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Independent University Bangladesh (IUB) Assignment 02 Course title: Quantum Mechanics 1 Course code: PHY304 Instructor: Dr. Jewel Kumar Ghosh Problem 01 (5 + 5 = 10 points) Consider the following Hamiltonian: Ĥ1 = − ~2 d2 + V (x). 2m dx2 (1) Suppose ψ0 (x) is the ground state wave function of this Hamiltonian with ground state energy 0. That means: ~2 d2 ψ0 (x) Ĥ1 ψ0 (x) = − + V (x)ψ0 (x) = 0. (2) 2m dx2 Now let us write the Hamiltonian as: Ĥ1 = Â† Â (3) where ~ d ~ d + W (x), Â = √ + W (x). Â† = − √ 2m dx 2m dx W (x) is called the superpotential in the literature. a) By equating two different expressions of the Hamiltonian: ~ d ~ d † √ Ĥ1 ψ0 (x) = Â Âψ0 (x) = − √ + W (x) + W (x) ψ0 (x) 2m dx 2m dx ~2 d2 ψ0 (x) + V (x)ψ0 (x), =− 2m dx2 find the relation between the potential V (x) and the superpotential W (x). b) Argue that for ~ d ~ d √ Ĥ1 ψ0 (x) == − √ + W (x) + W (x) ψ0 (x) = 0 2m dx 2m dx (4) (5) (6) the superpotential is given by: ~ ψ00 (x) W (x) = − √ . 2m ψ0 (x) The ground state wave function of an infinite potential well of width a is given by: r π 2 Φn (x) = sin x where n = 1, 2, 3, · · · a a with the ground state energy E0 = π 2 ~2 . 2ma2 (7) (8) Write down the expression of the superpotential W (x). 1 Problem 02 (5 + 5 + 5 = 15 points) a) For three operators Â, B̂, Ĉ, show the following: h i h i h i ÂB̂, Ĉ = Â B̂, Ĉ + Â, Ĉ B̂. (9) b) For a function f (x) show that [f (x), p̂] = i~ df . dx c) For simple harmonic oscillator show that h i Ĥ, â± = ±~ωâ± . 2 (10) (11)