Lecture Notes PH 411/511 ECE 598 A. La Rosa INTRODUCTION TO QUANTUM MECHANICS

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1 Lecture Notes PH 411/511 ECE 598 A La Rosa INTRODUCTION TO QUANTUM MECHANICS CHAPTER-6 QUANTUM BEHAVIOR of (electrons, photons) PARTICLES and the HEISENBERG S UNCERTAINTY PRINCIPLE Experiments suggesting a lack of trajectories in the particles motion 61 Shortcomings of classical concepts for describing the behavior of electrons passing through two slits 61A The concept of Probability An experiment with bullets 61B An experiment with light 61C An experiment with electrons 61D Attempts to track electrons trajectories 61E Discarding the idea that electrons move along paths 61F Experiments with more massive particles 6 The Heisenberg s Uncertainty Principle 6A Uncertainty in the position x and linear momentum p 6B Relationship between the uncertainty of the energy content E of a pulse and the time t required for the measurement 6Ba Gratings, phasors (addition of multiple waves), conditions for constructive interference, evaluation of the sharpness of the peak of order m 6Bb Gratings and spectral resolution 6Bc Condition for the minimum length time t required for the measurement of the energy E with a resolution E 6C More examples 6Ca 6Cb The resolving power of a microscope and the uncertainty principle The size of an atom 63 Contrasting the (deterministic) classical and (probabilistic) quantum mechanics descriptions 631 Deterministic character of classical mechanics 63 Absence of trajectories in the quantum behavior of particles 633 Classical mechanics as a requirement and as a limiting case of quantum mechanics 634 The less detailed quantum mechanics description 64 The non-relativistic quantum mechanics description 1

2 References Richard Feynman, The Feynman Lectures on Physics, Volume III, Chapter 1 Volume I, Chapters 30 and 38 L D Landau and E M Lifshitz, Quantum Mechanics (Non-relativistic Theory), Chapter 1, Butterworth Heinemann (003)

3 CHAPTER-6 QUANTUM BEHAVIOR of (electrons, photons) PARTICLES and the HEISENBERG S UNCERTAINTY PRINCIPLE: Experiments suggesting a lack of trajectories in the particle s motion In chapter 5 we combined i) the De Broglie wave-particle duality principle with ii) the Fourier analysis of a wave-packet (describing the motion of a free particle), to illustrate the relationship between the uncertainty x in the spatial position of a particle and the uncertainty p of its linear momentum Such mathematical analysis led us to gain an understanding of the Heisenberg s uncertainty principle that underlines any quantum mechanics formulation (ie that we need more than one single Fourier s frequency component to describe the motion of one particle) In this Chapter we emphasize again on such an important principle but with examples where the particle interacts with another object: the diffraction of electrons and photons though an impermeable screen in which two slits are cut The examples to be presented here are based on the experimental fact that when an electron beam passes through a crystal, the emergent beam exhibits a pattern of alternate maxima and minima of intensity, similar to the diffraction pattern observed in the diffraction of electromagnetic waves That is, under certain circumstances the behavior of material particles (ie particles of mass different than zero), in this case electrons, display characteristics that have typically been ascribed only to waves Diffraction is certainly not observed with macroscopic particles To gain some understanding of this markedly different (quantum mechanics) behavior of small particles, one typically uses imaginary experiments that are an idealization of the experiment of electron diffraction by a crystal Thus one considers a uniform beam of electrons (or photons) incident on an opaque screen that has two small apertures The results of these experiments, to be described in this chapter, reveal two striking characteristics: a) the inability of the observer to discern the specific path followed by each diffracting particle, and b) that the diffraction persists even when just one single particle is incident on the screen at a time These two features are unique to the quantum mechanics behavior of electrons and photons, and therefore require a quite different description than classical mechanics Indeed, the concept of trajectories is very much rooted in classical mechanics, but (based on the examples provided in this chapter) it cannot be integrated in a quantum mechanics formulation The experiments described in this chapter will tell us that such a concept of trajectory cannot be ascribed to a quantum particle This is in part reflected by the Heisenberg s principle, in that there is a relationship between the x and p uncertainties: the exact location cannot be discerned, unless the uncertainty of the momentum is huge (A classical trajectory does not have such restrictions over x and p) Explicitly, 3

4 x and p uncertainties lack of trajectories in the motion of quantum mechanics particles In addition to the x and p uncertainties, it turns out that, in a given experiment, there is also a relationship between the uncertainty E of the particles energy and the time t needed to measure the particle s energy This Heisenberg s uncertainty will be illustrated in this chapter through the particular case analysis of a refraction grating: In order to discern the energy content of an incident radiation beam with a given specified precision E, a corresponding minimum measurement time t is needed E uncertainty of incoming radiation minimum time-measurement and t needed to discern the energy content The examples presented in this chapter will highlight the need of a proper quantum mechanics formulation to describe the motion of a particle How to describe the motion of particles, where the concept of trajectory does not exists? How such new formulation could be made compatible also with the description of classical particles? Such tasks will be addressed separately but in the next chapter In this chapter we only describe their difficulty to understand to be explained in classical terms 61 Shortcomings of classical concepts for describing the behavior of electrons passing through two slits 61A The concept of Probability An experiment with bullets A gun shoots a stream of bullets, randomly and over a large angular spread Between the gun and the screen there exists a wall with two slits big enough to let one bullet to pass through A detector at the screen serves to count the number of bullets arriving at each position x Some bullets may go directly straight through any of the holes, or bounce on the edges, etc Thus there is some uncertainty about the exact position x at which a particular bullet will arrive at the screen One way to deal with uncertain quantities is to use the concept of probability Let s introduce the concept of probability through the analysis of different possible scenarios that can occur when using the experimental setup presented below 4

5 X X 1 x P 1 P 1 Bullets P Wall Observation P 1 =P 1 +P screen Fig 61 Schematic of the experimental setup to monitor the passing of bullets through two slits The traces on the right show the distribution of bullets arriving at the screen corresponding to different test scenarios: one of the apertures blocked (P 1 and P ), or both unblocked (P 1 ) Let s block the slit #1 What would be the probability P (x) that a bullet passing through the slit # arrives at the position x? The answer requires to make an experimental procedure: If out of a total number N of incident bullets, the number of bullets hitting the screen at total x is N ( x), then the probability P (x) is defined as the ratio P ( x) N( x) / Ntotal But for the concept of probability to acquire a useful meaning the total trial number N has to be very large Symbolically, total P N ( x) Definition of probability (1) ( x) lim Ntotal N total Fig 61 above shows a schematic of the expected behavior of P (x) Let s block slit # What would be the probability P 1 (x) that a bullet passing through the slit #1 arrives at the position x? N1( x) P( x) lim () 1 N total N total When the two holes are open, what would be the probability P 1 that a bullet, passing through either of the two holes 1 or on the wall, will arrive at the position x on the screen? Classical particles typically behave in such a way that, it turns out, the probabilities are additive; that is, P 1( 1 x x) P( x) P ( ) (3) 5

6 Consequently, we affirm that this phenomenon (that deals with relatively large particles) lacks interference; that is to say, the opening or closing of one hole does not affect the individual probabilities associated with one aperture In other words, if a fraction P ( x) = 15% of the particles arrive at x when the aperture #1 is closed, then when the two apertures are open the fraction of the particles that arrive at x after passing through aperture is still 15% 61B An experiment with light A source emits monochromatic light waves, which pass first through a couple of slits, and then arrives at a screen A detector on the screen measures the local intensity I (energy per unit time per unit area) of the incident light We would like to understand the intensity distribution I (x) established across the screen When one of the slits is covered, a rather simple intensity distribution (either I 1( x) or I ( x) respectively) is observed, as displayed in the Fig 6 below But when the two slits are open, a rather distinct wavy pattern I ( ) is observed Definitely I x) I ( x) I ( ) 1 x 1( 1 x Consequently, we affirm that this phenomenon with light displays interference since the opening or closing of one slit does influence the effect produced by the other slit In other words, one slit appear to behave different depending on whether the other slit is open or closed) X X 1 x I 1 I 1 Light source I Wall Observation I 1 I 1 + I screen Fig 6 An experiment similar to the one depicted in Fig 61, but with light Wave interpretation We could attempt to explain this result by considering the interference between the light beams that pass through the slit 1 and the slit If the beams arrive at the observation screen in phase they will produce a bright spot; if they arrive out of phase a dark spot will result 6

7 Notice, this plausible interpretation needs the simultaneous participation of two entities (two beams) What about if, in addition, we take into consideration the constituent granularity of light? How does this new information affect our conclusion made in the paragraph above? In such a case, we can still argue that photons passing through the two different apertures would interfere at the screen However, if we deem the intensity of the light source such that only one photon is emitted at a time (and employ a large exposure time so that we still collect a lot of them at the detection screen), what would be the result? Based on the plausible wave interpretation stated above, the prediction is that the wavy interference pattern I 1( x) would not be formed, for a single photon passing through a slit would have no other photon to interact with However, this prediction is wrong This is what happens when only one photon at a time is emitted from the light source: If a photographic plate is placed on the screen, and the exposure time is chosen as short as to receive only one photon at a time, it is observed that: Each photon produces a localized impact on the screen (4) That is, a single photon does not produce on the screen a weak interference pattern like I 1( x) (that is, a single photon does not spatially spread out); it produces just a single localized spot When the exposure time of single photons is drastically increased, as to capture a large number of photon, a progressive formation of a wavy pattern I ( ) is observed! This was demonstrated (although in an experimental set up different than the one being described here) by G I Taylor in 1909, who photographed the diffraction pattern formed by the shadow of a needle, using a very weak source such that the exposure lasted for months It can be concluded then that interference does not occur between photons, but it is a property of single photons (5) The photons arrive to the photographic plate in a statistical fashion and progressively build up a continuous-looking interference pattern This has been confirmed in more recent experiments performed in 1989 by A Aspect, P Grangier, and G Roger It turns out that materials particles, such as electrons, exhibit a similar behavior, as described below in Section 61C 1 x Why the pattern is affected so drastically when the two slits are open? 1 7

