Question

1. [3 marks] Show that in general, under stationary conditions where V(r) is time inde- pendent, the solutions Ψ(,t) of the tine dependent Schrodinger equation can be written as Ψ(r, t-/(r) . g(t), what does g(t) look like for a constant total energy E = T+ V

Answer 1

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Time dependent schrodinger equation i hwithstroke partial differential psi(r, t)/ partial differential t = -hwithstroke^2/ partial differential m nabla^2 psi(r, t) v(r) psi (r, t) rightarrow (1) if we write psi(r, t) = f(r)g(t) [i hwithstroke partial differential g(t)/partial differential t] f(r) = g(t) [- hwithstroke^2/ partial differential m nabla^2 f(r) + V(r) f(r)] divide both side by f(r) g(t) i hwithstroke/g(t) partial differential g(t)/partial differential t = 1/f(r)[- hwithstroke^2/partial differential m nabla^2 + V(r)] f(r) now LHS is function of 't' only & RHS is function of 'r' only. Therefore the two side must be equal to constant E. 1/g(t) partial differential g(t)/ partial differential t = -i E/ hwithstroke rightarrow (2) - hwithstroke^2/2m nabla^2 f(r) + V(r) f(r) = E f(r) rightarrow (3) time independent schrodinger equation therefore psi(r, t) can be written as f(r) g(t) now on solving equation (2) g(t) = exp(- i epsilon_t/ hwithstroke) form of g(t) for a constant total energy E.