Infinite wells

Problem:

The figure below shows one of the possible energy eigenfunctions ψ(x) for a particle bouncing freely back and forth between impenetrable walls located at x = -a and x = +a. The potential energy equals zero for |x| < a.  If the energy of the particle is 2 eV when it is in the quantum state associated with this eigenfunction, find the energy when it is in quantum state of lowest possible energy.

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Solution:

Problem:

A particle in a 1-D box has a minimum allowed energy of 2.5 eV.
(a)  What is the next higher energy it can have?  And the next higher after that?  Does it have a maximum allowed energy?
(b)  If the particle is an electron, how wide is the box?
(c)  The fact that particles in a 1-D box have a minimum energy is not completely unrelated to the uncertainty principle.  Using the minimum energy,  find the minimum magnitude of the momentum of a particle, with mass m, trapped in a 1-D box of size L in classical mechanics.  How does this compare with the momentum uncertainty required by the uncertainty principle, if we assume Δx = L?

Solution:

Problem:

An electron (m = 9.11*10-31 kg) moves with a speed v = 3.8*106 m/s (non-relativistic) back and forth inside a one-dimensional box (U = 0) of length L.  The potential is infinite elsewhere, hence the electron may not escape the box.
(a)  If the electron were a classical particle, what would be its energy?
(b)  Now treat the electron quantum-mechanically but assume it has the energy you found in part (a).  If the quantum number associated with the state of the electron is n = 2, what is the length of the box?
(c)  What is the energy of the ground state?
(d)  Write down the wave function for the first excited state.

Solution:

Problem:

For the infinite well shown, the wave function for a particle of mass m, at t = 0, is
ψ(x, t = 0) = (2/a)½sin(3πx/a).
(a)  Is ψ(x, t = 0) an eigenfunction of the Hamiltonian?
(b)  Calculate <X>, <Px>, and <H> at t = 0. U

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Solution:

Problem:

A particle of mass m moves in one dimension in a square well with walls of infinite height a distance L apart.  The particle is known to be in a state consisting of an equal admixture of the two lowest energy eigenstates of the system.  Find the probability as a function of time that the particle will be found in the right-hand half of the well.

Solution:

Problem:

Consider a particle of mass m in an one-dimension infinite square well of length L.  Assume that the particle is in the nth eigenstate (n = 1, 2, 3, … ).
(a)  The momentum is measured.  Show that the probability distribution Pn(k) for measuring a momentum p = ħk is Pn(k) = [2πLn2(k2L2 – n2π2)2] [1 + (-1)n+1cos(kL)].
(b)  What outcome of a momentum measurement is most likely?  Does your result agree with your intuition?
(c)  Which momenta cannot be the result of a momentum measurement?  Why is that so?

Solution: