Some Fundamental Experiments

  • Hans Lüth
Part of the Graduate Texts in Physics book series (GTP)

Abstract

Some selected experiments are described in this chapter, both from the early days of quantum physics and from recent time, the interpretation of which is not possible on the basis of classical physics. The interpretation of these experiments requires novel unconventional ways of thinking because of the observed particle-wave duality. Special emphasis is laid on the photoelectric effect, the Compton effect, the diffraction of particle waves and the disappearance of interference patterns by gaining additional “Which Way” information about the particle path.

Keywords

Interference Pattern Pixel Detector Photoelectric Effect Compton Effect Double Slit Experiment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

It is interesting to follow the development of today’s quantum physics by considering difficulties in the interpretation of important experimental results. In particular, around the end of the 19th and the beginning of the 20th century empirical facts accumulated which demonstrated the limits of interpretations on the basis of classical physics, Newton’s mechanics and Maxwell’s theory of electromagnetic fields. Such a historic approach is not intended in the present book. Instead, I want to select some few fundamental experiments, which indicate directly the peculiarities of atomic systems. The experiments are chosen such that they intuitively motivate the basic assumptions of quantum mechanics.

2.1 Photoelectric Effect

When a metal surface is irradiated with light of frequency ω (ultraviolet or visible for alkali metals), electrons are emitted from the metal. In an appropriate experiment, the electron emitting metal can be the cathode in a vacuum tube and the electrons are sucked up by a positively biased anode (Fig. 2.1). This set-up is the basic element of every secondary electron multiplier in which a series of additional electrodes amplifies the electron beam in a sort of avalanche process before it reaches the last anode and is detected.
Fig. 2.1

ae Photo-effect: a Experimental set-up. By light irradiation (photon energy ħω) electrons are emitted from a photo-cathode; they produce a photo-current I under the action of a bias voltage U. b Photo-current I as function of light frequency ω. c Photo-current I as function of applied voltage U. Positive bias defines the illuminated electrode as cathode. U max is the maximum negative bias which can be overcome by the emitted electrons due to their kinetic energy. The saturation current height I s depends on the irradiated light intensity. d Maximum deceleration energy eU max as function of light frequency ω. From this plot the natural constant ħ is obtained as slope; the onset of the curve (straight line) at ω=0 yields the work function W of the cathode material. e Explanation of the photo-effect by means of the potential box model of free metal electrons (shaded). The photon energy ħω of the irradiated light is sufficient for the electrons to overcome the energy barrier of the work function W; on top they carry an additional amount of kinetic energy E el

Also at negligible acceleration voltage and even under de-acceleration bias (illuminated metal positive) electrons are emitted under illumination. The emitted current vanishes not before a certain maximum de-acceleration voltage U max is exceeded (Fig. 2.1c). Thus, the energy of the emitted electrons can be determined from the energy difference eU max which can be overcome by the propagating electrons. With v as electron velocity one has eU max=mv 2/2. According to classical electrodynamics the energy flux density in the light beam is given by the Pointing vector \(\mathbf{S}=\boldsymbol{\mathcal{E}}\times \mathbf{H}\). For low light intensities one would, thus, expect that only after sufficient time enough energy for the emission of electrons has been transferred to the metal. Furthermore, the energy eU max of the photoelectrons determined from the de-acceleration voltage should increase with growing radiation power. This is not observed in the experiment. The energy of the photoelectrons does not depend on light intensity, that is, radiation power. Instead, a characteristic dependence of the effect on the light frequency ω is observed. A lower frequency limit ω lim=2πν lim does exist, below which electrons are not emitted from the metal (Fig. 2.1b). This frequency limit is specific for the material. Furthermore, the emission of electrons starts already at very low light intensities though with very low emission currents, i.e. very small numbers of emitted electrons. A plot of the energy E el of the emitted electrons (=eU max, determined from de-acceleration voltage) versus light frequency exhibits a linear dependence:
$$ E_{\text{el}} =eU_{\max} =\frac{1}{2}m v^2=\hbar \omega -W, $$
(2.1a)
W is the so-called work function of the metal which has to be overcome by the evading electron before it reaches the vacuum. The constant
$$ \hbar =h/2\pi =6.6\times 10^{-16}~\text{eV}\,\text{sec} $$
(2.1b)
is Planck’s constant, which can be measured in the described way by the photoelectric effect.

