Question

A normalized wave function would not be of much use if it did not stay normalized. Indeed, it can be shown that time evolution of the wave function via the Schrödinger equation does not change the norm of the wave function: -00 So that the total probability density is a constant of the motion. However, there are cases where this fact is computationally inconvenient, and one would like to introduce a mechanism for the wave function to decay (or leave a region of interest) without rigorously including all of the physics required to do this. As you will show below for a simple case, one can do this by letting the potential function become compler. Some examples where complex absorbing potentials are used are spontaneous emission, particle decay, an<d problems solved numerically on a finite sized grid where one would like the particle to leave the grid without reflection from its boundaries. For the latter case, the complex absorbing potential is sometimes called a "Gobbler", since it is placed at the edge of the boundary to eat the wave function when it gets to the boundary. Consider a particle governed by the 1D time dependent Schrödinger equation, except now with a complex constant added to the potential, viz where Γ is a real positive constant a) Show that instead of equation (3.1), you now get (3.2) (3.3) dt b) Solve for P(t) and find the lifetime of the particle in terms of Γ.

Answer 1

When we have h_0 = p^2/2m + v_0 wave function of form Psi (x, t) = Psi (0) exp (-i epsilon_0 t/h) So, Vertical-bar Psi (0) Vertical-bar ^2 = Vertical-bar Psi (t) Vertical-bar ^2 v_0 rightarrow v = v_0 - i gamma New Eigen values of this Hamiltonian are changed as E_0 rightarrow E - E_0 - i gamma So, wave function, Psi (t) = Psi (0) exp (-i elementof_0 t/h) exp (-gamma t/h) So, (Psi (t))^2 = (Psi (0))^2 exp (-2 gamma t/h) differentiate w.r.t t d (Psi (t))^2/dt = (Psi (0))^2 exp (-2 gamma t/h) (-2 gamma/h) = > -2 gamma/h (Psi (t))^2 So dP/dt = -2 gamma/h P proved For P, dP/p = -2 gamma/h dt = > integral dp/p = -2 gamma/h integral dt ln p = -2 gamma/h t p (t) = e^-2 gamma/h t Life time, T = 1/lambda here lambda = 2 gamma/h so T = h/2 gamma