lec22 - 22. Lecture 22 22.1 Photons As already discussed,...

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Unformatted text preview: 22. Lecture 22 22.1 Photons As already discussed, the correct explanation of the photoelectric effect is that electromagnetic waves are made of small quanta called photons. Each photon has energy and momentum given in terms of the frequency and wave-length by: Eph. = ￿ω = hf, Pph. = Eph. c = h λ (22.1) That photons have momentum can be seen by shining light on a reflective surface and observing that light exerts a pressure over the surface, namely transfers momentum to it. The value of ￿, the Planck constant is given ￿c = 197M eV f m (22.2) where c is the speed of light and as usual 1M eV = 106 eV , 1f m = 10−15 m. Notice that ￿ has units of Energy × time as expected from the formula Eph. = ￿ω . It turns out that usual electromagnetic waves are constituted by huge number of photons and for that reason the particle nature of light is not apparent. However, when interacting with electrons, each photon interacts with an individual electron and the quantum nature of light becomes manifest. 22.2 Hydrogen spectrum and Bohr atomic theory Now we consider the interaction of light with the simplest atom, namely hydrogen, which is made out of a proton and an electron orbiting around it. When a photon hits the electron it can break it free from the atom in the hydrogen version of the photoelectric effect. However, if the energy is not enough the electron will jump to another orbit and stay bounded to the proton. Similarly, once excited it can decay to a lower orbit and emit a photon. According to classical mechanics the energy of the emitted photon can take any value, the spectrum of energies is continuous. However, very surprisingly at the time, it was observed that when heated up, hydrogen emits photon of very well defined frequencies (or wave-lengths) as shown in fig.120. The same happens with absorption, the same frequencies which are emitted are also the ones that are absorbed by the atom. In fact, experimentally it is seen that the energy of the emitted or absorbed photons fit the very simple formula: ￿ ￿ 1 1 Eph. = 13.6 eV − (22.3) n2 n2 1 2 – 142 – where n1,2 are two positive integers such that n2 > n1 . As Bohr pointed out, the only explanation is that the electron cannot be in any arbitrary orbit but only in orbits with energy: 13.6eV Eel. = − , n = 1, 2 , 3 , . . . (22.4) n2 The negative sign means that the electron has less energy than a free electron, namely it is bounded to the proton. When an electron jumps from one orbit to another it can only emit or absorb photons of energies equal to the difference in energy between two of these orbits thus explaining eq.(22.3). Although this explains the hydrogen spectrum it is quite extraordinary. The first surprising point is that the lowest energy is attained for n = 1 called the ground state. Classically the electron would radiate all its energy and get stuck to the proton in a state of infinite negative energy. In quantum mechanics this is not what happens, there is a minimum energy that is attained. An explanation for this fact is given in the next subsection in terms of Heisenberg’s uncertainty principle. Moreover, above the minimal energy only very specific energies are allowed as we will explain in terms of the de Broglie proposal that particles behave also as waves. To summarize, we are in a realm where Newtonian mechanics does not apply any more and we have to discuss which principles allow us to understand the physical reality at the atomic scale. 22.3 Uncertainty principle One of the basic principles of quantum mechanics is the uncertainty principle proposed by Heisenberg. In classical mechanics the state of a system is determined by giving the position and momentum (or velocity) of each particle. In a quantum mechanical state, however, the position and momentum of a particle are not simultaneously well defined. If the particle is localized at a point then the momentum is completely undetermined and vice versa if the momentum is well defined, the particle is completely unlocalized. For that reason in quantum mechanics we talk about the probability distribution of position and momentum. If the particle is localized in a region of size ∆x and the momentum is in the range (p, p + ∆p) then the uncertainty principle establishes that ∆ x∆ p ≥ ￿ 2 (22.5) We can make ∆x small at the expense of making ∆p large and vice versa. Consider now the case of the hydrogen atom. The Coulomb potential is attractive and tries to localize the electron as close to the proton as possible. However, if we localize the electron very close to the proton, the momentum distribution is very spread. p2 Since the kinetic energy is given by K = 2m that means that the average value of the – 143 – Figure 120: When light is emitted by hydrogen only certain wave-lengths are present as seen in this spectrum where hydrogen light is split using a grating or other similar device. kinetic energy will be large. This is the way in which the Heisenberg principle operates. The potential energy tends to localize particles at the minimum of the potential. On the other hand the kinetic energy prefers that the particle is spread. For example in metals the conduction electrons are spread all over the metal and conduct electricity whereas there are other electrons which are localized around the atoms and do not contribute to the current. The more we want to localize a particle the stronger the potential we need. For that reason to study the physics at very small scales large energies per particle are required. Going back to the hydrogen atom we can make a quantitative prediction if we write the total energy as 1 e2 p2 e2 ¯ E = mv 2 − = − (22.6) 2 4π￿0 r 2m r where we used that the momentum is p = mv and also defined e2 = ¯ e2 = 1.44 M eV f m 4π￿0 (22.7) The last value is computed using the electron charge e = −1.6 × 10−19 C . It has the correct units of energy × length. If we localize the electron in a region of size r, the – 144 – Figure 121: Bohr proposed that only certain discrete orbits are possible based on a quantization principle. momentum will be spread ∆p ∼ ￿ r and therefore we estimate: p2 ∼ The total energy therefore is ￿2 e2 ¯ E￿ − = e2 ¯ 2 2mr r ￿ ￿2 r2 ￿2 1 − 2 r2 2me ¯ r where we defined r0 = ￿2 2me 2 ¯ (22.8) ￿ e2 ¯ = r0 ￿ 2 r0 r0 − r2 r ￿ (22.9) (22.10) The radius r0 has the value r0 = ￿2 ( ￿c ) 2 1972 M eV 2 f m2 = = = 2.7 × 104 f m = 2.7 × 10−11 m (22.11) 2 2 e2 2me ¯ 2mc ¯ 1M eV 1.44 M eV f m – 145 – where we used that mc2 = 0.5M eV , the energy equivalent of the electron mass. We see now more quantitatively what happens. If r is very small the kinetic energy grows, 1 1 if r is large then the potential energy grows. We plot the function f (x) = x2 − x where x = rr0 in fig.122 and see that it attains a minimum at x = rr0 = 2. If we replace r ￿ 2r0 in the previous computation of the energy we obtain ￿ ￿ e2 1 1 ¯ e2 ¯ E￿ − =− = −13.6eV (22.12) r0 4 2 4 r0 in very good agreement with the experimental result in eq.(22.4) for n = 1, the ground state. To be completely honest the full agreement is a coincidence. The uncertainty principle only allows us to get an estimate and this should work well. Namely we expect the energy to be of order of tens of electron volts and it is. That it comes exactly equal is as we said a coincidence and does not always work that way. In any case, summarizing, the uncertainty principle tells us that we need energy to localize a particle therefore there is a compromise radius where the particle is localized so that the Coulomb energy is low but not too localized that the kinetic energy will grow. Given the mass of the electron and the strength of the interaction (given by the charge) the radius is of order 10−10 m and the energy of order 10 eV . This principle allows us to understand why there is a minimum energy but we still do not have a principle that tells us that the higher energy states are quantized. We now have to introduce the idea of wave mechanics. – 146 – Figure 122: The function f (x) = with f (2) = −1/4. 1 x2 − 1 x is plotted and seen to have a minimum at x = 2 – 147 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University-West Lafayette.

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