Module 02: Wave-particle duality, de Broglie waves and the Uncertainty principle

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1 PG Pathshala Subject: BIOPHYSICS Paper 0: Quantum Biophysics Module 0: Wave-particle duality, de Broglie waves and the Uncertainty principle Principal Investigator: Prof. Moganty R. Rajeswari Professor, Department of Biochemistry All India Institute of Medical Sciences, New Delhi Co-Principal Investigator: Prof. T. P. Singh Distinguished Biotechnology Research Professor All India Institute of Medical Sciences, New Delhi Paper Coordinator and Content Writer: Prof. Rabi Majumdar Emeritus Professor, Department of Natural Sciences West Bengal University of Technology, Kolkata Formerly Professor and Head, Biophysics Division Saha Institute of Nuclear Physics, Kolkata Content Reviewer: Prof. Manju Bansal Professor, Molecular Biology Unit Indian Institute of Science, Bangalore 1

2 Introduction Basic fact which leads to development of Quantum mechanics is wave-particle duality not only for light but also for matter. Wavelengths for electrons are obtained from de Broglie hypothesis and the Bohr orbits are pictured as standing electron waves. Heisenberg s uncertainty principle and its consequences are also discussed. Learning Outcome Planck s law of black-body radiation is based on the particle nature of light. It helps us to calculate the temperature of a hot object by measuring the wavelength for peak intensity of radiation it emits. Electron exhibits diffraction pattern showing that like light it has a dual nature too. It can behave like a particle as well as like a wave. The wavelength of an electron depends on its energy and can be obtained from the de Broglie relation. The de Broglie wavelength for electrons helps us to view Bohr orbits in atoms as a system of standing electron waves. Electron microscope is a practical application of the wave nature of electrons. Its resolving power can be enhanced considerably by using high energy electrons. Heisenberg s uncertainty principle and its consequences are described. In view of the uncertainty relation, the motion of a particle should be described by a wave packet rather than by a single wave..1 Particles of Light: Photons It is well known that light is an electromagnetic wave, but can also appear as photons, which are special kind of particles having zero mass and moving only with speed c = 3.0 x 10 8 m/s in vacuum. Thus light exhibits dual character. There are specific examples, which may be cited in support of this dual character. For example, the usual interference or diffraction pattern is produced in the laboratory when light is passed through closely spaced narrow slits (such as the Young slits or a grating). This is characteristic of the wave nature of light. There are other examples, which support the particle nature of light, such as the photoelectric effect, Compton scattering and Planck s formula for Black-body radiation. These are summarized below. Photoelectric Effect Einstein explained photoelectric effect by assuming that the incident light consists of photons, which knock out electrons from the surface of a metal. A part of the photon energy h is utilized in overcoming the work function W of the metal and the remaining energy (h - W) accounts for the maximum kinetic energy of the ejected electrons. Thus 1 m max ( h W ) ev S (.1) Maximum velocity of the ejected electrons is determined by applying a stopping potential V S. The observed value agrees very well with this result based on the particle nature of light.

3 Compton Scattering In Compton scattering, when x-rays are passed through a metallic foil, a change in wavelength is observed, which depends on the angle at which the x-rays are scattered by electrons in the foil according to the formula h mc ( 1 cos ) (.) where m is the mass of the electron and (h/mc) is called the Compton wavelength. The derivation of this formula is based entirely on the particle nature of light. The observed change in wavelength is due to elastic collision of the x-ray photons with electrons. What is Back-body Radiation? In physics, black body ideally represents an object, which absorbs light of all wavelengths falling on it. Since no light is reflected or transmitted, particularly in the visible region, the object appears black when it is cold. However, on heating, the black body emits light of different wavelengths, the intensities of which depend on the temperatures. This is called black-body radiation as shown in Fig..1. At room temperature, a black body emits mostly infrared wavelengths. In real life, an object like human body can be treated approximately like a black body at temperature 310 K, which emits infrared light. The same happens in the case of an iron rod, for example, at room temperature. On heating, the emission maximum of a black body shifts towards shorter wavelengths (Fig..1), so that it starts emitting visible light and becomes red, orange, yellow, and so on, until at still higher temperatures, it emits substantial ultraviolet radiation. The sun can also be approximated by a black body. The outer region of sun having a temperatures around 5000 K emits wavelengths in the visible region, which appear as white light. The ultraviolet light comes from the interior of sun, which has much higher temperatures. Planck s Formula The spectrum or the intensity distribution of black-body radiation at different temperatures is shown in Fig..1. As mentioned already, please note that the emission maximum shifts towards shorter wavelengths on heating. Planck s law of black-body radiation accounts for the entire spectrum by the famous formula 8 h u(, T) d d (.3) 3 h / kt c e 1 This is known as Planck s formula, where is the frequency of emission and u(,t) is the corresponding energy density at temperature T. Substituting = c/, the formula becomes hc 1 u(, T) d d 5 hc/ kt (.4) e 1 The energy density u(,t) is a measure of the intensity of emission for different wavelengths at any given temperature T. 3

4 The results predicted by this formula are in excellent agreement with the curves of Fig..1. The figure also indicates how classical theory, based on the wave nature of light, fails to account for the entire intensity spectrum. The classical curve is based on Rayleigh-Jeans formula (not shown here). At higher temperatures, it gives fairly good results for higher wavelengths, but the intensity tends to blow off for shorter wavelengths, leading to an imaginary situation known as the ultraviolet catastrophe. Fig..1 Black-body radiation spectrum at different temperatures (Planck s Law) Planck s formula was empirical and he made it acceptable by postulating that radiation consists of harmonic oscillators, which could take only discrete energy values that are multiples of a fundamental energy ε = hν, where ν is an oscillator frequency and h is now known as Planck s constant. Temperature dependence of the oscillator energy distribution led to the desired formula. Famous Indian physicist S.N. Bose later derived Planck s formula, on the basis of particle nature of light, by assuming that light consists of photon gas. Such an assembly of photons obeys Bose statistics, which was later generalized by Einstein to Bose-Einstein statistics. Planck s law is considered to be starting point of quantum theory. Wavelength for Peak Intensity Wavelength for peak radiation intensity at a given temperature can be obtained from u(, T) 0 The resulting equation is solved numerically to obtain 4

5 max hc 4.96kT.9x10 T 6 nm where T is the absolute temperature. This is known as Wien s displacement law in classical theory, which was used to obtain the temperature of hot objects like stars by observing the wavelength for peak radiation intensity. EXAMPLE: Let us check that human body at temperature 310 K mostly radiates in the infrared region at 9000 nm, while sun s outer surface at 5800 K radiates mostly in the visible region around 500 nm. Solution: Just plug in the temperatures T in the expression for λ max to obtain the desired results. Photon wavelength The examples cited above exhibit particle nature of light and confirm the existence of photons. Please remember that photon is a special kind of particle, which is without any mass (strictly speaking it has zero rest mass in Einstein s theory of relativity) and always moves with velocity c = 3.0 x 10 8 m/s in vacuum. The energy of a photon is given by the well known formula Using c =, the photon wavelength is given by E h (.5) hc (.6) E Here is characteristic of wave propagation. This is the energy-wavelength relation for photons. This equation enables us to calculate the wavelength of light, which varies inversely as the photon energy. Thus the UV light, having wavelengths shorter than those of the visible light, has higher energies. The penetrating x-rays have still shorter wavelengths and hence still higher energies. Check that a typical x- ray photon of wavelength 1.0 nm has energy 1.5 kev.. Wave-Particle Duality: Wave Nature of Electrons Like photon, does the electron also have a wave character? The answer is yes. According to the famous French physicist Louis de Broglie, an electron or any other particle of mass m also exhibits wave character. Remarkably, this idea was a part of his Ph. D. thesis. It was proposed that the wavelength associated with a particle of momentum p is h h (.7) p m Introducing the wave number k = /, we may express this in the form p k (.8) 5

6 The wavelength is usually called the de Boglie wavelength. The idea of wave-particle duality was later put to practice by Davisson and Germer, who demonstrated that the electrons, like light, are capable of producing a diffraction pattern under certain conditions. Fig.. shows a typical diffraction pattern of electrons scattered from aluminum foil and recorded on film. Fig... A typical electron diffraction pattern The total energy of the electron can be expressed as a sum of the kinetic and potential energies. Thus the total energy is 1 p E m V( r) V( r) (.9) m This relation between E and p is also called the energy-momentum relation. For a free particle, V = 0, so that this equation reduces to E p / m (.10) Combining equations (.7) and (.10), we get h (.11) me This is the energy-wavelength relation for electrons (or any other material particles), which is the analogue of the equation (.6) for photons. The equation (.11) enables us to calculate the de Broglie wavelength of electrons having mass m = 9.1 x kg and energy E. Clearly, the electrons with higher energies have shorter wavelengths but, in this case, the wavelength varies inversely as the square root of the energy. This led to discovery of the electron microscope..3 Electron Microscope: A practical application of wave -particle duality How does an electron microscope differ from an ordinary optical microscope? An ordinary microscope generally uses visible light, whose wavelength determines the resolving power of the microscope. We know from our study of high school optics that the resolving power (RP) of a microscope with numerical aperture D: 6

7 1.D RP (.1) Thus for shorter wavelengths, the resolving power is higher, so that we can distinguish between objects closer together. In an electron microscope, however, much shorter wavelengths can be achieved, according to equation (.1), by using high energy electrons. Fig..3. Simplified Diagram of a Transmission Electron Microscope (TEM) Fig..4. Picture of polio virus. Electrons in a TEM are accelerated to energies of a few kilo electron-volts (kev), where their wavelength becomes small enough to give a much higher resolution than what is observed with an ordinary light 7

8 microscope. The electron beam is focused on a thin sample of the object by a magnetic lens arrangement. A fluorescent screen or a special photographic film is used to register the image. A photograph processed under the electron microscope is called an electron micrograph. Fig..4 shows the picture of polio virus, where the magnification is a few hundred thousand. A living cell can also be pictured by a TEM..4 Quantum Condition for Allowed Orbits We have learnt that the electrons have to satisfy the wave-particle duality condition as described by the equation (.7), namely h p h m In the case of an electron confined in a hydrogen atom with circular orbits, let us assume that the circumference of an allowed orbit must be an integral multiple of the debroglie wavelength of the electron in that orbit, so that r n (.13) where n is a positive integer. If a wave does not close on the circle, it interferes destructively with itself (as it goes round) and rapidly dies out. This happens for orbits that do not satisfy the condition (.13). If all the allowed orbits in an atom are constrained by the condition (.13), then the electron orbits in an atom are viewed as a series of standing waves, which do not change with time and forms a stationary pattern. Fig..5 illustrates two such orbits for n = 3. Fig..5. A standing wave in third Bohr orbit of an atom. The correctness of this standing wave picture of the allowed orbits can be verified by eliminating from equations (.7) and (.13). We get nh mr n (.14) where = h/. This means that the angular momentum L = mvr of the electron in an allowed orbit is an integral multiple of h/. It must be recognized that Bohr used this condition as a historical postulate in his proposed model for atoms. 8

