee is a smallest undividable enegy package called a quantum. We will see that in contast to classical systems whee the accessible enegies ae continuous, quantum systems can often only adopt specific enegy values wheeas othes ae fobidden, i.e. do not exist. This popety is what is descibed by the tem enegy quantization. An expeimental indication fo the existence of quantized systems is the photoelectic effect. 7
9 Fig. 3.9 Photoelectic effect. Upon exposing a metallic suface to ultaviolet o xay adiation, electons ae emitted if the enegy of the incident light is above a given theshold that is chaacteistic of the metallic mateial (Figue 3.9). If the fequency of the light is inceased futhe the kinetic enegy of the emitted electons inceases linealy. The photoelectic effect can only be explained if one assumes that the incident light has a given enegy and that one needs a well defined enegy to emove an electon fom a metallic suface atom. The enegy of the photon is absobed by the electon and, if sufficient, the electon can escape fom the mateial with a finite kinetic enegy. A single photon can only eject a single electon, as the enegy of one photon may only be absobed by one electon. By consevation of enegy, fo incident light of fequency ν the elation I kin I 1 hν = E + E = E + mev holds whee E I is the ionization enegy 8 needed to emove an electon fom an atom. Question 3.3: Upon adiation an electon is emitted fom a sodium suface (E I =.8 ev) with a velocity of 100 m/s. What was the wave length of the incident light? 8 In the case of a solid, the minimal enegy needed to emove an electon is also called the wok function. 8
10 Answe 3.3: 544nm. The photoelectic effect also shows that the electons of the metal atoms with the lowest ionization enegies occupy a well defined, discete enegy level E I. The discovey of the photoelectic effect contibuted significantly to the fomulation of quantum mechanics The Basic Axioms of Quantum Mechanics Quantum mechanics is based on few basic axioms 9. The fist axiom: wavefunctions and pobability distibutions Velocity Locatio (a) Tajectoy Fig (a) tajectoy of a classical paticle; (b) pobability distibution of a quantum paticle. The fist axiom of quantum mechanics states that evey system can be descibed by a wavefunction ψ(,t) that is a function of all the paticle coodinates and possibly the time. The squae of the wavefunction gives us the pobability of finding a paticle at position in space. Fo a timeindependent stationay system containing one paticle, the pobability P() to find the paticle in an infinitesimal small volume element dv aound is: P( ) =Ψ dv ( ) If the paticle is an electon, the squae of the wavefunction is also called the electon density ρ (i.e. the numbe of electons pe volume element). el ( ) ( N ρ = ) = Ψ dv ( ) 9 An axiom is a postulate that foms the basis of a theoy but cannot be deived o fomally poven. 9
11 The wavefunction is nomalized in such a way that the total pobability to find the paticle anywhee in space integates to 1: ( ) Ψ dv = V This tells us that a quantum system behaves fundamentally diffeent fom a classical system. Fo a classical paticle, we can detemine its position and velocity at any instant in time. In fact a classical paticle follows an accuately detemined tajectoy such as the one shown in Fig. 3.10(a). Fo a quantum paticle instead, we can only say that the paticle is at a given position with a cetain pobability. We can epesent the position of a quantum paticle with a pobability distibution which defines egions in space whee we find it with high pobability and othes whee the pobability to find the paticle is vey low (Fig.3.10(b)). Question 3.4: A quantum paticle is confined in a linea box of length L suounded by infinitely high potential walls. The gound state of this system is descibed by the wavefunction 1. L x L. Ψ 1( x) = sin π What is the pobability to find the paticle at a given position x? At which position is the pobability maximal? What is the total pobability to find the paticle in the box? Answe 3.4: Click hee to see the solution. The second axiom: obsevables and opeatos The second axiom deals with the question of how we can calculate the popeties of a quantum system. It states that evey obsevable 10 A in classical mechanics can be descibed by a coesponding opeato Ô A in quantum mechanics that acts on the wavefunction to poduce the value of the obsevable A multiplied by the wavefunction (a so called eigenvalue equation ): Oˆ A Ψ = AΨ. 10 An obsevable is any popety of the system that can be measued such as e.g. the position, the velocity, the linea momentum o the enegy. 30
12 Some examples: Obsevable Expession in classical opeato in quantum mechanics mechanics = x, y, z ˆ = x, y, z Position ( ) Momentum p = mv ( ) ˆ δ δ δ p = ih = ih,, δx δy δz With these two tansfomations, one can constuct the coesponding quantum opeatos of all classical quantities such as e.g. the opeato fo the kinetic enegy Ê kin p E kin = m ˆ E kin h = m Question 3.5: What is the linea momentum of a fee paticle that is descibed by the wavefunction Answe 3.5: ħk Ψ ( x) = e ikx.. The thid axiom: expectation values and uncetainties We have seen that the position of a quantum paticle is not descibed by a single shap value but by a pobability distibution. We cannot ask whee the paticle is but only what is the most pobable position to find it. The most pobable value of a pobability distibution is called the expectation value (o the mean value o fist moment of the distibution). Fo instance, the expectation value of the paticle position <> is = Ψ ( ) dv P i ( i). V The second equality is obtained afte a discetization of the space, which consists in dividing it in an infinite numbe of small cubes (labelled with the index i) of volume dv placed at position i. This means that we take evey possible value of and do a weighted aveage ove all values with the weight of each given by its pobability (i.e. the squae of the wavefunction at ). In the same way as fo the paticle position, we have to calculate any popety A of the quantum system as expectation value ove the pobability distibution given by the squae of the wavefunction: i 31
