QuantizingRadiation

QuantizingRadiation - Quantizing Radiation Michael Fowler,...

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Quantizing Radiation Michael Fowler, 5/4/06 Introduction In analyzing the photoelectric effect in hydrogen, we derived the rate of ionization of a hydrogen atom in a monochromatic electromagnetic wave of given strength, and the result we derived is in good agreement with experiment. Recall that the interaction Hamiltonian was () ( ) 1 0 0 cos . 2 ikr t t e Hk r t A mc e eeA mc ωω ω ⋅− − ⋅− ⎛⎞ =⋅ ⎜⎟ ⎝⎠ =+ GG p p G G G G G G and we dropped the it e term because it would correspond to the atom giving energy to the field, and our atom was already in its ground state. However, if we go through the same calculation for an atom not initially in the ground state, then indeed an electromagnetic wave of appropriate frequency will cause a transition rate to a lower energy state, and e is the relevant term. But this is not the whole story. An atom in an excited state will eventually emit a photon and go to a lower energy state, even if there is zero external field. Our analysis so far does not predict this—obviously, the interaction written above is only nonzero if A G is nonzero! So what are we missing? Essentially, the answer is that the electromagnetic field itself is quantized. Of course, we know that, it’s made up of photons. Recall Planck’s successful analysis of radiation in a box: he considered all possible normal modes for the radiation, and asserted that a mode of energy could only gain or lose energy in amounts = . This led to the correct formula for black body radiation, then Einstein proved that the same assumption, with the same , accounted for the photoelectric effect. We now understand that these modes of oscillation of radiation are just simple harmonic oscillators, with energy = ( ) 1 2 n + = , and, just as a mass on a spring oscillator has fluctuations in the ground state, 0 x = but 2 0 x , for these electromagnetic modes 0 A = G but 2 0 A G . These fluctuations in A G mean the interaction Hamiltonian is momentarily nonzero, and therefore can cause a transition. Therefore, to find the spontaneous transition rate (as it’s called) for an atom in a zero (classically speaking) electromagnetic field, we need to express the electromagnetic field in terms of normal modes (we’ll take a big box), then quantize these modes as quantum simple harmonic oscillators, introducing raising and lowering operators for each oscillator (these will be photon creation and
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2 annihilation operators) then construct the appropriate quantum operator expression for A G to put in the electron-radiation interaction Hamiltonian. The bras and kets will now be quantum states of the electron and the radiation field, in contrast to our analysis of the classical field above, where the radiation field didn’t change. (Of course, it did, really, in that it lost one photon, but in the classical limit there are infinitely many photons in each mode, so that wouldn’t register.) Our treatment follows Sakurai’s Advanced Quantum Mechanics , Chapter 2 (but we stay with
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QuantizingRadiation - Quantizing Radiation Michael Fowler,...

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