8 61C An experiment with electrons Let s consider an experiment in which mono-energetic electrons strike a wall having two slits; beyond the wall a screen full of detectors (only one is shown in the figure) allows recording the spatial pattern distribution of electrons-arrival along the screen The arrival of an electron at the detectors is sensed as a detector s click Electrons arrive at the detector in identical pieces (one click per electron) When one electron at a time is emitted by the source, only one detector s click is detected at a time X X 1 Detector x x o P 1 P 1 Filament (Electron source) P Wall Observation P 1 P 1 + P screen Fig 63 Schematic of the two-slit experiment using electrons The results sketched in the diagram above indicate that, electrons behave like light (6) Indeed interference pattern due to incoming single electrons was observed in 1989 by A Tonomura, J Endo, T Masuda, T Kawasaki, and H Ezawa in two-slit experiments in which the accumulation of the interference pattern due to incoming single electrons was observed A Tonomura,J Endo, T Matsuda, T Kawasaki and H Ezawa, Demonstration of single-electron buildup of an interference pattern Am J Phys 57, 117 (1989) (More recently interference using more massive particles have been observed; see Section 61F below) In summary, the observed interference effect in the detection of electrons and photons (as described above) indicate that the following statement (plausible in the classical world) is not true: 8

9 Each electron either goes through aperture 1 or through the aperture (7) For: When we close aperture-1 (so we know for sure the electron passed by the aperture-) the detector at x=x o (see figure 63 above) collects a given number of electrons, let s say, N (x o )= 348 When we close aperture- (so we know for sure the electron passed by aperture 1) the detector at x=x o collects a given number of electrons N 1(x o )= 1965 How could it be that when we leave both apertures open the detector at x=x o collects none? The wavy pattern P 1( x) (as shown in the Fig 63 above) is then not compatible with the statement (7) When the two apertures are open, we do not know which aperture a specific electron passed through This type of uncertainty is rooted in the quantum behavior of atomic objects Further, it appears that Quantum Mechanics protects this uncertainty as to defeat any attempt to overcome it In the next section we will describe those attempts in a more formal detail 61D Attempts to track the trajectory of the electrons motion Watching electrons with a light source Fig 64 below shows that a light source has been added to the previous set up in an attempt to watch what specific aperture electrons choose to pass through Procedure: When an electron passes aperture-1, for example, we should observe photons being scattered from a region nearby the aperture 1 These electrons (type 1) will be tabulated under the table-column P 1 Similarly, for electrons scattering from a region near the aperture a second tabulated column will be generated When this experiment is performed, the result, it turns out, is: Result: P 1 = P 1 + P Just as we expect for classical particles (That is, the interference pattern, obtained when we do not watch the electrons, does not occur any more) Conclusion When we watch the electrons, they behave different than when we do not use light to watch them (8) 9

10 X X 1 Detector x P 1 P 1 Filament (Electron source) P Wall Observation P 1 = P 1 + P screen Fig 64 Schematic of the two-slit experiment with electrons while watching them passing through the slits using a light source What about using more gentle photons? Since, as we know, light is an electromagnetic entity (which interact with charged particles), probably we should expect electrons to behave different when using photons to watch them The motion of the delicate electrons may be perturbed by the linear momentum p h / λ carried by the photons An alternative would be to use more gentle photons; those of longer and longer wavelength We will encounter a limitation in this approach, however Since using large wavelength worsen the resolution As we know, when using light there is a restriction on the ability to distinguish how close two spots are apart from each other (and still be seen as two separate spots): If the separation between the objects is smaller than the wavelength of the radiation being used, we will not able to distinguish them Thus, in our two-slit experiment with electrons, when using longer and longer wavelengths (as to minimize the perturbation on the electron s motion) our ability to distinguish the slits becomes worst and worst We will not be able, then, to see which slit a given electron passed through The above observation helps illustrate the unavoidable perturbation role played by a measurement (We knew this, but we did not expect to be so drastic when working with small particles like electrons) It is impossible to arrange the light in such a way that one can tell which hole the electron went through and at the same time not to disturb the pattern (9) 61E Discarding the idea that electrons move along paths 10

11 Trying to predict the results of the passage of electrons through two slits, one expected, based on classical intuition, that the patter formed on the detection screen would be a simple superposition of the two patterns obtained when one slit, one at the time, is covered: each electron, moving in its path, passes through one of the slits and has no effect on the electrons passing through the other slits The experiment however shows an interference pattern, no corresponding at all to the sum of the pattern produced by each slit separately It is clear that this result can in no way be reconciled with the idea that electrons move in paths Thus, a new description, the quantum mechanics description, must be based on ideas of motion which are fundamentally different from those of classical mechanics In QM there is not such a concept as the path of a particle This forms the content of what is called the uncertainty principle, one of the fundamental principles of quantum mechanics, discovered by W Heisenberg in F Double-slit experiments with more massive particles Following the demonstrations of the double-slit interference with electrons, a quest continued for demonstrating the quantum nature of massive particles Such double-slit experiments are regarded as the ultimate demonstration of the quantum nature of matter (1994) Fragile clusters of He, H, and D were selected and identified nondestructively by diffraction from a transmission grating W Scholikopf and J P Toennies, Nondestructive Mass Selection of Small van der Waals Clusters, Science 66, (1994) (00) The dual quantum character of the famous double-slit experiment is explained with a large and most classical object, the fullerene molecules The soccer-ball-shaped carbon cages C 60 are large, massive, and appealing objects for which it is clear that they must behave like particles under ordinary circumstances The authors present the results of a multislit diffraction experiment with such objects to demonstrate their wave nature O Nairz, M Arndt, and A Zeilinger, Quantum interference experiments with large molecules, Am J Phys 71, 319 (003) (01) In contrast to classical physics, quantum interference can be observed when single particles arrive at the detector one by one The authors record the full two-dimensional build-up of quantum interference patterns in real time for phthalocyanine molecules and for derivatives of phthalocyanine molecules, which have masses of 514 AMU and 1,98 AMU respectively Wide-field fluorescence microscopy detected the position of each molecule with an accuracy of 10nm and revealed the build-up of a deterministic ensemble interference pattern from single molecules that arrived stochastically at the detector T Juffmann, A Milic1, M Mullneritsch, P Asenbaum, A Tsukernik, J Tuxen, M Mayor, O Cheshnovsky, and M Arndt, Real-time single-molecule imaging of quantum 11