13 V ˆ A A = ΨO ΨdV whee we can think of the opeato Ô A as an opeato that poduces the expectation value fo the popety A. If we do many measuements of a popety A, the width of the esulting pobability distibution detemines how likely it is to find values that deviate fom the mean value <A>. This uncetainty ΔA is given be the standad deviation of the pobability distibution. Δ A = N i = 1 P ( A i N i i = 1 P i A Question 3.6: What is the expectation value when you thow an ideal dice? Answe 3.6: 3.5 (fo an explicit solution click hee). Question 3.7: An electon is descibed by a Gaussian wavefunction of the fom Ψ () x = σ π 1/ 1/4 e ( x μ) /σ What is the expectation value fo its position? What is the standad deviation? Answe 3.7: μ,σ (fo an explicit solution click hee). The fouth axiom: the Hamilton opeato and the Schödinge equation All popeties of a quantum system can be detemined once its wavefunction is known. The equation that detemines the wavefunction and its time evolution is the timedependent Schödinge equation ˆ δψ HΨ (, t) = ih δ t 3
14 Fo stationay states (i.e. nontime dependent systems) one can use the timeindependent Schödinge equation H ˆ Ψ( ) = EΨ( ) whee Ĥ is the Hamilton opeato of the system. The Hamilton opeato is the quantum opeato fo the total enegy E of the system ˆ ˆ ˆ h H = E ( ) ˆ kin + V = + V( ). m The lowest of its eigenvalues (E 0 ) is the gound state enegy of the system Heisenbeg s Uncetainty Relation The Heisenbeg uncetainty pinciple states that the uncetainties of an infinitely pecise measuement of the paticle position x and its momentum p ae coelated. If the paticle state is such that the fist measuement yields a standad deviation of values Δx, then the second measuement will have a distibution of values whose dispesion Δp is at least invesely popotional to Δx. The constant of popotionality is equal to Planck's constant divided by 4π. ΔxΔp h π. Question 3.8: What is the uncetainty of the momentum of paticle whose position can be measued with infinite accuacy (Δx = 0)? Answe 3.8: Compaison between classical and quantum mechanics Wheeas classical mechanics fails to descibe quantum phenomena, quantum mechanics is a geneal theoy that educes to the laws of classical mechanics fo systems with high masses, enegies, and tempeatues, o fomally, if the Planck constant becomes infinitely small h 0. The following table 33
15 summaizes once moe the pincipal diffeences between classical and quantum mechanics: Classical Mechanics position and linea momentum of a paticle have shaply defined values, p Continuous enegy specta E kin (T=0) = 0 Time evolution is given by the classical equations of motion: Newton s equation f v = ma = m δ δt Quantum Mechanics ΔΔp ħ/ ψ(,t); P(,t) = ψ * (,t)ψ(,t) Quantized enegies E kin (T=0) > 0 (zeo point enegy) Time evolution is given by the timedependent Schödinge equation H ˆ Ψ(, t) = ih δψ δt Example of a Quantum System: Paticle in a Box V 0 L Fig Paticle in a box. Conside a paticle with mass m in a one dimensional box of length L that is suounded by infinitely high potential walls. The potential that the paticle expeiences inside the box is 0. 0 V ( x) = fo fo x < 0 0 < x < L and x > L Since the potential is zeo inside, the only enegy that the paticle has is kinetic enegy E kin. If we eplace the momentum in the classical expession fo the kinetic enegy p E kin = m with the de Boglie wavelength p = h λ, we get an expession fo the enegy of the system 34
16 E tot = h 1 m λ that is invesely popotional to λ. What ae possible λ fo a paticle in a box? We know that the wavefunction has to fulfil the bounday condition that it is zeo at the walls (x=0 and x=l). A simple tigonometic function such as the one shown in Fig fulfils these conditions. In fact all hamonic waves whose peiod is such that they become zeo at the wall ae acceptable wavefunctions fo this system. All possible wavelengths of this system thus satisfy the condition that intege multiples of half a wavelength ae equal to the box length L λ n = L whee n = 1,,3 Whee n is called a quantum numbe i.e. a numbe that specifies each possible state of the system (in this case fom n = 1 to infinity). If we substitute the expession fo λ = L/n in the expession fo E tot we get E tot = h n 8m L i.e. the enegy of the system depends invesely on the squae of the box length L and is popotional to n. Figue 3.1 shows the possible enegy level fo each value of n with the coesponding wavefunction. Fig. 3.1 Enegy levels and wavefunctions fo a paticle in a box. 35
17 Note that, the system can only adopt these discete enegy values wheeas fo a classical system all enegy values would be allowed. The state with the lowest possible enegy (n =1) is called the gound state. All othe possible states with n>1 ae excited states. 36
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