An explanation of these phenomena is obviously not possible on the basis of classical Maxwell’s electrodynamic theory, it became possible by means of Einstein’s light quantum hypothesis [1] (1905, Nobel prize 1921). In Einstein’s revolutionary new assumption light consists of small particles, the photons, which carry the energy ħω=. Energy can be transferred from the light beam to the metal only in portions of these quanta. Each electron which leaves the metal with an energy E el (2.1a) has taken over the energy of a photon. The intensity of a light beam with frequency ω is proportional to the number of photons with energy ħω in the beam. Thus, the emission current is also proportional to the number of photons. These assumptions consistently explain the photoelectric effect (Fig. 2.1).

Further properties of photons can be derived by means of relativity theory, where the speed of light is the absolutely highest possible velocity, and this in all inertial systems moving against each other with certain velocities. Photons as light quanta, thus, move with the speed of light c in the direction of light propagation described by the light wave vector k. From the existence of a maximum constant light velocity relativity theory gives an expression for the energy of a mass m moving with a momentum p:
$$ E=\sqrt{p^2c^2+m^2c^4}. $$
(2.2)
Light, that is, also its constituting particles, the photons, have no mass. Together with the light dispersion relation ω=ck one obtains from (2.2)
$$ E=\hbar \omega =\hbar ck = pc. $$
(2.3)
We, thus, must attribute a momentum p=ħk to the mass-less photons. We conclude that the electromagnetic field being continuous on the macroscopic scale is built up by small particles, the photons, to which we attribute the specific photon energy
$$ E =\hbar \omega =h\nu $$
(2.4a)
and a momentum
$$ \mathbf{p} =\hbar \mathbf{k}. $$
(2.4b)
The continuous field of classical Maxwell theory obviously has a granular character in reality which is not seen in phenomena on macroscopic scale.

2.2 Compton Effect

The particle character of electromagnetic radiation is also very clearly seen in the Compton effect, which was detected by Compton and Simon [2] in 1925. When X-rays with photon energies between 103 and 106 eV are scattered on free or weekly bound electrons, beside elastically scattered Rayleigh radiation (equal wavelength λ as incident radiation) there appears a second contribution of scattered radiation which is shifted in wavelength by Δλ, independent on the material of the scattering target (Fig. 2.2). In the elastic Rayleigh scattering process the oscillating electric field of the incoming X-rays excites electron oscillations (e.g. in the field of the positive nuclei) with the X-ray frequency. These electrons then again emit secondary radiation of the same frequency, the Rayleigh radiation. The additionally emitted radiation whose wavelength is shifted against the Rayleigh scattered one exhibits a characteristic dependence of the wavelength shift Δλ on the scattering angle ϑ (Fig. 2.2). This phenomenon can be explained quantitatively only under the assumption of an elastic collision with energy and momentum conservation between the electron and the light particle, the photon. We try this approach and write down the following ansatz for momentum conservation in x- and y-direction (Fig. 2.3b).
$$\frac{hv}{c}=\frac{h{v}'}{c}\cos \vartheta +m\text{ }\!\!\upsilon\!\!\text{ }\,\text{cos}\varphi \text{,}$$
(2.5a)
$$o=\frac{h{v}'}{c}\sin \vartheta -m\text{ }\!\!\upsilon\!\!\text{ }\,\text{sin}\varphi \text{.}$$
(2.5b)
Hereby, m is the mass of the propagating electron which is related to its rest-mass m 0 according to relativity theory by
$$ m=m_{0}\bigl(1-v^{2}/c^{2} \bigr)^{-1/2}. $$
(2.6)
As in the interpretation of the photoelectric effect (2.4b), the photon carries the momentum p=ħk=h/λ=/c. The observed frequency shift Δν=νν′ of the X-rays after scattering can therefore be related to a momentum change of the X-ray photons during the collision with an electron. Apart from momentum conservation (2.5a), (2.5b), also the relativistic energy conservation must hold for the particles, that is, with (2.4a) the energy of the photons must obey the relation
$$ h\nu +m_0 c^2=h\nu'+mc^2. $$
(2.7)
Hereby, the electron was assumed to be at rest (rest mass m 0) before the collision. By squaring (2.7) and by using (2.6) one obtains the following expression for the frequency change Δν: In (2.5a), (2.5b), sinφ and cosφ can be eliminated by using the relation sin2 φ+cos2 φ=1. After some calculation, one obtains A comparison of (2.8) and (2.9) shows the equality also of the left sides of the equations which yields
$$ m_0 c^2h\Delta \nu =h^2 \nu ( {\nu +\Delta \nu } ) ( {1-\cos \vartheta } ), $$
(2.10)
with
$$ \Delta \lambda =\frac{c}{\nu }-\frac{c}{\nu +\Delta \nu }= \frac{c\Delta \nu}{\nu ( {\nu +\Delta \nu } )}. $$
(2.11a)
Equation (2.10) yields
$$ \Delta \lambda =\frac{h}{m_0 c} ( {1-\cos \vartheta } )= \lambda_C ( {1-\cos \vartheta } ). $$
(2.11b)
The constant λ C is called Compton wavelength, it amounts to:
$$ \lambda_C =\frac{h}{m_0 c}=3.86\times 10^{-11}\ \text{cm}. $$
(2.12)
The Compton wavelength depends only on natural constants and is, therefore, of general interest. The quantum energy of radiation with a wavelength λ C corresponds just to the rest mass m 0 of the electron:
$$ \frac{hc}{\lambda_{C}}=h\nu =m_0 c^2=511\ \text{keV}. $$
(2.13)
Equations (2.11a), (2.11b) describe quantitatively the frequency or wavelength shift Δν or Δλ as a function of scattering angle ϑ as it is observed in the Compton effect.
Fig. 2.2

