Final answer:
The probability a particle stays bound after shifting from one potential to another is determined by solving the Schrödinger equation, finding initial and final state wave functions, and calculating the square of the modulus of the overlap integral between these wave functions.
Step-by-step explanation:
To calculate the probability that a particle stays bound after shifting from one constant to another delta function potential, one must apply the principles of quantum mechanics, specifically the time-independent Schrödinger equation.
By solving this equation for the given potential, we can find the wave functions associated with the initial and final states of the particle. The probability of the particle remaining in a bound state can then be obtained by calculating the square of the modulus of the overlap integral between the initial and final state wave functions. This process involves mathematical skills such as differential equations and complex analysis along with a good understanding of quantum mechanics concepts such as barriers, tunneling, and wave function normalization.
The initial wave function before the change in potential must be determined, as well as the set of possible final wave functions corresponding to bound states after the change in potential. The square of the absolute value of the overlap integral of the initial wave function with each of these final wave functions gives the probability for the particle to end up in that particular final bound state.