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Unformatted text preview: Matter Waves Towards the end of the 19th century, most physicists believed that physics had arrived a mature state, and that all fundamental physics was well understood, such that only precise measurement and clarifying some as yet unexplained eﬀects would be needed. These unexplained eﬀects where 1. Black–body radiation 2. The photoelectric eﬀect 3. atomic spectra It was generally believed that these eﬀects could be explained using classical mechanics and Maxwell’s electromagnetism, and that these theories could explain all natural phenomena. When these eﬀects are eventually understood, it turned out that an entire new physics emerged, namely Quantum Mechanics. With hindsight, there were other hints for Quantum Mechanics known at the end of the 19th century, speciﬁcally Brownian motion and the speciﬁc heat and the equipartition of energy of multi–atomic gases. The photoelectric eﬀect The photoelectric eﬀect is our ﬁrst example for an eﬀect that occurs when our energy measurement sensitivity is suﬃcient to detect a single photon. (All other optical phenomena we have considered —either geometrical or physical optics— have the energy sensitivity such that it is much higher than the energy of a single photon.) The photoelectric eﬀect was discovered accidentally by Hertz in the same experiment that veriﬁed Maxwell’s prediction that electromagnetic waves existed. In his experiment (1887), Hertz found that ultraviolet light from sparks generated at transmission was aﬀecting the receiver’s sensitivity. Hertz was careful enough to take notice of this eﬀect, and over the following 18 years more detailed description of it became available. It is one of the ironies of science that the same experiment that proved the existence of electromagnetic waves, also found the ﬁrst evidence that the same radiation is made of particles, that later were dubbed photons. In the photoelectric eﬀect we have an open electric circuit, and lights irradiated a cathode, releasing electrons, that are then collected by an anode, thus eﬀectively closing the circuit. One can control the current in the circuit by changing the voltage diﬀerence between the cathode and anode. The prediction of Maxwell’s theory is that as the electromagnetic waves irradiate the cathode, they endow energy to it’s electrons, each wavelength imparts more energy to the electrons, 111 until enough energy is given to the electrons, eventually releasing them from their binding to the cathode. The predicted rate of electron emission is rate of electron emission = number of electrons energy delivered / time = time energy needed to release one electron energy delivered × area per particle time × area = energy needed to release one electron I × d2 ∼ Eeject where I is the light intensity. Speciﬁcally, we ﬁnd that the electron emission rate is proportional to the light intensity. If I is low it will take a long time until the ﬁrst electron is eventually ejected, and this ejection time is T Eeject 1 ∼ ×. e d2 I The ejected electrons have a distribution of kinetic energies. As we vary of voltage, we ﬁnd Vstop , the smallest voltage for which there is no current ﬂowing through the circuit. When V = Vstop even the most energetic ejected electrons are turned back to the cathode and never reach the anode, and the maximal kinetic energy of the electrons is Kmax = eVstop . Classical theory predicts that as we increase the intensity I , Kmax should also increase, because with greater I more energy is given to the electrons. In practice, the following results are found in the experiment: 1. At suﬃciently high I and ﬁxed frequency f , the rate of ejected electrons is proportional to I . (This is indeed what classical theory predicts.) 2. Electrons are ejected essentially instantly, no matter how low the intensity I , with typical time scale at the order of 10−8 sec. (This result is especially troubling, as with very low I the delay in electron ejection should be very long.) 3. For ﬁxed I , the rate of ejected electrons decreases with increasing frequency f for all materials. (In classical theory there is no dependence on f .) 4. Below a certain (material dependent) cutoﬀ frequency f , no electrons are ejected, no matter how large is I . (This result is in direct conﬂict with the predictions of classical theory: ﬁrst, why is there a cutoﬀ frequency, and second, why are no electrons ejected even if we allow for enough time for energy to build up?) 112 5. For ﬁxed f , the maximal kinetic energy of the ejected electrons, Kmax , in independent of I . (This result conﬂicts with our previous prediction that Kmax ∝ I . One may liken this result to a 50ft tsunami making the same impact as a 1ft ankle splasher.) 