Question

A particle of mass m is released at t = 0 in the one-dimensional double square well shown in Fig. in such a way that its wave function at t = 0 is just one sinusoidal loop (half a sine wave with nodes just at the edges of the left half of the potential as shown.

(a) Find the average value of the energy at t = 0 (in terms of symbols defined above).

(b) Will the average value of the energy be constant for times subsequent to the release of the particle? Why?

(c) Is this a state of definite energy? (That is, will a measurement of the energy in this state always give the same value?) Why?

(d) Will the wave function change with time from its value at t = 0? Explain how you would attempt to calculate the change in the wave function. If how explain why not.

(e) Is it possible that the particle could escape from the potential well (from the whole potential well, from both halves)? Explain.

Answer 1

(a) The normalized wave function at t = 0 is \(\psi\)(z, 0) = \(\sqrt{\frac 2\alpha} \sin\frac{\pi x}\alpha\). Thus

(b) \(\langle \hat H\rangle\) is a constant for t > 0 since \(\partial \langle\hat H\rangle/\partial t \) = 0.

(c) It is not a state of definite energy, because the wave function of the initial state is the eigenfunction of an infinitely deep square well potential with width a, and not of the given potential. It is a superposition state of the different energy eigenstates of the given potential. Therefore different measurements of the energy in this state will not give the same value, but a group of energies according to their probabilities.

(d) The shape of the wave function is time dependent since the solution satisfying the given conditions is a superposition state:

The shape of \(\psi\)(z, t) will change with time because E_n changes with n.

(e) The particle can escape from the whole potential well if the following condition is satisfied: h²π²/2ma > V₀. That is to say, if the width of the potential well is small enough (i.e., the kinetic energy of the particle is large enough), the depth is not very large (i.e., the value of V₀ is not very large), and the energy of the particle is positive, the particle can escape from the whole potential well.