Original measurement data of Compton effect [2]. A graphite sample is irradiated by K α radiation from Mo under different angles (0° to 135°) with regard to the direction of incidence. The radiation is scattered elastically (λ=0.71 Å) without λ shift, partially inelastically with increased wavelength λ

Fig. 2.3

ac Scheme of Compton effect. a Experimental set-up. b Explanation of scattering parameters and particle parameters of X-rays ( photon energy, /c photon momentum) as well as of scattered electron (mv 2/2 energy, mv momentum). c Momentum conservation in a Compton scattering experiment

Someone who feels stressed by the relativistic calculation (2.5a)–(2.12) can obtain the result for the limit of small frequency changes by a non-relativistic treatment (Fig. 2.3c) where the electron mass is approximated by its rest mass (mm 0). Inspection of Fig. 2.3c easily shows that for the limit νν′ the momentum vectors of the incident and the scattered light are almost equal (/c′/c). By considering the two rectangular triangles SCB and SCA momentum conservation (mv=AB) yields the following relation:
$$ \frac{1}{2}mv=\frac{h\nu }{c}\sin \frac{\vartheta }{2}. $$
(2.14)
By means of (2.14), the change of the kinetic energy of the electron (initially at rest) which scatters the photon can be expressed by a change of photon energy (′):
$$ \frac{1}{2}mv^2=\frac{1}{2} \frac{ ( {mv})^2}{m}=\frac{4h^2\nu^2\sin^2\vartheta /2}{2mc^2}=h\nu -h\nu'. $$
(2.15)
With νν′ and by dividing nominator and denominator by 2, one obtains:
$$ \frac{2h}{mc^2}\sin^2\frac{\vartheta }{2}= \frac{\nu -\nu'}{\nu^2}\approx \frac{1}{\nu'}-\frac{1}{\nu }. $$
(2.16)
Written as a wavelength change, Δλ it follows
$$ \Delta \lambda =\lambda'-\lambda = \frac{2h}{mc}\sin^2\frac{\vartheta }{2}=\frac{h}{mc} ( {1-\cos \vartheta } ) . $$
(2.17)
In the non-relativistic limit (mm 0), this equation is identical with the more general relativistic relation (2.11a), (2.11b).

2.3 Diffraction of Massive Particles

While photoelectric and Compton effect can only be interpreted on the basis of the particle character of electromagnetic radiation, there are meanwhile numerous diffraction experiments (typical for waves) with all kinds of massive particle beams as electrons, neutrons, atoms, molecules etc. which doubtlessly demonstrate the wave-like propagation of these particles.

