Modern Physics Part 3: Bohr Model & Matter Waves

Transcription

1 Modern Physics Part 3: Bohr Model & Matter Waves Last modified: 28/08/2018

2 Links Atomic Spectra Introduction Atomic Emission Spectra Atomic Absorption Spectra Bohr Model of the Hydrogen Atom Emission Spectrum of Hydrogen Rydberg Formula Rutherford: Atoms are Solar Systems Bohr: Angular Momentum Quantized Bohr Radius Energy of Electron Example Atomic Spectra Explained Problems Summary Matter Waves de Broglie Wavelength Example 1 Example 2 Wave-Particle Duality de Broglie and the Bohr Model A Ball in a Box Example 1 Example 2 Summary Quantum Mechanics (*) Wave Functions & Probability Hydrogen Atom Superposition & Schrödinger s Cat Starred (*) sections will not be included in exam.

3 Introduction In a previous lecture we saw how the QUANTIZATION of LIGHT allowed us to understand Black Body Radiation and the Photoelectric Effect - two phenomena that otherwise could not be explained using Classical Physics. As we ll see, this idea of quantizing quantities that in Classical Physics are considered to be continuous (i.e. take any value) will be useful in understanding other phenomena as well. First we will look at another of the experimental puzzles from the late 1800 s - Atomic Spectral Lines.

4 Atomic Emission Spectra When white light passes through a prism, it is found to consist of the many different colours of the rainbow, each with equal intensity. White Light If we heat a solid object (e.g. a chunk of metal), it will emit light (Black Body Radiation) which we find, like white light, to include all the colours of the rainbow, but with a temperature dependent intensity distribution. Black Body Radiation

5 However, when we measure the light emitted by a heated gas of a single element, we find a surprise - only a limited number of wavelengths of light are emitted. For example, for Hydrogen this emission spectrum contains only four visible colours: Emission Spectrum of Hydrogen (observed) H There are also wavelengths observed in the ultraviolet and infrared ranges. These observed wavelengths are NOT dependent on the temperature of the gas. A hotter gas emits MORE light, but at the SAME distinct wavelengths.

6 The wavelengths of the black lines here correspond exactly with the lines seen in the emission spectrum. Atomic Absorption Spectra If we direct white light (i.e. all colours) through a gas, we expect some of the light to be absorbed. Classical physics predicts that all wavelengths will be absorbed to some extent. Classical Physics Absorption (prediction) gas white light Again our observations are inconsistent with the Classical Physics prediction. For Hydrogen gas, only four visible wavelengths are absorbed: Absorption Spectrum of Hydrogen (observed) white light H

7 The observed emission and absorption spectra for other elements are similar. Only a limited number of wavelengths are emitted or absorbed. Here are a few further examples of emission and absorption spectra (again only visible wavelengths are shown): Helium Sodium Mercury Emission Absorption Emission Absorpton Emission Absorpton In all cases, the absorbed wavelengths are the same as the emitted wavelengths.

8 Every element has a distinctive spectrum and this can be used like a fingerprint for identification. In fact Helium was first identified by observation of its spectral lines in the spectrum of light from the Sun (and so named after Helios, the Greek god of the Sun). Astronomers can measure the different wavelengths in the light from a star, and by comparing these to emission spectra, determine the elements present in that star. The composition of galactic dust clouds can be determined by observing the missing (i.e. absorbed) wavelengths of light passing through them and comparing these with absorption spectra. To try to understand what is going on here, let us take a more detailed look at the lightest element - Hydrogen.

9 Emission Spectrum of Hydrogen Atom The full emission spectrum of Hydrogen contains many more wavelengths than shown above: visible wavelength (nm) Note the colours used here are not the real colours - everything outside the area with blue background is invisible to the eye. Wavelengths are grouped together in what are called series, named after their discoverers - Lyman series, Balmer series, Paschen, Brackett etc.

10 Rydberg s Formula In 1888, Johannes Rydberg showed that the measured wavelengths fit the formula: 1 λ = R ( 1 H n 1 ) 2 m 2 where R H = m 1 is the Rydberg constant, and n, m are positive integers with m > n. The Lyman series corresponds to the wavelengths with n = 1, the Balmer series n = 2 and so on for the other series. Why this formula worked was, at the time, a complete mystery.

11 Rutherford: Atoms are Solar Systems In 1911, Ernest Rutherford s experiments determined that atoms (all atoms, not just hydrogen) consist of a small, dense, positively charged nucleus orbited by negatively charged electrons. The centripetal force on the electrons is provided by the Coulomb attraction between the opposite charges. v electron proton F = ke2 r 2 r The Hydrogen atom is a single proton orbited by a single electron.

12 Bohr: Angular Momentum is Quantized This solar system model has some problems: Why doesn t the electron lose energy and spiral toward the proton? (Classical electromagnetism predicts an accelerating charge will emit EM radiation and hence lose energy over time.) What causes the observed spectral lines? Niels Bohr took the next step to understanding the H atom in He proposed that the electron could only exist in specially allowed orbits, in which they did not radiate. He further suggested that these allowed orbits corresponded to particular values of angular momentum. In other words: Angular Momentum of the orbiting electron is QUANTIZED.

13 Angular Momentum Reminder The ANGULAR MOMENTUM vector L of a mass with momentum p = mv around a pivot is defined to be: L = r p where r is the position vector of the mass, relative to the pivot. z L y r p x For a circular orbit we know r v, thus L = rp = rmv

14 Bohr Radius Bohr s quantization condition is that for the allowed electron orbits: L = m e vr = n ( ) h 2π n n = 1,2,3... The common combination h/2π is usually shortened to (pronounced h-bar ) Applying Newton s second law to the electron: ke 2 r 2 = m v 2 e r (Coulomb force = centripetal force) ke 2 r = m e v 2 = m e [ n m e r ] 2 (using above quantization condition) r = n2 2 m e ke 2 = n2 (a bunch of constants)

15 The allowed orbits have the radii: ) r n = n 2 ( 2 m e ke 2 n 2 a 0 n = 1, 2, 3... where a 0 = nm (the Bohr radius) is the radius of the smallest possible orbit. What are the radii of the n = 4 and n = 5 orbits? r 4 = 4 2 a 0 = = nm r 5 = 5 2 a 0 = = nm

16 Energy of Electron in n th Bohr orbit Total Energy = Kinetic Energy + Potential Energy = 1 ( ) ke 2 m ev 2 + ( e) is the potential due to the proton) = 1 2 = 1 2 ( ke 2 r n ) r n ( ) ke 2 r n ( ke r n ( ) ke 2 = 1n ( ) ke 2 2 r n = 1 n 2 constants 2a 0 (using m ev 2 = ke2 r from 2 pages back) (using r n = n 2 a 0) E n = 1 n 2 E 1 = 1 n 2 ( ev)

17 The negative sign indicates that the electron is in a bound state. The lowest energy orbit (the ground state) corresponds to n = 1, and the energies get progressively larger (i.e. closer to zero) as n increases. What are the energies of electrons in the n = 2, 3, 4, 5 orbits? E 2 = E 1 = E 3 = E 1 = E 4 = E 1 = E 5 = E 1 = = 3.40 ev = 1.51 ev = 0.85 ev = 0.54 ev

18 The Rydberg formula can now be understood if we consider an electron dropping from orbit m to a lower orbit n, so the change in its energy will be: ( 1 E = E m E n = m 2 1 ) n 2 E 1 Where does this energy go? It appears as a photon with wavelength λ given by Planck s formula E = hc λ. And so: ( 1 1 λ = n 2 1 ) E1 m 2 hc Comparing this to the Rydberg formula seen earlier, we predict : R H = E 1 hc which agrees with the experimental value R H = m 1.

