Quantum Mechanics: Derivation of Schrödinger's Equation and Ground State Transition - Prof, Study notes of Mathematics

Download Quantum Mechanics: Derivation of Schrödinger's Equation and Ground State Transition - Prof and more Study notes Mathematics in PDF only on Docsity! Math 260: Perturbation theory Miguel Carrión Álvarez 1. The perturbation series. Starting with ψ(t) = ∑ n≥0 ∫ t 0 ∫ tn 0 · · · ∫ t2 0 e−i(t−tn)H0(−iV )e−i(tn−tn−1)H0 · · · e−i(t2−t1)H0(−iV )e−it1H0ψdt1 · · · dtn−1 dtn we get ∂tψ(t) = ∑ n≥1 ∫ t 0 ∫ tn−1 0 · · · ∫ t2 0 (−iV )e−i(t−tn−1)H0 · · · e−i(t2−t1)H0(−iV )e−it1H0ψdt1 · · · dtn−2 dtn−1+ + ∑ n≥1 ∫ t 0 ∫ tn 0 · · · ∫ t2 0 (−iH0)e−i(t−tn)H0 · · · e−i(t2−t1)H0(−iV )e−it1H0ψdt1 · · · dtn−1 dtn =− i(V +H0)ψ(t) which is Schrödinger’s equation. 2. The perturbation series in the interaction picture. ψ(t) = ∑ n≥0 ∫ t 0 ∫ tn 0 · · · ∫ t2 0 e−i(t−tn)H0(−iV )e−i(tn−tn−1)H0 · · · e−i(t2−t1)H0(−iV )e−it1H0ψdt1 · · · dtn−1 dtn = =e−itH0 ∑ n≥0 ∫ t 0 ∫ tn 0 · · · ∫ t2 0 [ eitnH0(−iV )e−itnH0 ] · · · [eit1H0(−iV )e−it1H0 ]ψdt1 · · · dtn−1 dtn implies ψint(t) = ∑ n≥0 ∫ t 0 ∫ tn 0 · · · ∫ t2 0 [ −iV (tn) ] · · · [ −iV (t1) ] ψdt1 · · · dtn−1 dtn 3. Ground state transition amplitude. Using the facts that H01 = 0, and H0z = z, and the representation a ∼ ∂z and a∗ ∼ z, 〈1, e−i(t−t1)H0(−iV )e−it1H01〉 = λ i √ 2 〈1, e−i(t−t1)H0(a+ a∗)e−it1H01〉 = λ i √ 2 〈1, e−i(t−t1)H0(a+ a∗)1〉 = = λ i √ 2 〈1, e−i(t−t1)H0(0 + z)〉 = λ i √ 2 〈1, e−i(t−t1)z〉 = λ i √ 2 e−i(t−t1)〈1, z〉 = 0. Similarly, 〈1, e−i(t−t2)H0(−iV )e−i(t2−t1)H0(−iV )e−it1H01〉 =−λ 2 2 〈1, e−i(t−t2)H0(a+ a∗)e−i(t2−t1)H0(a+ a∗)1〉 = = −λ2 2 〈1, e−i(t−t2)H0(a+ a∗)e−i(t2−t1)H0(0 + z)〉 = = −λ2 2 e−i(t2−t1)〈1, e−i(t−t2)H0(a+ a∗)z〉 = = −λ2 2 e−i(t2−t1)〈1, e−i(t−t2)H0(1 + z2)〉 = = −λ2 2 e−i(t2−t1)〈1, (1 + e−i(t−t2)2z2)〉 = = −λ2 2 e−i(t2−t1)〈1, 1〉 = −λ 2 2 e−i(t2−t1).