6. For f > fcutoﬀ , Kmax ∝ f , with the same proportionality coeﬃcient for all materials. (In classical theory there is no dependence of the frequency.) Einstein’s (1905) explained all these observations (although some of them where as yet unknown at the time) by his photon hypothesis (the term photon was dubbed later), namely a light particle that connects to the wave picture through the concept of frequency. Speciﬁcally, light is made of photons, each photon being a “bundle of energy,” with energy E = hf , where h is Planck’s constant. Higher intensity of light of the same frequency means larger number of photons of the same energy each. When a photon collides with an atom in the metal, it will eject an electron only if the photon carried more energy than the least binding energy of electrons to the metal, a quantity called the work function Φ. Any surplus energy will be endowed to the electron in the form of kinetic energy. If the energy needed to eject the electron is greater than the work function, the ejected electron’s energy is lower than the maximal kinetic energy it can obtain, so that Ephoton ≡ hf = Kmax + Φ . Planck’s constant has the value h = 6.63 × 10−34 J · s = 4.15 × 10−15 eV · s. A very useful and frequently found combination is the product hc = 1, 240 eV · nm. It is important to notice that the dimensions of Planck’s constant are those of angular momentum. All the observational results are readily explained with the photon hypothesis: The photon’s energy E is transmitted to a single electron, and no matter how large the intensity I may be, no more than Kmax can be endowed to the electrons in kinetic energy. The intensity I is related to the number of electrons, not to the properties of individual photons. If the frequency f is too low (below the cutoﬀ frequency), no electrons can be ejected, simply because the transmitted energy is smaller than the work function. Notice, that it is highly improbably that more than just one photon will be giving energy to the same electron. hf = eVstop + Φ h Φ Vstop = f− e e so that when Vstop is plotted as a function of f , the slope is the same for all materials, and only the intersection point with the frequency axis is diﬀerent (and is determined by the work function of the material): Φ = hfcutoﬀ . 113 How does Einstein’s photon hypothesis explain all the observational facts? We list here the explanations for the same six observational facts above: 1. Say the intensity is low, i.e., light is very dim: there are few photons irradiating the surface. It is not very likely then that the photon will collide with any atom in the material, and therefore will not eject any electron. If the light is very bright, there are many photons, and a certain fraction thereof collides with atoms and ejects electrons. If we now double the intensity, we also double the number of incident photons, and therefore also double the number of collisions and correspondingly the number of ejected electrons. (Recall that this experimental result is partially explained also by classical theory.) 2. When a photon collides with an atom it ionizes it and releases an electron on the atomic transition time scale. There is no need to “build up” the energy transferred to the material, and all the energy is imparted at once. 3. For ﬁxed I , if we increase f the number of photons n decreases (for the product nhf to remain constant). With fewer photons, there are also fewer ejected electrons. 4. Below the cutoﬀ frequency the photon does not carry enough energy t release the electron, regardless of how many photons there are. Recall that electrons are released by a single photon. (The probability of multi–photon interaction is very small.) 5. Increasing the intensity means increasing the number of photons, but the energy imparted by a single photons is unchanged. 6. When the frequency f increases, the photon carries more energy. The work function remains unchanged, so that the additional energy is converted to a higher kinetic energy, i.e., higher Kmax . As we showed above Vstop ∝ f with the same proportionality coeﬃcients for all materials (h/e), and kmax = eVstop . The Compton eﬀect If light is made of particles, then photons should collide with other particles like particles do. We therefore analyze the elastic collision of two particles, letting one of them have energy according to Einstein’s hypothesis (E = hf ), and allow for velocities to be relativistic. We take the particle to be static before the collision, and the photon moving before the collision in the x–direction. We denote the angle the scattered photon makes with the x–axis by φ and the angle the scattered particle makes with the same axis by θ. 114 ...
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- Spring '09