19 What is the wavelength of the photon emitted when an electron in a H atom drops from the third excited state (i.e. n = 4) to the ground state (n = 1)? E = ( ) = ev λ = hc E = ( ) ( ) ( = 97 nm ) alternatively, using the Rydberg formula: ( 1 1 = λ ) 4 2 λ = 97 nm

20 Bohr Model & Atomic Spectra The Bohr model neatly explains why we see only certain wavelengths in the spectra of Hydrogen. Heating a gas will increase the kinetic energy of the atoms and when these atoms collide some of this energy can be absorbed by an electron which will then move to a higher energy orbit. Some time later it will drop down to a lower energy orbit, emitting a photon as it does so. Note that there are usually multiple pathways for an electron to lose energy. For example an electron in the n = 4 state can drop to any of the lower states n = 1, 2, 3 - each transition has a unique photon wavelength, which is then seen in the emission spectrum. If a photon of exactly the right energy strikes the atom, it can be absorbed by an electron which will move to a higher orbit. Photons with the wrong wavelengths will not be absorbed. These wavelengths are of course exactly the same as those emitted. The spectra of other elements are presumably caused similarly, but the wavelengths cannot be calculated in the Bohr model.

21 Bohr Model Problems Though it successfully explains some simple properties of the Hydrogen atom, the Bohr model has shortcomings, for example: It doesn t work for atoms with more than one electron (most of them!) Though it correctly gives wavelengths of the spectral lines, it does not explain why some lines are seen to be brighter than others. When we look very closely, we find that rather than being a single line, many of the lines in the Hydrogen spectrum are actually several lines, very close together. This is known as the fine structure of the spectrum. Though useful for simple calculations, the Bohr model is incomplete.

22 Bohr Model: Summary Only a limited number of electron orbits are possible - those with radius r n (a 0 = nm is the Bohr radius): r n = n 2 a 0 for n = 1,2,3... The energy of the electron in the n-th orbit is: E n = 1 n 2 ( ev) Electrons can jump between allowed orbits. Dropping from the initial orbit with n = n i to the final orbit with n = n f, a photon will be emitted with wavelength λ given by the Rydberg formula: ( 1 λ = R 1 H nf 2 1 n 2 i )

23 The series observed in the spectrum are the groups of photon wavelengths where the transition ends in the same final orbit: Lyman series: n i > 1 n f = 1 + n=1 n=2 n=3 Balmer series: n i > 2 n f = 2 Paschen series: n i > 3 n f = 3 and so on for n = 4, 5, 6...

24 de Broglie: Particles are Waves Too! As we have seen, light, which is classically considered to be a wave, can behave more like a stream of particles with momentum p = h/λ. In 1924, Louis de Broglie asked the question: Since this wave can act like a particle, could the reverse also be true? Can a particle act like a wave? The answer is YES. The connection between the de Broglie wavelength λ of a massive particle and its momentum p is the same as the photon formula: λ = h/p It isn t entirely clear what the medium for this wave will be. We ll come back to this point later.

25 de Broglie: Example 1 What is the de Broglie wavelength of a car (mass = 1000 kg) travelling at 30 m/s (just over 100 km/hr)? λ = h/p = h/(mv) = /( ) = m! This is so ridiculously small that we will never notice any wavelike behaviour of this car. (Remember, as we have just seen, the diameter of a Hydrogen atom is about m) We will find a similar result with all everyday objects and speeds. In the Classical Physics world, the wave-ness of a particle is not significant.

26 de Broglie: Example 2 What is the de Broglie wavelength of an electron (mass = kg) travelling at m/s? (kinetic energy 100 ev) λ = h/p = ( )/(( ) ( )) = m 2.5 a 0 This wavelength is comparable to the size of an atom, so in the interaction between this electron and an atom, the electron must be thought of as a wave.