Already in 1919, Davisson and Germer detected intensity modulations in the reflection of low energy electrons from crystalline surfaces as a function of the observation angle [3]. The explanation of these observations became possible by De Broglie’s hypothesis that the propagation of electrons obeys the laws of waves [4]. In analogy to the photon, the mass-less light particle, De Broglie assumed the validity of the fundamental relation p=h/λ (2.4b) between momentum and wavelength also for massive particles as electrons. Relating the momentum p=mv to the kinetic energy E kin=mv 2/2 of a moving particle, one calculates the wavelength of a propagating electron as
$$ \lambda =h ( {2mE_{\text{kin}} } )^{-\frac{1}{2}}, $$
(2.18a)
that is, electrons which have been accelerated by a voltage U possess a wavelength
The experiments of Davisson and Germer have initiated the development of a standard characterization method for the atomic structure of solid surfaces, the LEED technique (low energy electron diffraction). The experimental set-up for LEED studies is meanwhile found in every surface science laboratory around the world. The schematic representation of such an experiment is shown in Fig. 2.4. The solid surface under study is arranged in front of a curved fluorescent screen in a vacuum vessel, usually an ultrahigh vacuum (UHV) chamber with background pressure below 10−10 Torr. Through an opening in the screen, an electron beam with well defined kinetic particle energy E kin=eU obtained by acceleration in a bias between 30 V and 200 V is irradiated on the crystal surface. The electrons backscattered from the sample surface have to pass an acceleration grid in front of the fluorescent screen and an acceleration voltage of some 1000 V in order to have enough energy to become visible on the fluorescent screen. When the sample surface under study is crystalline one always observes more or less bright intensity peaks on the screen, the so-called LEED reflexes. In Fig. 2.5, the LEED reflexes observed on a clean ZnO surface prepared in UHV are shown. The interpretation of this reflex (LEED) pattern is only possible by attributing the propagating electrons in the primary beam a wave. When this electron wave hits the surface atoms of the sample, each atom in the lattice emits a spherical wave. All these spherical waves superimpose and interfere constructively in certain directions and destructively in others. Since electrons with low energies in the order of 100 eV are scattered preferentially on the uppermost atomic layer, the scattering target is 2-dimensional to first approximation. According to Fig. 2.4b, the path difference between two partial waves originating from atoms A and B is Δs=asinϑ with a as the interatomic distance within the surface. For constructive interference, Δs must equal a multiple of the electron wavelength λ, which yields the condition
$$ a \sin\vartheta =n \lambda . $$
(2.19)
Fig. 2.4

ab Scheme of a LEED diffraction experiment (LEED is Low Energy Electron Diffraction) with slow electrons. a Experimental set-up in ultra-high vacuum (UHV). b Schematic representation of the diffraction of an incident electron on the upper most atomic layer of a crystal. The atoms A and B are the origin of scattered spherical waves which interfere constructively (Bragg reflection peak) or destructively depending on path difference Δs. a is distance between atoms

Fig. 2.5

LEED diffraction pattern of electrons of a kinetic energy eU=140 eV on a \(\mathrm{ZnO}(10\bar{1}0)\) surface. The electrons are incident normal to the crystal surface; bright spots are Bragg reflection spots due to constructive superposition of waves. The dark shadow in the diffraction pattern is due to the crystal holder

Diffraction intensity is thus expected on a cone with opening angle (π/2−ϑ) around the atom row along A and B. Since the arrangement of scattering atoms is 2-dimensional a second condition for constructive interference, analogously to (2.19), must be fulfilled in a direction normal to AB in the surface. The two conditions together limit the spatial range for constructive interference to only one direction, that is, the direction of a particular LEED reflex (bright spot in Fig. 2.5). The different diffraction spots in Fig. 2.5 belong to higher diffraction orders, that is, to different numbers n in (2.19) and the corresponding second equation. For the interpretation of a LEED pattern as in Fig. 2.5, one calculates the electron wavelength from the kinetic energy of the primary electrons, or respectively from the acceleration voltage according to (2.18a), (2.18b). By means of (2.19), the observation angle for a particular LEED reflex yields information about the interatomic distance, more accurately the periodicity interval, within the sample surface. LEED is meanwhile a standard analysis technique in surface science. Each LEED experiment, many times performed around the globe, demonstrates the wave character of propagating electrons.