12 interference, NATURE NANOTECHNOLOGY 7, 97 (01) 6 The Heisenberg s Uncertainty Principle It was suggested by Heisenberg that the new law of Nature could only be consistent if there were some basic limitations in our experimental capabilities not previously recognized Two important cases will be discussed with some detail in the sections below; one pertaining the position and momentum variables, the other Energy and time (indeed, the uncertainty principle involves a pair of variables) 6A Uncertainty in the position x and linear momentum p Originally Heisenberg stated: If you make the measurement of any object, and you can determine the x-component of its momentum with an uncertainty p, you cannot at the same time, know its x- position more accurately than Δx ( / ) /( p) Here h / (10) Let s consider a specific example to see the reasons why there is an uncertainty in the position and the momentum if quantum mechanics is right Example: Particles (electrons or photons) passing through a single slit We will focus our interest in the vertical component of the particles linear momentum, p y As shown in Fig 66, since the particles (photons) come from a source far away from the slit, they can be considered to arrive with a linear momentum oriented practically horizontal Thus, Before arriving to the slit: p y =0 However, we do not know with certainty the vertical positions of the particles; ie y = 1

13 p o a Y Particles distribution Particle-waves arrive to the screen essentially horizontally p o p y p y = p o Fig 65 Trying to localize better the particles in the Y-direction, we make them pass through a slit of size a Let use an aperture of size a as an attempt to localize better the vertical position of some of the particles contained in the incident beam That way, After the particles had passed through the aperture we would know the y-position with a precision equal to Δ y a (11) However, notice that for the particles passing through the slit we lose precise information about their vertical momentum (it was p y =0) The reason is that, according to the wave theory, there is a certain probability that the particles will deviate from the straight incoming direction, and rather form a spread diffraction pattern In other words, there is a probability that the passing particles acquire a vertical momentum A representative measurement of the angular spread is given by the angle at which the diffraction pattern displays a minimum (see the inclined dashed-line in the figure above) From the diffraction theory, the angular position at which this minimum occurs is given by, a A corresponding measurement of the spread of the vertical momentum values is then no better than p o ; i e, py po po / a Introducing the quantum mechanics concept that the momentum p y h/a p o is given by p o h /, Notice, if a smaller value of a were chosen (in order to get a better precision of the particles vertical position) a corresponding larger spread of p is obtained From, (11) and (1), y (1) 13

14 p y y h in reasonable agreement with the ultimate limit / given in expression (10) above Wider slit Narrower slit a S Less uncertainty in the vertical momentum p o a S Larger uncertainty in the vertical momentum Fig 67 The probability that a particle ends up at S increases as the slits gets narrower which agrees with the statement in (10) That is, the better precision y in the particle s vertical position (using a narrower the slit) the greater the spread in the vertical linear momentum (the greater the probability that a passing particle ends up having a very large vertical linear momentum) 6B Relationship between the uncertainty of the energy content E of a pulse and the time t required for the measurement In this section, the working principle of a grating is used to illustrate the energy-time uncertainty principle First, we familiarize with the functioning of a grating, and emphasize its inherent limited spectral resolution (inability to distinguish waves of slightly different frequency,) which in principle has nothing to do with quantum mechanics) Then, by associating the radiation frequency (used in the description in the functioning of the grating) with the energy of the radiation, as well as properly interpreting the time needed to make a measurement of the energy content of a pulse with a desired precision E, the energy-time uncertainty principle results If you are already familiar with the functioning of a grating, you can skip Section 6Ba and go directly to Section 6Bb What is important is to learn the spectral resolution power of a grating, and how this limited resolution relates to the energy-time uncertainty principle 6Ba Gratings Let s consider a fully illuminated grating of length L and composed of N lines equally separated by a distance d 14

15 L k k= N lines separated by a distance d from each other Grating m = 1 m = 0 Fig68 For radiation of a given wavelength, several maxima of intensity (m=0, 1,, ) are observed at different angular positions as a result of constructive interference by the N lines sources from the grating Calculation of the maxima of interference by the method of phasors 3 Case: Phasor addition of two waves Wave-1 = A Cos(kx) phasor-1 = 1 = Ae i [ kx ] Wave- = A Cos(kx+k) phasor- = = Ae A Cos(kx) = Real( 1 ) A Cos(kx+k) = Real( ) Notice, i [ kx + k ] k = k = 15

16 1 d Wave-1 x Wave- d Sin 1 = Ae i [ kx ] phase = Ae i [ k (x + ] i[ k x + k ] = e Fig 68 Real-variable waves are represented by their corresponding complexvariable phasors Phases (complex variable): 1 A e i [ k x ] e i [ k x + k ] Real variable: Real ( ) Real( 1 + ) A Cos( kx ) + A Cos( kx + k) Im z k Im z k 1 kx Real z kx Real z Fig 69 Addition of two phasors in the complex plane Case: Phasor addition of N waves = 1 n = = Ae ikx Ae i( kx+k Ae i( kx+k i( kx+(n-1)k + Ae = e i kx e ik e ik + e i(n-1)k = N e i kx 1 s 0 e isk (13) Fig 610 shows the graphic interpretation of the result (13) 16