27 This is not just a theoretical formula - wave behaviour of particles CAN be observed experimentally, in a version of Young s experiment. The separation of atoms in a crystal will be in the order of several times the Bohr radius, which as we have just calculated is about the same size as the de Broglie wavelength of a 100 ev electron. When we direct a beam of such electrons at a crystal, the spacing of the atoms act like the thin slits in Young s experiment. Electrons following different paths throught the crystal will interfere with each other and produce fringes of large and small numbers of electrons. detected electrons form interference pattern electrons atoms in crystal

28 Wave-Particle Duality This split personality of light (seen in previous lecture) and massive particles (just now) is known as wave-particle duality. Light and matter will behave as either a wave or a particle, but not both at once. Which behaviour we expect depends on the scale of the interactions occuring. For light, particle behaviour (i.e. photons) is seen when light is interacting with other objects with similar energies to the photon. e.g. the photoelectric effect. Wave behaviour dominates for larger objects e.g. light being reflected from a mirror. For massive particles, wave-ness is only observed when the de Broglie wavelength is comparable to the size of the relevant environment, e.g. inside an atom. The same electron travelling across a room will always be seen to act like a particle. Particles larger than an electron are rarely seen to show wave-like behaviour.

29 Bohr Model: Electrons Interfere Thinking of electrons as waves gives us a new way to understand the Bohr model. If the electron is a wave following a circular path around the proton, then this wave will meet up with, and thus interfere with itself. Most such paths will result in destructive interference, and the electron cannot exist in these paths. This wave is plotted with a different colour for each new cycle of the orbit. Imagine adding these amplitudes together. The result will be zero.

30 Some of these paths though, will constructively interfere and form non-zero waves around the nucleus. The circumference of these orbits must be an integer number of wavelengths, and using the de Broglie formula gives: 2πr = nλ = nh/p pr = L = n This is the same condition for quantization of angular momentum that the Bohr model simply assumed. The constructive/destructive interference of the electron waves explains why only certain orbits are allowed.

31 A Ball in a Box Consider a ball moving inside a box. Assuming it collides elastically with the walls of this box, then it will bounce backwards and forwards between the walls with a constant speed v. v If we take de Broglie seriously, then we should instead think of the ball as a wave, and instead of it bouncing off the walls, it will be reflected. The reflected wave will interfere constructively and destructively with the original wave and will form a limited number of standing waves, with nodes located at the walls: L

32 The calculations will follow exactly the same path as we previously saw for standing waves on a string. The allowed wavelengths are related to the size of the box: λ n = 2L n n = 1, 2, 3... The de Broglie equation connects wavelength to momentum. Limiting the wavelengths also limits momentum. In other words, the momentum of the ball is quantized, with allowed values: p n = h ( ) h = n = np 1 n = 1, 2, 3... λ n 2L The (kinetic) energy of the ball is thus also quantized: ( ) E n = p2 n h 2 2m = n2 8mL 2 = n 2 E 1 n = 1, 2, 3...

33 Ball in Box: Example 1 Calculate the ground state energy (i.e. lowest possible energy) for a golf ball of mass 50 g in a box of width 1 m. The lowest energy is found for n = 1 (and is not zero as expected Classically): ( ) h 2 E 1 = 8mL 2 = ( ) = J = ev Even in electronvolts this is ridiculously small! Yet again, we find that for everyday objects, quantization is not noticable.

34 Ball in Box: Example 2 Calculate the energies of the ground state and the first two excited states for an electron (m = kg) in a box of width 10 nm. ( ) h 2 ( ) 2 E 1 = 8mL 2 = 8 ( ) ( ) 2 = J = ev And for the excited states: E 2 = 2 2 E 1 = ev E 3 = 3 2 E 1 = ev 10 nm is about the size of the wires in modern computer chips, and the energies here correspond to voltages of 3.8 mv, 15.1 mv and 34 mv respectively, which though small, are similar to voltages found in such circuits. The wave-ness of electrons is very relevant to engineers designing these chips.

35 Matter Waves: Summary A particle with momentum p has a de Broglie wavelength λ, given by: λ = h p where h is the Planck constant. For everyday objects, this wavelength is far too small to be important. For smaller, lighter particles, particularly the electron, wave behaviour can be very important, generally when the de Broglie wavelength is similar in size to the particle s environment. A bound (e.g. electron in the hydrogen atom) or confined (ball in the box) particle will interfere with itself, giving a limited number of allowed wavelengths, thus only certain allowed values of momentum and energy. i.e. Energy is Quantized

36 A particle moving freely in space is not quantized and can still have any energy. Particularly in systems with many possible energies, such as an atom or molecule, it can be helpful to draw energy level diagrams. For example: Hydrogen Atom Ball in Box n = 4 E 0 n = 4 3 n = 2 n = 3 n = 2 n = 1 n = 1 E = 0 For such systems, there will be an emission spectrum of photons given by the transition n m where n > m, and an absorption spectrum for the reverse transitions m n.