Not only moving electrons but also other particles obey the laws of wave propagation. Already in 1930 Estermann and Stern demonstrated that He and H2 beams undergo diffraction phenomena on solid surfaces [5]. A clear example from recent time are diffraction experiments with He beams on clean, UHV prepared Pt surfaces [6]. The Pt surfaces exhibit a series of regularly spaced monoatomic steps (distance a=2 nm) which are produced by cutting the crystal at the appropriate angle and annealing in vacuum. The atomic He beam used in the experiment is produced by a supersonic expansion of the gas from a nozzle. The interaction between the atoms in the expanding gas produces a velocity distribution that is significantly sharper than the Maxwell distribution present before the expansion. The energetically sharp He beam is irradiated on the Pt surface under UHV conditions (background pressure below 10−10 Torr). In Fig. 2.6a, the diffracted intensity of He atoms is shown as a function of the scattering (reflection) angle ϑ r with a fixed angle of incidence ϑ i =85° against the surface normal. The intensity maxima correspond to the diffraction orders of the periodic lattice of terraces, that is, steps on the Pt surface rather than from the lattice of individual atoms. The steps act as scattering centers, they form a 1-dimensional array. Thus, for the interpretation of the scattering distribution (Fig. 2.6a) relation (2.19) can directly be applied. Only the path difference between two neighboring scattered beams contains the amounts Δs i and Δs r of the incident and the reflected (scattered) wave. The position of the diffraction maxima is thus given by
$$ a ( {\sin \vartheta_i -\sin \vartheta_r} )=n\lambda . $$
(2.20)
From (2.18a) the wavelength of the He atoms in the beam is obtained as 0.56 Å. With the step distance a=2 nm the intensity maxima numerated by n=0,1,2,3,… in Fig. 2.6a are calculated. The agreement between theory and experiment is excellent. As in the case of an optical echelon grating, the direction corresponding to specular (mirror) reflection from the terraces (maxima 3 and 4) is favored in the intensity distribution.
Fig. 2.6

a, b Diffraction of a He atom beam on a Pt surface with a regular step array, step distance a=2 nm [6]. Like for an Echelette grating in light optics maximum diffraction intensity is obtained in diffraction orders which appear under specular direction with regard to the interaction potential. a Diffracted intensity as function of scattering angle ϑ r ; angle of incidence ϑ i =85° with regard to the Pt surface normal. The reflection angles indicated by 0,1,2,…,5 are calculated for a step distance a=2 nm. b Scheme of the diffraction geometry. The path differences Δs i and Δs r determine the reflection angle, under which the diffraction peak appears

Neutrons interact only extremely weakly with matter because of their missing charge. They penetrate relatively thick solid samples without a significant loss of beam intensity. But also in this case, neutrons which are irradiated on a solid crystalline sample, produce, beside the directly transmitted beam, well-defined sharp beams of neutrons which are diffracted into certain angles with respect to the primary beam direction (Fig. 2.7). The interpretation of the experimental results is based, similarly as in the case of electrons or He atoms, on the assumption of the propagation of neutron waves and their diffraction on the regularly arranged atomic nuclei in the crystal [7].
Fig. 2.7

a, b Neutron diffraction on a FeCo alloy [7]. a disordered (left) and ordered (right) phase of FeCo. b Neutron diffractogram of the ordered and disordered phase of FeCo. Because of low counting rates in neutron diffraction long measurement times are needed

All these experiments with particle beams demonstrate clearly and doubtlessly, that the propagation of massive particles as electrons, neutrons, molecules etc. must be described in terms of wave expansion. Otherwise, we could not understand the occurrence of diffraction and interference phenomena observed with these particles and which are used meanwhile worldwide in standard characterization and analysis techniques in solid state and surface physics. Present cutting edge research in this field aims at the physical limits for the observation of particle interference with bigger and bigger particles.

Interference patterns have already been observed even with Huge Bucky ball molecules cinsisting of 60 carbon atoms (C60). Experiments with viruses are on the way. The interesting question is, at what particle size is the quantum character lost and the particle starts to behave classically.