17 1 N N N Im z 1 Fig 610 Addition of N phasors in the complex plane kx k Real z Condition for constructive interference This occurs when contiguous interfering waves travel a path difference exactly equal to an integer number of wavelengths = m m = 0,1,, 3, (14) which take place at the angles m that satisfies: d sin m = m (14) Equivalently, since k=, the max of constructive interference occurs when the phase difference k between two contiguous waves is exactly equal to an integer number of ; k = m Condition for max of constructive interference where ( m = 0,1,, 3, ) See the geometrical interpretation in Figures d j+1 j m d Sin m m Conditions for max of intensity of order m j kx j+1 k m 17

18 Fig 611 Condition in which the phase difference between two contiguous phasors is m They contribute to the max of order m (see also Fig 61 below) 1 = m m Im z N N N m 1 kx Max, m k m Real z max m = 1 first-order peak Grating m max m = 0 zero-th order peak Fig 61 Condition to produce a maximum of intensity (of order m) Top: All the N phasors add up as to give a phasor of the largest possible amplitude Bottom: Visualization of the corresponding maxima of order m (m=0, 1,, ) Condition for evaluating the sharpness of the zero-th order peak Around a peak of maximum intensity, the intensity drops to zero, thus defining a line thickness We would like to calculate the angular broadening associated to that thickness of line intensity Lets consider first the max of order m=0 Using the method of phasors, we realize that, 18

19 A path difference gives a phase difference of (between contiguous radiators) k = If k =/N For N oscillators the total accumulated phase would be ; But Fig 610 indicates that when the total accumulated phase is the phasor sum would be zero In other word, an intensity of zero amplitude wave would be obtained if = Condition for zero intensity (15) Since d sin ( expression (15) determines the angle min that locates the minimum of intensity (next to the maximum of order m=0), or d sin (, d sin ( min,0 = Condition for the angular (16) position (next to the maximum of order m=0) for which the light intensity is zero Grating min,0 min max m = 0 N lines separated by a distance d Conditions for minimum of intensity 19

20 1 N min,0 N-1 N 1 Fig 613 Condition for finding the angular position of the minimum of intensity closest to the maximum of order m=0 (hence, giving a measurement of the angular sharpness of the peak of order m=0) k kx Condition for evaluating the sharpness of the m-th order peak The angular location min,m of the minimum of intensity closest to the m-th order maximum of intensity can be calculated in a similar fashion Indeed, in the case m=0 we chose a path difference of =/ N between two contiguous waves For the case m 0 let s choose = m + / N (ie a path difference a bit bigger than the one that gives the peak of order m) The corresponding phase rotation is k= ( m + / N m + / N Notice, however, that the latter rotation is equivalent to a net rotation of / N (since a rotation of mbrings the phasor to the same place) k / N Thus we are choosing a net phase difference between two contiguous waves equal to / N Since we have N phasors, after N rotations we will have completed a full rotation, which brings the total phasor to zero This is illustrated in Fig 15 0

21 m = 1 Grating max,1 min,1 m = 0 1 N Nm =m =m max, m 1 N Nm '= m min, m Fig614 For a given line-space d and, the diagrams (top and bottom figures) show the angle max,m that determines the intensity-maximum of m-th order (d Sin max,m m) As the angle increases a bit, a minimum of intensity will occur at the angle min,m, which is determined by the condition d Sin( min,m m(see also Fig 615 below) 1

22 1 = m min,m N Nm N 1 k kx Fig 615 The diagram shows the condition that determine the angle min; m (closest to the angle max;m ) for which the intensity at the far located screen is zero This occurs when d Sin=m The condition = mor d sin( max, m ) m gives the angular location of the max of order m (17) The condition ' = mn (18) or equivalently, ' = N d sin( min, m = m N gives the angular location of the minimum immediately next to the max of order m From (17) and (18), N d sin( min, m N d sin( max, m = [m N- [N m N d [sin( min, m sin( max, m ]= The left side is approximately equal to, N d [Cos max, m ] = where max, m min,m width m width m = [Nd Cos max, m ] ½ width of the peak (19) of order m

23 Question: Given, and having found its peak of order m at the angular location max, m, whose width is width m = [Nd Cos max, m ], what would be the value ' of another incident radiation such that it peaks at = max, m + width m width,m m th peak Grating max,m min,m Fig616 Illustration of the width of the maximum of order m For we want a maximum of order m to occur at = max,m + width m This requirement is expressed by, d sin( max, m + width m = m' (Condition of max of order m for ') Using a Taylor expansion, sin( + sin( [cos( d sin( max, m d width m Cos( max, m = m' Let s set '= + d sin( max, m d width m Cos( max, m = m(+ The condition for max of order m for implies dsin( max,m m; accordingly, d width m Cos( max, m = m Replacing the vaue of width m d/ [Nd Cos max, m ] } Cos( max, m = m which givesn = m or =mn (0) The interpretation of this result is: if is increased to ' ( mn), 3