9 .5 Thermal Neutrons and Free Electrons Gas Thermal neutrons are normally used in nuclear fission, while free valence electrons in metals are responsible for thermal and electrical properties of the metals. Let us compare their de Broglie wavelengths at room temperature (T = 300 K). It is well known from kinetic theory of ideal gases that the average thermal energy E of a particle is 3 E kt where k is the Boltzmann constant. We assume that same applies to thermal neutrons (neutron gas) or free electrons (electron gas) at T = 300 K. Plugging this into equation (.11), the wavelength has the form h 3mkT EXAMPLE: Compare the values of de Broglie wavelengths for thermal neutrons and free electrons gas at T = 300 K. Solution: Using neutron mass m = 1.67 x 10-7 kg, we get 0.15 nm = 1.5Å for neutron. In the case of electron, we put m = 9.1 x kg and get = 6.0 nm = 60 Å, which is much larger than that of the neutron. The electron wavelength at normal temperatures is also much larger than the usual distances between atoms in metals..6 The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle is yet another basic concept on which modern quantum mechanics is based. According to this principle, a particle like electron can never be located with certainty in space. Any attempt to observe an electron will involve light or bombardment with photons that will disturb the electron, leading to uncertainty in its position. Suppose x denotes this inherent uncertainty of the electron s position in one-dimensional space, as if the electron is surrounded by an uncertainty cloud, beyond which its position cannot be determined. According to uncertainty principle, such an uncertainty in position x is always accompanied by a minimum uncertainty in momentum p, such that x p (.15) The exact relation is usually stated to be ΔxΔp ħ/ We will find that an electron is represented by a wave function (denoted by), which tells us about the uncertainty in its position more precisely in terms of probabilities. A Thought Experiment To make uncertainty principle plausible, imagine a thought experiment with microscope, where we measure the position of an electron with photons. It is natural to assume that an object can be seen at best to an accuracy of about the wavelength of the radiation used. Thus the uncertainty in electron s position is limited to 9

10 x But a photon with shorter wavelength has larger momentum (since photon energy E = cp, which gives p = E/c = h/c = h/) and it will transfer this momentum to the object. Assuming a complete momentum transfer, so that p h xp h This result clearly makes the uncertainty relation (.15) plausible. WERNER HEISENBERG Ground State Energy of Hydrogen Atom Let us now see how the uncertainty relation puts a natural constraint on the size of a hydrogen atom and also helps us to estimate its ground state energy. In a hydrogen atom, let the minimum possible distance of the electron from the nucleus be a. This is the radius of the atom in its ground state. In this case, the entire momentum p of the electron is expected to arise as a result of the uncertainty principle. Thus putting x = a and p = p in equation (.15), we get p h / a (.16) It is interesting to note that this is equivalent to de Broglie relation (.7) for a ground state electron in hydrogen atom, since equation (.14) gives a = for the ground state. Using equation (.9), the total energy of the electron in a hydrogen atom is 10

11 h e E k (.17) 8 ma a The system will settle to a state of lowest energy for the value of a to be determined from the condition de 0 da Applying this condition, we get for the size of the hydrogen atom, h a (.18) 4 kme kme This is the same as the Bohr radius. Putting this into equation (.17), the lowest possible energy of the electron in a hydrogen atom is 4 k me E (.19) h This is the ground state energy of the hydrogen atom. As expected, this equation is identical to that obtained from Bohr s theory for ground state of the hydrogen atom. Wave packet In view of Heisenberg s uncertainty principle, it feasible to describe the motion of an electron by a single wave only? Since xp ħ, there is an inherent uncertainty x in localizing an electron in space and this always causes an uncertainty in its momentum p. Since the de Broglie wavelength = h/p, this uncertainty p clearly implies a spread in the electron s wavelength. For the numerical value of this spread, we get h p As a result, the uncertainty relation gives x 1 (.0) Thus the uncertainty x in electron s position implies a spread in its wavelength.so, in real life, the motion of an electron is best described by a wave packet, which represents a superimposed group of waves having wavelengths between and +. It can be shown that the resulting pilot wave has a maximum, which moves with a group velocity or the velocity of the particle it represents. Imrerested students may consult ref. []. 11

12 Summary Planck s law is the beginning of Quantum mechanics. It is based on particle nature of light. Electrons behave like waves as shown by its diffraction pattern. De Broglie wavelength for electrons helps us to view Bohr orbits as standing electron waves. Electron microscope is a very useful practical application of wave-particle duality of electrons. Heisenberg s uncertainty principle and its implications are described. Thank you! 1