2.4 Particle Interference at the Double Slit

Interference experiments with a double slit, that is, the appearance of diffraction fringes on a screen after a light beam has passed the double slit arrangement, lead Th. Young already in 1802 to the interpretation of light as a wave. Instead of a double slit A.J. Fresnel used a bi-prism (Fig. 2.8a) for the demonstration of double slit interferences. In this particular set-up a monochromatic light beam originating from a single slit S illuminates a double prism with small prism angles. This bi-prism splits the primary beam into two partial beams which are superimposed on a remote screen. As is seen from Fig. 2.8a, the two partial beams seem to originate from two virtual slits S′ and S″. The interference pattern observed on the screen, thus, is identical with one produced by a double slit arrangement as in Young’s experiment. The intensity I of the interference pattern reaches a maximum when the path difference between the two partial waves from S′ and S″ equals a multiple of the light wavelength λ. Destructive interference, that is, intensity minima appear on the screen for path differences of odd multiples of λ/2. These types of double slit interferences can only be explained in terms of wave propagation, a non-local phenomenon. An interpretation on the basis of a particle picture is excluded.
Fig. 2.8

ac Double slit diffraction of light and of electrons. a Set-up for the observation of optical double beam interference with monochromatic light. The two light beams are produced by an optical biprism. b Analogue equipment for the observation of electron double beam interference. The biprism is realized by a positively charged metal filament in an electron microscope column. c Electron double slit interference pattern produced by the experimental set-up in (b) [8, 9]

2.4.1 Double Slit Experiments with Electrons

Already in 1956, G. Möllenstedt and H. Düker performed a double slit experiment with electrons by means of a bi-prism [8]. The bi-prism for electrons in this experiment consisted of a positively charged metallic filament arranged between two planar electrodes on ground potential (Fig. 2.8b). This set-up is incorporated into an electron microscope column, where an electron beam is focused in a focal point F (Fig. 2.8b). The double prism arrangement splits the electron beam into two partial beams, similarly as in the optical analogon, and deflects the two beams to the center again. The electric field of the positive filament is proportional to r −1 (r distance from filament). An electron passing the wire in close vicinity is strongly deflected horizontally, but only for a short time. An electron passing further away experiences a smaller force, but this for a longer time. The total deflection angle of the electrons in the field of the wire surprisingly depends only on the electron energy and not on the distance from the wire. Thus the two partial electron beams are focused and superimposed on a photosensitive screen behind. An interference pattern with bright and dark fringes is observed (Fig. 2.8c). Electrons with a fixed energy thus behave as light waves passing Fresnel’s bi-prism or Young’s double slit, a further demonstration of the wave character of electrons.

The experiment of Möllenstedt and Düker was repeated by Tonomura et al. [10] in 1989 with more sophisticated experimental tools. A particular advancement was the use of extremely sensitive, space resolving (imaging) semiconductor detectors. A whole field of highly sensitive pixel detectors enables the detection of one single electron at one pixel and, thus, the computer aided construction of an image of the spatial distribution of the electrons having passed the double slit. The results of the experiment (Fig. 2.9) clearly show the unexpected and weird behavior of electrons propagating in space.
Fig. 2.9

ad Successive formation of a two-beam (double slit) electron interference pattern. The diffraction experiment has been performed by means of a biprism set-up as depicted in Fig. 2.8b [10]. The electron density in the beam is such low that only one single electron passes the electron microscope column at a time. Only single distinct electrons are detected, one after each other, on the 2-dimensional spatially resolving pixel detector screen. The diffraction patterns (a) to (d) are recorded after increasing electron numbers have passed the apparatus

Electrons expand in space according to the laws of waves, they produce interference patterns, just as light does. But the interference fringes become visible only after the observation of a sufficiently large number of electrons. The observation of only 10 electrons which have passed the bi-prism (Fig. 2.9a) yields a random flash of one pixel somewhere on the screen. An interference pattern can not be recognized. Collecting 100, 3000, or 70,000 events of electrons which have passed the bi-prism builds up step by step the double slit interference pattern (Fig. 2.9d). Only for an ensemble with huge numbers of electrons the laws of wave propagation are valid. One single electron behaves randomly; totally unexpected and statistically the response of a pixel on the screen is caused by an impinging electron which transfers its kinetic energy to the point-like pixel detector.