24 the angular position of the peak of order m corresponding to' will lie on the same angular position of the minimum of intensity closest to the max of order m corresponding to Below, we will obtain this result again (in the context of finding the resolution power of a grating) 6Bb Gratings and Spectral Resolution The figure shows incident radiation of two different wavelengths We want to evaluate the ability of a given grating to distinguish them 1 L N lines Fig 617 Radiation of different wavelength (or frequency) produces maxima of intensity a different angular positions Each radiation will produce its own independent spectrum (a set of peak of different orders) Let s focus on a particular order, let s say the order m We know that the angular location of the peak increases with, dsin( max,m m Accordingly, if the value of is too close the value of 1 then their m-order peaks of intensity (which have some width) may superimpose to the point that it would be hard to distinguish them from each other in the recorded spectrum How close are and 1 are allowed to be and still be able to distinguishing from each other in the grating spectrum? Raleigh s criterion of resolution offers a standard metric: The smallest difference = - 1 allowed for this two wavelength is the one for which the angular position of minimum of intensity for 1 (the minimum closest to its maximum of order m) occurs at the same angular position of the m- order 4

25 The Raleigh s criterium is illustrated in Fig 616 for the cas m=1 1 N m m max, m N m 1 N m + /N (m + /N) min, m N m + Order m = Grating max,1 min,1 Order m = 0 Fig 618 Raleigh s criterion of resolution illustrated for the case m=1 For the same angle, we want to have i) a maximum of order m for the incident light of wavelength d sin( = m or Nd sin( = Nm (1) ii) a minimum of intensity for the wavelength 1 (a minimum right after its max of orde m) N d sin( = mn 1 1 (1) and () imply, Nm = Nm Hence, Nm( - 1 ) = 1 Using = - 1 the Raleigh s criterion implies, 5

26 Light containing radiation of wavelength is incident on a grating of N lines, producing a spectral line of order m If the incident light contains radiation of wavelength that differs from by a quantity, it will be distinguished (according to the Raleigh criteria) if satisfies, λ λ N m (3) Equivalently, to distinguish the presence of two waves whose wavelength difference is, requires a grating of the following number of lines λ N 1 (when using the maxima of m-th order) (4) m λ Since λ λ ν, this expression can be written also as ν ν Δν (Δν) min (5) m N This expression gives the minimum frequency-difference between two waves that a grating of N lines is able to distinguish as, in fact, two distinct incident frequencies (when working around the maximum of order m) 6Bc Condition for the minimum length time t required for the measurement of the energy E with a resolution E Consider a grating of N lines (separated by a distance d from each other) Let (or equivalently a given frequency ) for which the grating produces a maximum of intensity of order m According to expression (14) this maximum occurs at an angular orientation max, m given by d SIN( max,m m Notice in the graph below that in order to fully exploit the coherent interference from the N oscillators and obtain the maximum spectral resolution (the greater the N the greater the resolution, as indicated by expression (5) ) we have to require that the spatial -l extent of the incident wave-train must exceed a minimum length equal to l = AQ = mn (maximum of intensity of order m) 6

27 Now, we notice something mathematically curious The time T AQ needed by the light to travel the distance AQ is, T AQ AQ' c Nmλ c N m ν Using the result in (5), the expression above for T AQ can also be written in terms of min (the smallest frequency difference the grating can resolve), T AQ 1 1 (6) ν /(Nm) ν min P P C 1 l Q N Q A l =mn (a) P and Q will become part of the same wave-front after passing the grating P P 1 C S S l Q N A Q l =mn (b) If the spatial extent of the wave-train were made smaller than l, then only a reduced number of grating lines contribute to the constructive interference (hence, according to (5), worsening the resolution of the grating) Fig 619 Given a grating composed of N lines, for the N oscillators to participate in the m-th order coherent interference the spatial extent of the incident wave-train radiation has to exceed 7

28 a distance l = mn (Otherwise, a shorter wave-train will exploit only a fraction of the N lines in the grating) This result, lead us to an important experimental interpretation, as illustrated in Fig 619 In order to resolve the frequency content of the incident radiation with a resolution ν, the measurement has to last at least a time T AQ =1/ ν, or greater That is, 1 ( t) measurement (5) ν (Otherwise, if the incident radiation were spatially shorter than l = mn, then not all the N lines would participate, and the effective resolution of the grating would be less) 1 N N m m m max, m 1 N N lines m + /N N m + (m + /N) min, m 1 L C A Q N m Fig 60 Aiming to distinguish the two frequencies 1 and ( = - 1 ) a setup to detect the maximum of order m, is chosen Such constructive interference requires the simultaneous participation of the N wave components in the interfering wavefront CQ As illustrated also by Fig 618 above, the formation of such wavefront CQ requires a pulse duration t of greater than, or at least equal to, the time required by the light to travel the distance AQ, which is t = AQ/c It turns out AQ/c = 1/ We have arrived to a very interesting interpretation: In order to find out whether an incident radiation contains harmonic wave components whose frequencies differ by, the minimum time duration (6) of the pulse has to be 1/ The less uncertainty we want to have concerning the frequency components in the pulse, the longer the pulse need to be (ie a longer measurement-time will be required) We would need more (measurement) time if we want to find out the spectral content with more precision (ie with less uncertainty) 8