37 Quantum Jumps The transition of particles between energy levels gives a glimpse of the weirdness of the Quantum world. Normally (i.e classically), we would expect that if the n = 2 electron in a hydrogen atom was moving from, for example, the first excited state (n = 2) with r = 4a 0 to the ground state (n = 1) with r = a 0, then it n = 1 must move through all the values of r between these values, taking a time t to do so. + a0 4a0 t All of the usual definitions of velocity, acceleration etc will apply during this time interval.

38 However, in the Bohr model the intermediate values of r between the states are not allowed. One instant, the electron will be in the excited state, then the next it will be in the ground state, without having ever having been at any point in between. This transition is known as a quantum jump or quantum leap between the states. A photon is produced with energy equal to the difference in energies of the two levels. n = 2 n = 1 + photon t = 0 The absorption of a photon of the correct energy is similar, with the electron quantum jumping instantly from a lower energy level to a higher. The Classical Physics idea of particles following predictable, continuous paths is not true at the Quantum level.

39 Wave Functions & Probability If we are to really take the idea of the electron as a wave seriously, we need to represent it by a wave function, (Ψ) (similar to the formula for y for a wave on a string) which is the solution of a wave equation. For matter waves, this equation is called the Schrödinger Equation, which is a differential equation with first and second derivatives of Ψ. This can be challenging to solve mathematically. The solution of the Schrödinger Equation for atoms and molecules, is known as Quantum Mechanics and from the late 1920 s was used successfully to describe the behaviour of atoms, molecules etc. non-examinable But what is the meaning of Ψ? Remember in the case of a wave on a string, y is the transverse displacement, and for a sound wave it is the change in pressure etc. In Quantum Mechanics, the square of the magnitude of the wavefunction Ψ(x) 2 is proportional to the probability of the electron being at point x.

40 Quantum Mechanics calculations always involve probabilities. We can never calculate exactly when a particle in an excited state will drop to a lower state, or exactly which state it will finish in. This is an important difference with Classical Physics, where we can always (at least in principle) determine the exact motion of a particle. non-examinable Consider the following (totally made-up) example: A Quantum Mechanics calculation tells us that for an electron in the n = 3 state of a quantum system, that after 5 minutes, there is a probability of: 60% that the electron has dropped to the n = 1 state, emitting a photon of energy E 3 E 1 and wavelength λ = hc E 3 E 1, 60% 30% that it is in the n = 2 state, after emitting a photon with wavelength λ = hc E 3 E 2, 10% that it remains in the n = 3 state. λ λ 30% n = 3 n = 2 n = 1 E = 0

41 For a single electron, this information is not very useful. In a real world situation though, we usaually have billions upon billions of such electrons, and so probabilities become more meaningful. Consider a large number N of electrons in the n = 3 state of our example system: N electrons n = 3 n = 2 n = 1 E = 0 5 minutes 0.1N 0.3N 0.6N n = 3 n = 2 n = 1 E = 0 non-examinable Using probabilities, we predict that in a 5 minute interval we will see twice as many photons of wavelength λ emitted as photons of the other wavelength λ. (In this example system, there would also be photons emitted when electrons drop from the n = 2 level to n = 1. Determining the probability for this would require another calculation.)

42 Quantum Mechanics & the Hydrogen Atom Like the Bohr model, Quantum Mechanics predicts that electron energy is quantized in the Hydrogen atom. The possible electron energy levels are labelled by the principal quantum number, n = 1, 2, 3... Also, angular momentum is quantized: L = l. BUT, unlike the Bohr model, the n th energy level can have multiple values of l: 0 l < n. For each value of l there are 2l + 1 solutions for the electron wavefunction. These solutions are known as orbitals or shells. non-examinable For n = 1, the only possible value for l is l = 0. This l = 0 solution is spherically symmetric about the nucleus, which can be visualized as a spherical cloud (not a fixed orbit as predicted by Bohr). This is known as an s-shell. + s-shell The lowest energy electron in the H atom is in the 1s shell, which has an energy very close to the n = 1 state in the Bohr model.

