# Group theory: the application to quantum mechanics by Paul H E Meijer

By Paul H E Meijer

Upper-level undergraduate and graduate scholars obtain an creation to problem-solving through eigenfunction transformation houses with this article, which specializes in eigenvalue difficulties during which differential equations or obstacles are unaffected through definite rotations or translations. 1965 version.

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Vols. : Lectures on Quantum Optics. : Quantum Optics. Springer Verlag, Berlin (1994) 4. States of the Electromagnetic Field I In this chapter, we study the coherent states and the thermal radiation. The coherent states were introduced by Glauber [1] and Sudarshan [2] deﬁned as the eigenstate of the annihilation operator. 1) where α is a complex number. 3) n=1 ∞ cn | n , =α n=0 from which we get the following recursion relation √ cn n = αcn−1 . 4) gives αn cn = √ co . n! The coeﬃcient co is found from normalization α | α = 1 =| co |2 n | α |2n =| co |2 exp | α |2 n!

N! 23) is saying that the probability of having n photons in a coherent state obeys a Poisson statistics. 24) n2 = α | a† aa† a | α = α | a† a | α + α | a†2 a2 | α =| α |2 + | α |4 , so that (Δn)2 = n2 − n statistics. 6 Coordinate Representation We would like to ﬁnd the quantity: q | α . 25) by q | from the left, we get √ q | (ωq + ip) | α = 2 ωα q | α ∂ = (ωq + ) q |α . 26) is d q |α = q |α 2ω α− ω q dq . 28 was obtained using the normalization condition +∞ 2 −∞ dq | q | α | = 1, and deﬁning α = r exp(iφ), with φ = 0.

1. 35). References 1. : Lasers, Theory and Applications. Plenum Press, NY (1981) 2. : The Quantum Theory of Light. Clarendon Press, Oxford (1983) 3. : Elements of Quantum Optics. Springer Verlag, Berlin (1990) 4. : Lasers in Applied and Fundamental Research. : Foundations of Laser spectroscopy. : Introduction to Quantum Optics. Gordon and Breach, London (1973) 3. Quantization of the Electromagnetic Field In this chapter, we quantize the electromagnetic ﬁeld and ﬁnd the commutation relations between the various componets of the electric, magnetic ﬁelds and the vector potential.