29 T min 1 T min 1 T min has to be greater than T AQ A Q N m Fig 61 There is a close relationship between the precision with which we want to know the spectral content of a pulse, and the minimum time duration T that the pulse is required to have in order a measurement with such spectral precision: T min =1/ This is also illustrated in Fig 6 where it is assumed that the frequency 1 is contained in the pulse f (because it produces a big max of interference at 1 ), but the rest of the spectrum will be known with good precision if the incident pulses are longer in duration Short pulse f (t) Spectra Width of the spectral line (It is determined by the temporal duration of the incident pulse) t) f t 1 ( t) f It is not possible to know whether or not frequency-components from this frequency range are contained in the pulse f Fig 6 The shorter the pulse t) f, the larger the uncertainty in knowing the frequency content of the pulse This occurs because of the short time t) f available for the measurement Notice, the results just described have nothing to do with quantum mechanics It is a property of any wave It occurs, however, that quantum mechanics associate a wave character to the particles; hence the variables associated to the wave-particle (momentum, position, energy) become subjected to the frequency/time of position/momentum uncertainties 9

30 For example, applying the concept of quantum mechanics E hν, expression (5) can be expressed as, ( E)( t h (7) ) measuremen t 6C More examples 6Ca The resolving power of a microscope and the uncertainty principle First we obtain the condition of maximum resolution that can be obtained in a conventional optical microscope based on image formation principles Next, we obtain the same condition but using the uncertainty principle i) Resolving power of a microscope Image formation P focuses at T A condition for a clear focused image formation at T is that all rays leaving P and passing through the lens take an equal time to arrive (8) to T A point P a bit away from P will focus at a region very close to T If P is too close to P, the images of P and P image will superimpose and become too blur to distinguish them; they would appear to be the same point How far apart have to be P and P so that they can be distinguished (9) as two different point? We will answer this question in terms of a comparison between the time used by the rays PST and P ST S P P T Fig 63 A point P is imaged by the lens on point T How far apart have to be P and P so that they can be distinguished as two different points? Let s call t(pst) the traveling time of light to go from P to T passing through S A condition that will guide us to discern that a point P will be focus at a different point than T (and thus be distinguished from P) is the following: t(p ST) and t(pst) have to be different 30

31 But again, how much different do t (P ST) and t (PST) have to be so that we can conclude that these two paths do not belong to the set of rays (defined in expression (8) above) forming a clear image at T? The condition of resolution Starting with the path PST and its corresponding time t (PST), let s calculate the time t(p ST) as the point P moves out of axis away from P The condition of resolution (the ability to distinguish P and P ) states that, Two different point sources P and P can be resolved (ie can be distinguished) if t (PST) differs from t (P ST) by more than one period 5 t (P ST) - t (PST) one period of oscillation (of the radiation being used for imaging) (30) t (P ST) - t (PST) Using the diagram in Fig 63 ( zoomed in in Fig 64), the time difference [ t (P ST) - t (PST) ] is equal to the distance PM divided by the speed of light v, [ t (P ST) - t (PST) ] = PM/ v ( d sin ) / v where n is the index of refraction of the medium ( d sin ) / ( c/n) (31) S d P P M Fig 64 Diagram to calculate the difference in travel time by the two rays PST and P ST Here P M is perpendicular to PS Using (31), the condition of resolution in (30) can be rewritten as, 1 dsinθ /(c/n) ν Or, 31

32 λ Resolving power of a lens (3) n sinθ d 0 λ 0 is the minimum distance that two points P and n sinθ P need to be separated in order to be imaged as two different points The quantity in the denominator is defined as the numeral aperture (NA) of the lens, NA nsinθ (33) and the resolving power is typically expressed as λ esolving power R (34) NA R 0 where λ 0 is the wavelength in vacuum S P d 1 Object plane Fig65 Two arrangements with different lateral resolution capabilities The higher NA the finer the resolution ii) The uncertainty principle and the microscope resolving power We wish to measure the position and momentum of an electron using the optical microscope setup shown in Figure 66 below The moment a photon interacts with an electron, it recoils in a way that cannot be completely determined (there will be an uncertainty in the possible exact recoil direction) That is, the very act of observing the electron with photons disturbs the electron s motion Based on the description given in section i) above, if a photon is detected at point T, it is because it originated (on the object plane) at any point from a region of size x, (x) electron = λ / sinθ Uncertainty in the position of the electron (35) d 3

33 If we detect a signal through the microscope, it would because the photon of wavelength (and of linear momentum p=h/) has recoiled anywhere within the lens angle of view That is, the x-component of the photon s momentum can be any value between p Sinand -p SinThat is, the microscope collect photons whose x-component of their momentum is between 0 and p x photon ±p SinBy the conservation of linear momentum, the electron must have the same uncertainty in the x-component of its linear momentum, p x electron p x photon Thus, the uncertainty in the x-component of the recoiled electron s linear momentum is, ( p x ) psin ( h / λ) Sin (36) electron From (35) and (36) one obtains, (x) electron Uncertainty in the electron s momentum ( x ) h (37) p electron in reasonable agreement with the ultimate limit / given in expression (10) above p = h/ Photon Photon (x) electron x-component of the recoiled electron Fig 66 The numerical aperture nsin of the lens determines the uncertainty (x) electron = λ / sinθ At the same time, nsin determines the uncertainty in the electron s momentum p x p Sin 6Cb The size of an atom (Feynman Vol III, page -5) Suppose we have a hydrogen atom; ie we have a system composed of a proton and an electron) We must not be able to predict exactly where the electron will be; otherwise its momentum spread would be infinite Every time we look at the electron, it is somewhere; it has an amplitude-probability to be here or there Let s assume there is a spread in position of the order of a (38) That is, the distance between the electron and the nucleus is usually about a 33

