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lec23

# lec23 - 23 Lecture 23 23.1 De Broglie waves We have seen...

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Unformatted text preview: 23. Lecture 23 23.1 De Broglie waves We have seen that light appears to be a wave but, under careful examination, behaves as if composed of particles. Nevertheless all the wave properties of interference and diﬀraction are still valid. In view of this “particle-wave” duality, de Broglie proposed that particles should also behave as waves. The frequency and wave-length are given by the same relation as for photons: Eel. = ￿ω , Pel. = h λ (23.1) In the case of the photon we were more familiar with ω , λ and computed E, P . In the case of the electron we are more familiar with E, P and compute the associated angular frequency ω and wave-length λ. Although these relations are the same as for the photon, it should be emphasized that the relation between energy and momentum is diﬀerent: 2 Pph. Pel. Eel. = , Eph. = (23.2) 2m c This was a very strange proposal but it received spectacular experimental conﬁrmation when an experiment analogous to the two slit experiment showed an interference pattern for electrons exactly the same as for light. In fact any experiment of diﬀraction and interference of electrons can be analyzed in the same way as for light. We only need to compute the wave-length using λ = h . p Now we want to see how this helps us in the hydrogen atom. First think of a string in a violin or guitar. It is well known that such a string only vibrates at speciﬁc frequencies which is why is used in a musical instrument. In fact the lowest frequency is such that the length of the string is half a wave-length. We say that the vibrations of a string have a discrete spectrum of frequencies. The proposal is that the discrete spectrum of the hydrogen atom is due precisely to the wave-like behavior of the electron. More quantitatively, we require the the size of the orbit is an integer multiple of the wave length of the electron. Namely in each orbit an integer number of wave-lengths ﬁt: 2π r = nλ, n = 1, 2 , 3 , 4 , . . . (23.3) Now we use the Newtonian relation v2 e2 ¯ m =2 r r ⇒ – 148 – v= ￿ e2 ¯ mr (23.4) to compute the velocity of the electron and thus the momentum and wave-length: ￿ ￿ me 2 ¯ 2π ￿ r p = mv = , λ= = 2π ￿ (23.5) r p me 2 ¯ Using now eq.(23.3), namely 2π r = nλ we get: ￿ r 2 π r = 2π ￿ n⇒ me 2 ¯ r= n 2 ￿2 me 2 ¯ (23.6) We indeed ﬁnd a discrete set of orbits!. The energy is computed by replacing the velocity from eq.(23.4) into 1 2 e2 ¯ e2 ¯ 1 me 4 ¯ 1 E = mv − = − = − 2 2 = −13.6eV 2 2 r r n￿ n (23.7) where we replaced the known values of e, m and ￿. Now we have the energy of all ¯ possible orbits ﬁtting precisely eq.(22.4)!. This is a good check that the discreteness of the hydrogen spectrum is due to the wave-like nature of the electron. 23.2 Other results and applications We basically have ﬁnished what we wanted to discuss about quantum mechanics. For the last hundred years we have been improving our understanding of how quantum mechanics applies to diﬀerent physical phenomena. For example one can understand the periodic table of the elements, chemistry, material science, particle physics, etc. as ﬁelds where quantum mechanics is of paramount importance. However it has only been recently that experimental progress has been enough that one can start thinking of practical application in which controlling the state of a quantum system can be used to our advantage. One such possibility would be to use quantum mechanics to create better computers, a ﬁeld which is still in its infancy. For illustration we will discuss two devices which are practical applications of quantum mechanics: lasers and atomic clocks. A laser creates a beam of coherent light which is well collimated, namely does not spread much. In quantum mechanical terms we generate a large number of photons all in the same state. To understand how that is achieved we need another important property: if an atom is in an excited state it can transition to a lower energy state by emitting a photon with the corresponding energy. However, if there is already a number of photons of that energy present, the probability of decay is enhanced for the photon to go to the same state in which those photons are. A laser utilizes this by having a material whose atoms are excited by electric discharges, light, electric ﬁelds, etc. In ﬁg. 125 we see an example, the active zone, where the material is excited, is situated – 149 – Figure 123: The Bohr atom explained through the de Broglie hypothesis. An integer number of wave-lengths ﬁt into each orbit. between two mirrors. A decay produces a photon. The other atoms, when they decay prefer to emit the photon in the same state as the one already present. As more and more photons accumulate on a state, more likely is for others to join. In this manner light is ampliﬁed which gives its name to the device: Light Ampliﬁcation by Stimulated Emission of Radiation. The words stimulated emission of radiation makes reference to the idea that a photon already present stimulates the atoms to emit radiation in the same state. The mirrors can be thought as determining standing waves similarly as in a sound waves in a pipe. Alternatively we can think that a beam of light goes back and forth between them being ampliﬁed all the time. To extract the energy, one of the mirrors is partially transparent. It is clear that only photons moving along the cylinder, perpendicular to the mirrors are ampliﬁed. The others are lost since they are not reﬂected back into the active zone. Furthermore, all the photons are in the same state, so they are in phase and the light is coherent. This should be contrasted with – 150 – the usual thermal emission in which the same medium is heated and each atom decays independently of the others. In that case we have a large number of sources all emitting independently. In the laser they all contribute to the same wave generating a highly coherent pulse. The same idea is used with microwaves. A resonant cavity as it is called contains atoms and an alternating frequency is applied. When the external frequency agrees with the energy of an atomic transition a resonance occurs that can be easily detected. Since the frequency of the atomic transition is very precise, we can create an alternating voltage with a very precise frequency or period. But a periodic signal with very precise period is exactly what we need to build a clock. This type of clocks are called atomic clocks and are the most precise clocks available at the moment. They are used in numerous applications, for example atomic clocks aboard the GPS satellites produce timig signals that allow us to determine our position by getting timing signals from diﬀerent satellites. Finally, another phenomenon that we wanted to discuss is Compton scattering. When photoelectric eﬀect occurs but the energy of the photon is much larger than the binding energy of the electron, the electron can be considered as a free particle. In such situation, the photon cannot be absorbed and is deﬂected. If the deﬂection angle is θ, see ﬁg.124 then, conservation of energy and momentum give the relation between the wave-length λ of the incoming photon and λ￿ the wave-length of the outgoing photon: λ￿ − λ = h (1 − cos θ) mc (23.8) Here, m is the mass of the electron. In fact to obtain this result you need to use the ￿ relativistic relation between momentum an energy E = m2 c4 + p2 c2 . We leave this as an exercise for anyone who is interested. – 151 – Figure 124: Compton scattering is analogous to the photoelectric eﬀect but the electron either is not bound or the binding energy is small compared to the energy of the photon. The photon deﬂected by an angle θ and changes its wave-length. Figure 125: Schematic of a laser. A beam of light goes back and forth in an active medium which ampliﬁes light. One of the mirrors is partially transparent and lets the laser light go out. – 152 – ...
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