43 For n = 2, then there is one solution with l = 0 (again spherical) and three with l = 1, known as p-shells. These 2p states each have a dumbbell shape, lined up with the x, y and z axes. 2s p non-examinable The 2p orbitals have a slightly higher energy than the 2s, so the photons released in the 2p 1s and 2s 1s transitions have slightly different wavelengths which are seen in the fine structure of the spectrum. The different angular momenta of the 2s and 2p states will affect the probabilities of them dropping to the 1s state. The 2p 1s is more likely, and so we see the spectral line due to this transition as brighter. The Bohr model cannot explain why some lines are observed to be brighter than others.

44 For n = 3 there is one l = 0 state (3s), three l = 1 (3p) states and five with l = 2. The l = 2 orbitals are known as d-shells and have even more complicated shapes. Remember the diagrams indicate the most likely positions of the electron. It does not follow a fixed orbit, as our simpler models suggested. 3s d non-examinable 3p This pattern continues for increasing values of n. Other atoms are described similarly. With the addition of the Pauli Exclusion Principle (each orbital can contain a maximum of two electrons), Quantum Mechanics explains the structure of the Periodic Table and is vital to understanding chemical bonds and other atomic properties.

45 Superposition & Schrödinger s Cat Like other waves, solutions of the Schrödinger equation can be superposed (i.e. added). It is possible, for example, for an electron inside a Hydrogen atom to be in a superposition of the 1s and 2s orbitals, so its wavefunction is: Ψ = aψ 1s + bψ 2s where a and b are constants Exactly what this superposition meant physically was the basis of much discussion during the 1920 s. non-examinable Bohr (and others) argued that small particles like electrons are special and follow different rules to tennis balls, cars etc. They believed that the above electron is actually in both of the 1s and the 2s states at the same time. If we make a measurement of the electron energy for instance, our result wil be either the 1s energy or the 2s energy. Only when we actually measure it does the electron decide to be in one state or the other.

46 Einstein (and others, including Schrödinger) disagreed. They interpreted the superposition as an expression of our incomplete knowledge of the electron s state. We can only know the probability of measuring 1s or 2s energies. This probabilistic behaviour is not normally noticable in the everyday world, because we are dealing with enormous numbers of atoms. It is tempting then, to think of probability as only affecting the behaviour of atoms etc. and not relevant to the real world. non-examinable A famous thought experiment, known as Schrödinger s Cat requires us to think more about this. It demonstrates that it is possible for a large (or macroscopic) object to exist in an undetermined, probabilistic state. Imagine Fluffy is placed into a soundproof box with a container of poisonous gas. The container is connected to a device which willl release the gas if it detects a radioactive decay (which is a probablistic quantum process).

47 A Now we wait for a time t.... Is Fluffy now dead or alive? non-examinable We can use our knowledge of radioactive decay to determine Fluffy s wavefunction (a superposition of the two possible states): Ψ Fluffy = aψ alive + bψ dead where a and b will depend on time Bohr (and friends) interpret this wavefunction to be telling us that Fluffy is both alive and dead at the same time! If we open the box to check on Fluffy s health, one of these states then suddenly becomes true.

48 Schrödinger said this is nonsense - clearly Fluffy must be either alive or dead, not both at once. BUT, without opening the box, we can never know for sure what state Fluffy is in. Before opening the box, we can use Fluffy s wavefunction to determine the probabilities of the two outcomes: Probability Fluffy is alive = a 2 Probability Fluffy is dead = b 2 non-examinable (obviously it must be true that a 2 + b 2 = 1!) Even for a question like this, that we normally expect to have a definite answer, we can only ever express our knowledge in terms of probabilities. Remember, in Classical Physics, we can always calculate (at least theoretically) exactly what is going on.