Quantum Physics in the Nanoworld pp 9-26 | Cite as

# Some Fundamental Experiments

## 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 ExperimentIt 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

*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.

*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}=2

*πν*

_{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:

*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

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*=*hν*. 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).

*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*:

*ω*=

*ck*one obtains from (2.2)

*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

## 2.2 Compton Effect

^{3}and 10

^{6}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 Δ

*λ*, 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 Δ

*λ*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).

*m*is the mass of the propagating electron which is related to its rest-mass

*m*

_{0}according to relativity theory by

*p*=

*ħk*=

*h*/

*λ*=

*hν*/

*c*. The observed frequency shift Δ

*ν*=

*ν*−

*ν*′ 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

*m*

_{0}) before the collision. By squaring (2.7) and by using (2.6) one obtains the following expression for the frequency change Δ

*ν*:

*φ*and cos

*φ*can be eliminated by using the relation sin

^{2}

*φ*+cos

^{2}

*φ*=1. After some calculation, one obtains

*λ*

_{ C }is called Compton wavelength, it amounts to:

*λ*

_{ C }corresponds just to the rest mass

*m*

_{0}of the electron:

*ν*or Δ

*λ*as a function of scattering angle

*ϑ*as it is observed in the Compton effect.

*m*≈

*m*

_{0}). Inspection of Fig. 2.3c easily shows that for the limit

*ν*≈

*ν*′ the momentum vectors of the incident and the scattered light are almost equal (

*hν*/

*c*≈

*hν*′/

*c*). By considering the two rectangular triangles SCB and SCA momentum conservation (

*mv*=

*AB*) yields the following relation:

*hν*−

*hν*′):

*ν*≈

*ν*′ and by dividing nominator and denominator by

*hν*

^{2}, one obtains:

*λ*it follows

*m*≈

*m*

_{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.

*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

*U*possess a wavelength

^{−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 Δ

*s*=

*a*sin

*ϑ*with

*a*as the interatomic distance within the surface. For constructive interference, Δ

*s*must equal a multiple of the electron wavelength

*λ*, which yields the condition

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.

_{2}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 Δ

*s*

_{ i }and Δ

*s*

_{ r }of the incident and the reflected (scattered) wave. The position of the diffraction maxima is thus given by

*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.

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 (C_{60}). 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

*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.

### 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.

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.

*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.

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).

## References

- 1.A. Einstein, Ann. Phys.
**17**, 132 (1905) MATHCrossRefGoogle Scholar - 2.A.H. Compton, A. Simon, Phys. Rev.
**25**, 306 (1925) ADSCrossRefGoogle Scholar - 3.C.J. Davisson, L.H. Germer, Phys. Rev.
**30**, 705 (1927) ADSCrossRefGoogle Scholar - 4.L. de Broglie, C. R. Acad. Sci. Paris
**177**, 507 (1923) Google Scholar - 5.I. Estermann, O. Stern, Z. Phys.
**61**, 95 (1930) ADSCrossRefGoogle Scholar - 6.G. Comsa, G. Mechtersheimer, B. Poelsema, S. Tomoda, Surf. Sci.
**89**, 123 (1979) ADSCrossRefGoogle Scholar - 7.C.G. Shull, S. Siegel, Phys. Rev.
**75**, 1008 (1949) ADSCrossRefGoogle Scholar - 8.G. Möllenstedt, H. Dücker, Z. Phys.
**145**, 366 (1956) Google Scholar - 9.C. Jönsson, Z. Phys.
**161**, 454 (1961) ADSCrossRefGoogle Scholar - 10.A. Tonomura, J. Endo, T. Matsuda, T. Kaeasaki, E. Ezawa, Am. J. Phys.
**57**, 157 (1989) ADSCrossRefGoogle Scholar - 11.R.P. Feymann, R.B. Leighton, M. Sands,
*The Feymann Lectures on Physics—Quantum Mechanics*(Addison-Wesley, Reading, 1965) Google Scholar - 12.S. Dürr, T. Nonn, G. Rempe, Nature
**395**, 33 (1998) ADSCrossRefGoogle Scholar - 13.S. Dürr, G. Rempe, Adv. At. Mol. Opt. Phys.
**42**, 29 (2000) ADSCrossRefGoogle Scholar