2. Adiabatic expansion of infinite square well (based on Griffiths 2.8, 2.38, 2.39) (a) Suppose you have a particle "at rest", equally likely to be found anywhere in the well, at t = 0. What should its mean momentum be? What is its wavefunction? (b) If you measured the energy of the particle, what possible values could you obtain and what are their probabilities? (c) What is the expected value of the energy? (d) Write down the wavefunction at some later time t. (Leave it as an infinite sum.) (e) Show that at time t = 4ma² /nh, the wavefunction returns to its initial state. %3D (f) Suppose the well was somehow expanded to double the length, keeping the centre un- changed, without perturbing the wavefunction of the particle. Now, if you measured the energy, what possible values could you obtain and with what probabilities?
Infinite square well or particle in a box is a quantum mechanical model used to describe the motion of a particle confined in a small space surrounded by impenetrable barriers. It is used to illustrate the difference between classical and quantum systems.
For a classical system, the position of the particle inside such a box can be found at one position than another and can have any velocity. but for a quantum mechanical system where the distance is in nanometer-scale, the particle can only occupy certain energy levels and can also have zero energy. This means that in a quantum mechanical model the particle is never found to be still.
The wavefunction of a particle in a box of width a is given by,
Here n denoted the state of the particle.
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