It must be emphasized at this point that an electron–electron interaction can be excluded while the electrons pass the double prism arrangement to form the interference fringes. Two subsequent electrons do not “see” each other in space and time. The intensity of the electron beam current is so low that only after the detection of one electron in a pixel detector the next electron leaves the cathode of the microscope column.

Single electrons have the choice to take one or the other path—through this or the other slit—they are detected as point-like particles in a pixel detector, but randomly distributed over the screen. We do not know their individual history, but as an ensemble they build up the interference pattern without having information about each other. This particle-wave duality, which is absolutely counter-intuitive, weird in our imagination, is at the heart of quantum mechanics. Feynman [11] describes this behavior being apparent in the double slit experiment as “impossible, absolutely impossible to explain in any classical way, and has in it the heart of quantum mechanics”. We have to get familiar with the idea, that nature behaves completely different from our everyday experience on an atomic scale or below. For human beings, the natural length scale is that of centimeters and meters corresponding to the perception horizon in our macroscopic surrounding. It would be astonishing, on the other hand, if our sense organs and our brain, which have adapted during more than 100 million years of biological evolution to a macroscopic environment, could perceive the reality of the whole cosmos, the smallest and largest on subatomic and cosmological length scales. In these periods of adaptation it was much more important for human survival to correctly estimate the width of a creek or the distance between two branches of an arbor than the path of an electron. We should, therefore, not be surprised that the atomic and sub-atomic world as it appears in quantum physics is not accessible to our limited senses and imagination. We should, however, be surprised that mathematics opens the way to create an abstract picture of the atomic behavior which allows even quantitative predictions of experimental results. The most straightforward explanation is certainly that a structured reality does exist beyond human perception and imagination which obeys the laws of logic. Mathematics and logic obviously go beyond the reality accessible to our senses and enable the invention of theoretical systems as quantum theory which can correctly describe wide fields of reality extending much further than our meter and centimeter environment.

2.4.2 Particle Interference and “Which-Way” Information

The behavior of atomic and sub-atomic particles becomes even more strange when we ask the question through which particular slit has the particle moved in the double slit experiment (Sect. 2.4.1). Is this question for the detailed way of the particle compatible with the observation of the double slit interference pattern? Already in the early days of quantum mechanics, around 1920, this question was discussed extensively in gedanken (thought) experiments by Heisenberg, Einstein and others and later by Feynman [11]. The essential conclusion of all these discussions always was that the interference pattern can only be observed without additional experiments to elucidate the detailed path (“which-way” information) of the particles. Every measurement of the detailed way, e.g. by scattering of a photon (see Compton effect, Sect. 2.2) in front of one of the two slits transfers so much momentum p=ħk to the electron that interference of the electron waves is not possible anymore, the fringe pattern is washed out due to phase shifts. According to the arguments of Heisenberg and Feynman the photon energy of the probing light can be decreased to such an extent that its effect on the electron is negligible. But simultaneously one has to increase the wavelength of the light, because of p=ħk=h/λ, to an amount which does not resolve the spatial distance between the two slits anymore. Microscopic imaging of a structural dimension d, namely, requires λ<d. In the gedanken experiment, the measurement of the detailed particle path requires a light wavelength λ<slit distance, which simultaneously is accompanied by a momentum transfer to the electron high enough to destroy the interference pattern.

In recent time, now, experiments became possible, where the “which-way” information can be obtained without significant momentum transfer to the diffracted particle in a double slit experiment. But look, the interference pattern disappears without momentum transfer. The interference fringes can only be seen, when the detection apparatus for the “which-way” information is switched off. Dürr, Nonn and Rempe [12, 13] have performed an experiment with a beam of Rb atoms which are diffracted on a standing laser light wave. As we will see later in Chap.  8, high intensity standing light waves with their spatially fixed intensity maxima and knots (intensity=0) act as a diffraction grating for atoms, with a grating period of half the light wavelength, similarly as the periodic array of atoms in a crystal (Sect. 2.3). According to Fig. 2.10a, diffraction of the Rb atoms on a first standing wave produces, beside the transmitted beam C (0th order) a beam B diffracted in 1st order. These two atom beams hit a second standing light wave where they are diffracted into the beams D, E and F, G, which pair-wise interfere with each other. Thus, two interference patterns phase shifted against each other are produced in a space resolving imaging detector behind. Figure 2.10b shows the experimentally observed interference patterns for two different laser light wavelengths with knot distances (periodicity period) d=1.3 and 3.1 μm.
Fig. 2.10

