Lecture31

# Lecture31 - Physics 344 Foundations of 21st Century Physics...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 [email protected] TA: Dan Hartzler Office: PHYS 7 [email protected] Grader: Fan Chen Office: PHYS 222 [email protected] Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 – 4:00 in PHYS 154 Reading: on x-ray diffraction in section 25.2 of M&I Volume II, 3rd edition or on pages 847-850 in M&I Volume II, 2nd edition. Exam 2: Thursday, December 1 at 8:00pm in ARMS B061

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Single-Photon Interference Experiments: We’ve taken advantage of the discreteness of the modes of a mirrored cavity to develop a quantitative model of the single-single photon states of the electromagnetic field within such a cavity. ( ( , ) ( ) ( ) n n i t i t n n n n n n x t x i e c x e ϖ ϖ ψ β α ψ - - Ψ = - with ( 2 2 2 * 1 n n n n n c c c α β = = = A basic class of such states are the stationary states for which the photon has a definite energy. They correspond to a classical field state with a single mode, say, mode n , excited to a field energy equal to that of the corresponding quantum of energy, ħω n . The classical field’s structure is simply related to the corresponding photon wavefunction, These are called stationary states because their probability densities determining the probability of detecting the photon in the range x to x + x within the cavity are time independent, * * ( , ) ( , ) ( , ) ( ) ( ) n n n n x t x x t x t x x x x ρ ψ ψ = Ψ Ψ = which assures that the probability of detecting a photon with energy ħω n is 1. which is quite evenly distributed through the cavity. Its location is uncertain.
To represent a more localized and time-dependent electromagnetic state, like one produced when an atom in the cavity initially in an excited internal energy state decays to a lower energy state, we had to a use normalized superposition of stationary states with different mode energies, ( 29 n=1 n=1 ( , ) ( ) ( ) n n i t i t n n n n n x t i x e c x e ϖ ϖ β α ψ ψ - - Ψ = - Normalization means that 2 2 2 * m m m n n n P c c c α β = + = = ( 29 2 2 2 * n=1 n=1 n=1 1 n n n n n c c c α β = = = so that we can interpret as the probability that the photon detected in a measurement made on this state has energy ħω n . The photon in such a state has an uncertain energy that explains the natural width observed for such spectral lines. The Fundamental Principles (Rules) of Quantum Mechanics were abstracted from our experience with such states and account for all of the single-photon effects we’ve encountered and more. As required by the State Vector Rule, the state above is a normalized state vector represented by components c n , and in accordance with the Eigenvector Rule, for every possible photon energy value there is a normalized energy eigenvector ψ n (x) .

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The interpretation of |c m | 2 as the probability that the outcome of an energy measurement on the state is consistent with the Outcome Probability Rule
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