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By Rudolf Kingslake

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5). In fact, by partial integration we have b (φ, pψ) = dqφ ∗ (q)(−i h)∂ψ(q)/∂q ✏ a b = dq{i h∂φ ∗ (q)/∂q}ψ(q) + (−i h)φ ∗ (q)ψ(q)|ab ✏ a b = ✏ dq{−i h∂φ(q)/∂q}∗ ψ(q) = ( pφ, ψ) ✏ a where (a, b) denotes (−∞, +∞) or (0, L). The Hermiticity of q is obvious, whence that of the functional H ( p, q) ≡ p 2 /2m + V (q) follows immediately. 8) due to the Hermiticity of H . Namely, the normalization (ψt , ψt ) = 1 holds at any time if it does at t = 0. 5). Firstly, apply operator A to eq. 1) and multiply ψt∗ (q) from the left side.

11). This contradicts the definition of η0 as the smallest eigenvalue, leaving only possibility (ii), namely η0 = 0. In this way, we find that the eigenvalues of the operator n are given by η = 0, 1, 2, . . 2 Quantum mechanics of a harmonic oscillator 17 Hereafter n will be called the number operator since its eigenvalues η are nonnegative integers. One also finds from eq. 10), that bψ0 = 0. 14) While eq. 15) except for the normalization constant c, as is confirmed as follows: nψη ≡ b† b b† ψη = b† (b† b + 1)ψη = b† (η + 1)ψη = (η + 1)ψη .

Let us consider an incident photon flux within a small solid angle around a given direction of propagation k/k, polarization eλ and having a spectral energy density given by ✏ w(e,k) (ω) = n λhωλ ρ(ω) ✏ . 4), by (e,k) (e,k) (e,k) Wabs. 6) where (e,k) Bab = (π/ )(hωba )−2 |eλ · ( j k )ba |2 . 4) which is proportional to n λ represents the (e,k) induced emission and is denoted by Wind. em. It is exactly equal to eq. 6). 4) which is proportional to unity represents the spontaneous emission since it arises independently from (and hence even in the absence of) incoming electromagnetic waves.

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