Question

2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested in the time dynamics of the system. Note that t . he set-up is distinct from what Townsend discusses 0 Depicted are the two lowest energy eigenstates as discussed in class. We assume that the energy eigenvalue of the ground state is less than the height of potential energy barrier. Thus, we expect that the particle is mostly found in either of the potential wells. However, quantum mechanically there is a finite probability to find it in the classically forbidden zone, i.e. the potential energy is larger than the total energy of the particle (a) Which differential equation does the position representation of the wavefunc- tion in the classically allowed zone obey. Which does it obey in the classically 1 point forbidden zone? What do its generic solutions look like? We expect the ground state of the system to be composed of individual solutions similar to what we found in class for the infinite well. We denote these individual solutions as(TL) and (zlR (b) Sketch the wavefunction for a single finite potential well with width a (not depicted), i.e. how do (xL) and (xR) look? Write their functional form in the two classically forbidden zones as well as in the classically allowed zone as a piece-wise defined function. You do not need to determine the constants, but you need to convincingly argue which of the generic solutions (1 point) you want to keep For the full wavefunction, we make the ansatz Ψ(z) 〈x|Z)+ei"(XR)(see figure) If the potential barrier is sufficiently high (or wide), the wavefunction in the two wells are totally independent, hence, we introduced the phase p. However, for finite potential wells the wavefunction penetrates the classically forbidden zone The two generic wavefunctions (xV,m) are depicted in the figure, i.e. there exists a symmetric (p = 0) and an antisymmetric situation (Ø 7) (c) Which of the two energy eigenstates |V) is energetically favored? Why? (d) Tunnelling is defined as a process where a particle located at one location 1 point (e.g. the left well) can be found at a distinctly different location some times later. Explain why the energy difference between V+ leads to tunnelling 1 point (e) Assuming that the wavefunctions Ψ3M are identical for x 0 and x 〉 b, Ψ+lHV+ - ỹ.HV- is propor- show that the energy difference ΔΕ : tional to 2m(Vo - E) ДЕ ~ exp (3 points) (f) What does the scaling with the mass of the particle as well as the energy (1 point) where E is the energy of the particle and m its mass. and width of the barrier mean for our macroscopic world?

Answer 1

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