ac Two-beam interference of two Rb atom beams. An internal degree of freedom (spin orientation of the outer Rb shell electron) can yield information about the path of a single electron (“Which Way Information”) [12, 13]. a Scheme of the atom interferometer: By Bragg reflection on an intense standing laser light wave the incident atomic beam A is split into two partial beams B and C. A second standing laser light wave splits these two beams into the partial beams D and E (negative spatial coordinate), respectively F and G (positive spatial coordinate). These beams pair-wise interfere with each other. Irradiation of microwaves before entering the first diffraction grating (1st standing laser light wave) can excite the Rb atoms in an excited internal state. A 2nd microwave pulse irradiated between the two diffraction gratings (1st and 2nd standing laser light waves) allows the read-out of the “Which Way Information”, i.e., the detailed path of the two interfering atomic beams (see also Sect.  8.2.4 and Fig.  8.5). b Measured atomic beam interference pattern originating from the superposition of the partial beams D and E, respectively, F and G; for these measurements the Which-Way Information” was not recorded (no microwave pulses); results with two different grating periods (node distance of standing laser waves) d=1.3 μm and d=3.1 μm. The solid lines are calculated results. c Measured beam intensities upon superposition of partial beams D and E, respectively, F and G and recording the “Which-Way Information” using microwave pulses

A special property of this experiment is due to the fact that the diffracted Rb atoms are characterized, beside their spatial information, that is, the probability of being somewhere, also by internal degrees of freedom as spin excitations etc. We will be able to understand details of the described experiment only much later in this book (Sect.  8.2.4) after we have learnt a lot more about quantum theory. Nevertheless, it should be anticipated at this point, that irradiation of microwave radiation with a frequency of 3 GHz excites the Rb atoms into an excited state before entering the first diffraction grating (1st standing wave). A second microwave pulse irradiated after the splitting into the two partial beams B and C allows the distinction between the two possibilities if the interference pattern (beams D and E respectively, F and G) originates from an atom of the partial beam B or C.

In this experiment, the two beams of the double slit experiment are realized by the partial beams B and C. By means of microwave pulses before and after passing the first diffraction grating (1st standing light wave) one can distinguish between the ways B and C which could have been taken by the atom. It is easily estimated (Sect.  8.2.4) that a photon of 3 GHz microwave radiation can not transfer enough momentum to the relatively heavy Rb atom such that the interference pattern is washed out. Nevertheless switching on the microwave radiation as the measurement probe destroys the interference (Fig. 2.10c). Only a monotonous intensity background corresponding to the average Rb atom density in the beams D and E respectively, F and G is detected. This experimental result is found independently on the observation by a human experimentalist; only the read-out of the which-way information by the corresponding hard-ware probe is essential for the appearance or disappearance of the interference pattern.

What do we learn from this experiment? First, we see that not the human observer has an effect on the outcome of the interference experiment, only the switched on measurement probe for the which-way information is responsible for the destruction of the interference pattern.

Real world does not worry if it is observed by a human being (Realism instead of Idealism!). Furthermore, there must exist a correlation between the observed particle and the measurement probe, which can not be reduced to energy or momentum transfer between particle and measurement set-up. This phenomenon which is inherently of quantum mechanical character is typical for atomic and sub-atomic systems and beyond our macroscopic perception. It is called “entanglement” (Verschränkung in German, as Schrödinger called it), we will better understand what it means after having learnt more about quantum physics (Chap.  7).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hans Lüth
    • 1
  1. 1.Forschungszentrum Jülich GmbH, Peter Grünberg Institut (PGI)PGI-9: Semiconductor Nanoelectronics and Jülich Aachen Research Alliance (JARA)JülichGermany

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