34 We shall determine a by minimizing the total energy of the atom The spread in momentum is ~ / a, because of the uncertainty principle (see expression (37)) Then the kinetic energy is roughly ( 1/ ) m v = p / m = /(a m) ; K m a The more confined the electron,, the higher its kinetic energy The potential energy is, V = - ( 1/(4 ) ( e / ) 0 a And the total mechanical energy is, E a m e a E( a) (39) The value of a is unknown, but the atom is going to arrange itself so that the energy E is minimum de da - 3 a m e a e 1 - am 4 0 a Setting de/da 0 gives the value for a, a 4 0 e m Using 1/ 4 0 = Nm /C, e= C, = J-s, m= Kg a 1 9 ( )( ( ) ) (898)(16) 91-8 a = m This distance is called the Bohr radius (40) We learn that atomic dimensions are of the order of Angstroms 63 Contrasting the (deterministic) classical and (probabilistic) quantum mechanics descriptions 631 Deterministic character of classical mechanics 34

35 In classical mechanics, the state of a system composed by N point masses is completely described by 3N Cartesian coordinates (x 1, y 1, z 1 ), (x, y, z ),, (x N, y N, z N ) If the system is subjected to constrains the 3N Cartesian coordinates are not independent variables If n is the least number of variables necessary to specify the most general motion of the system, then the system is said to have n degrees of freedom The configuration of a system with n degrees of freedom is fully specified by n generalized position coordinates q 1, q,, q n The objective in classical mechanics is to find the trajectories q j = q j (t) j = 1,, 3,, n (35) Lagrangian approach to find the trajectories: One expresses first the Lagrangian L of the system in terms of the generalized position and the generalized velocities coordinates q and q (j=1,,, n) Typically L=T-V, where T is the kinetic energy and V the potential energy j j L( q, q, t ) (36) where q stands collectively for the set (q 1, q, q n ) The functional S, defined as t S (q) = L( q (t), q (t), t ) dt (37) t1 varies depending on the selected path q (t) that joins the coordinates q ( t 1 ) and q ( t ) Of all possible paths the system can take from q t ) to q t ), the system takes only one On what basis is this path chosen? Answer: The path followed by the system is the one that makes the functional S an extreme (i e a maximum or a minimum) S ( 1 q q = 0 (38) C ( 35

36 t t q(t ) t 1 q c q-space q(t 1 ) Fig 65 Possible motions for a system going from the fixed coordinate q ( t 1 ) to the fixed coordinate q ( t ) The motion q (t) that makes the functional S an extreme (and represented by the solid path in the figure) is the one that the system actually takes Applying the condition (38) to (37) leads to a set of n differential equations of second order d L L 0 = 1,, 3,, n (39) dt q q These are the equations to be satisfied by the classical trajectory q c (t) and the latter constitutes the solution we were looking for 63 Absence of trajectories in the quantum behavior of particles 6 In the quantum description, in contrast to the classical one, there are not trajectories It was emphasized at the beginning of this chapter that if assuming electrons followed trajectories it would lead to the collapse of an atom, which certainly it does not occur Notice that the experiment illustrated in Fig 66 above reinforces the concept that it is not possible to assign a trajectory of motion (in the classical sense) to the dynamics of a particle For example, if we attempted to measure the velocity of the incident particles a x=0, we would place two small apertures of size a at two close positions separated by But after passing the first aperture (which already poses an uncertainty a on the y-position at x=0), there is an uncertainty about y-position of the particle at x= of the order of (/a) or greater The definition of a y= y(t) trajectory would require choosing a very small value of a, but that would worsen the range of possible values for the y-coordinate at x= Hence, there is not possible to establish differences like (final vertical position) (initial vertical position), because such difference adopts scattered values; the smaller the value of a the more scattered the values of y (Note: A blurred and imprecise path could be obtained if the coordinates of the particle are 36

37 measured with a low degree of accuracy But what we are addressing here is the impossibility to define quantum mechanically, a trajectory of arbitrary precision) Hence, in quantum mechanics there is no such a concept as the velocity of a particle in the classical sense (ie limit of y/t as t tends to zero) The Heisenberg principle (stated in expression (10) above) prevents ascribing a classical trajectory (of arbitrary precision) to a QM particle p o Y a Spread of the vertical position created by the first aperture X Fig 66 Limitations to ascribe trajectories to the motion of a quantum particle The higher precision of the vertical position at x=0, the less precise its vertical position at later times, which makes impossible to apply the classical mechanics definition of velocity 633 Classical mechanics as a limiting case as well as a requirement for formulating quantum mechanics 7 Still, as we will see later, in quantum mechanics a reasonable definition of the velocity of a particle at a given instant can be constructed This is a reflection of the fact that the formulation of the quantum mechanics theory is such that it has the classical mechanics as a limit Also, it is in principle impossible to formulate the basic concepts of quantum mechanics without any reference to classical mechanics Thus Quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation The fact that a quantum object does not have a definite trajectory means that it has also no other (classically interpretable) dynamic motion characteristics Therefore, for a system composed only of quantum objects (system of atomic dimensions) it would be impossible to construct a logically independent mechanics theory It result plausible to conceive then that a quantitative description of the dynamics of a quantum system dynamics has to include the presence of a classical object; that way as a result of the interaction between the quantum system and the classical object, a classical signature will be registered on the latter which could reveal information about the former The nature and magnitude of the change on the classical object depends on the state of the quantum object, which can serve to characterize the latter quantitatively The classical object is usually called apparatus, and its interaction with the quantum object is referred to as measurement More specifically, 37